Integrand size = 30, antiderivative size = 507 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\frac {d^3 x \sqrt {a+b x^2}}{4 c (d e-c f)^3 \left (c+d x^2\right )^2}+\frac {d^3 (2 b c (d e-7 c f)-3 a d (d e-5 c f)) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) (d e-c f)^4 \left (c+d x^2\right )}-\frac {f^3 x \sqrt {a+b x^2}}{4 e (d e-c f)^3 \left (e+f x^2\right )^2}+\frac {f^3 (3 a f (5 d e-c f)-2 b e (7 d e-c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f) (d e-c f)^4 \left (e+f x^2\right )}-\frac {d^2 \left (48 b^2 c^4 f^2+3 a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )-4 a b c d \left (d^2 e^2-5 c d e f+28 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2} (d e-c f)^5}+\frac {f^2 \left (48 b^2 d^2 e^4+3 a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )-4 a b e f \left (28 d^2 e^2-5 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{3/2} (d e-c f)^5} \] Output:
1/4*d^3*x*(b*x^2+a)^(1/2)/c/(-c*f+d*e)^3/(d*x^2+c)^2+1/8*d^3*(2*b*c*(-7*c* f+d*e)-3*a*d*(-5*c*f+d*e))*x*(b*x^2+a)^(1/2)/c^2/(-a*d+b*c)/(-c*f+d*e)^4/( d*x^2+c)-1/4*f^3*x*(b*x^2+a)^(1/2)/e/(-c*f+d*e)^3/(f*x^2+e)^2+1/8*f^3*(3*a *f*(-c*f+5*d*e)-2*b*e*(-c*f+7*d*e))*x*(b*x^2+a)^(1/2)/e^2/(-a*f+b*e)/(-c*f +d*e)^4/(f*x^2+e)-1/8*d^2*(48*b^2*c^4*f^2+3*a^2*d^2*(21*c^2*f^2-6*c*d*e*f+ d^2*e^2)-4*a*b*c*d*(28*c^2*f^2-5*c*d*e*f+d^2*e^2))*arctanh((-a*d+b*c)^(1/2 )*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(5/2)/(-a*d+b*c)^(3/2)/(-c*f+d*e)^5+1/8*f^2 *(48*b^2*d^2*e^4+3*a^2*f^2*(c^2*f^2-6*c*d*e*f+21*d^2*e^2)-4*a*b*e*f*(c^2*f ^2-5*c*d*e*f+28*d^2*e^2))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/ 2))/e^(5/2)/(-a*f+b*e)^(3/2)/(-c*f+d*e)^5
Time = 13.28 (sec) , antiderivative size = 465, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (x \sqrt {a+b x^2} \left (-\frac {2 d^3}{c (-d e+c f)^3 \left (c+d x^2\right )^2}-\frac {d^3 (3 a d (d e-5 c f)+2 b c (-d e+7 c f))}{c^2 (b c-a d) (d e-c f)^4 \left (c+d x^2\right )}-\frac {2 f^3}{e (d e-c f)^3 \left (e+f x^2\right )^2}-\frac {f^3 (2 b e (7 d e-c f)+3 a f (-5 d e+c f))}{e^2 (b e-a f) (d e-c f)^4 \left (e+f x^2\right )}\right )-\frac {d^2 \left (48 b^2 c^4 f^2+3 a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )-4 a b c d \left (d^2 e^2-5 c d e f+28 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{5/2} (-b c+a d)^{3/2} (-d e+c f)^5}-\frac {f^2 \left (48 b^2 d^2 e^4+3 a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )-4 a b e f \left (28 d^2 e^2-5 c d e f+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{3/2} (d e-c f)^5}\right ) \] Input:
Integrate[Sqrt[a + b*x^2]/((c + d*x^2)^3*(e + f*x^2)^3),x]
Output:
(x*Sqrt[a + b*x^2]*((-2*d^3)/(c*(-(d*e) + c*f)^3*(c + d*x^2)^2) - (d^3*(3* a*d*(d*e - 5*c*f) + 2*b*c*(-(d*e) + 7*c*f)))/(c^2*(b*c - a*d)*(d*e - c*f)^ 4*(c + d*x^2)) - (2*f^3)/(e*(d*e - c*f)^3*(e + f*x^2)^2) - (f^3*(2*b*e*(7* d*e - c*f) + 3*a*f*(-5*d*e + c*f)))/(e^2*(b*e - a*f)*(d*e - c*f)^4*(e + f* x^2))) - (d^2*(48*b^2*c^4*f^2 + 3*a^2*d^2*(d^2*e^2 - 6*c*d*e*f + 21*c^2*f^ 2) - 4*a*b*c*d*(d^2*e^2 - 5*c*d*e*f + 28*c^2*f^2))*ArcTan[(Sqrt[-(b*c) + a *d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(5/2)*(-(b*c) + a*d)^(3/2)*(-(d*e) + c*f)^5) - (f^2*(48*b^2*d^2*e^4 + 3*a^2*f^2*(21*d^2*e^2 - 6*c*d*e*f + c^2* f^2) - 4*a*b*e*f*(28*d^2*e^2 - 5*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(e^(5/2)*(-(b*e) + a*f)^(3/2)*(d*e - c*f)^5))/8
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{d}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int -\frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {\int -\frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\int \frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 407 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}} f^2}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (\frac {\int -\frac {a d (d e-3 c f)-2 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{2 c (b c-a d) \left (d x^2+c\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b e (2 d e-c f)-a f (3 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {2 b d (d e-c f) x^2+a d (3 d e-7 c f)-4 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{4 c (b c-a d) \left (d x^2+c\right )^2}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int -\frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\int \frac {a d (d e-3 c f)-2 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b e (2 d e-c f)-a f (3 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {\int \frac {2 b d (d e-c f) x^2+a d (3 d e-7 c f)-4 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {\int \frac {2 b d (d e-c f) x^2+a d (3 d e-7 c f)-4 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {\int \frac {2 b d (d e-c f) x^2+a d (3 d e-7 c f)-4 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {\int \frac {2 b d (d e-c f) x^2+a d (3 d e-7 c f)-4 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {\frac {\int -\frac {8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) x \sqrt {b x^2+a}}{2 c (b c-a d) \left (d x^2+c\right )}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int -\frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\int \frac {8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\int \frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\left (8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\left (8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)\right ) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 c (b c-a d)}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\left (8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 407 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right ) f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\left (8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) f^2}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {(a d (d e-3 c f)-2 b c (d e-2 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {d \left (\frac {\left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right ) f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\left (8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\right )}{d}\) |
Input:
Int[Sqrt[a + b*x^2]/((c + d*x^2)^3*(e + f*x^2)^3),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2 ]), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/((a + b*x^2)*Sqrt[e + f*x^2]), x], x] - Simp[d/(b*c - a*d) Int[1/((c + d*x^2)*Sqrt[e + f*x^2]), x], x] / ; FreeQ[{a, b, c, d, e, f}, x]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^2)^p*(c + d*x ^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt Q[p, 0] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[b/(b*c - a*d) Int[(a + b*x^2)^p*(c + d*x^2)^ (q + 1)*(e + f*x^2)^r, x], x] - Simp[d/(b*c - a*d) Int[(a + b*x^2)^(p + 1 )*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && ILtQ[p, 0] && LeQ[q, -1]
Time = 2.95 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.78
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (-21 \sqrt {\left (a f -b e \right ) e}\, d^{2} \left (x^{2} d +c \right )^{2} \left (\frac {16 b^{2} c^{4} f^{2}}{21}-\frac {16 a b \,c^{3} d \,f^{2}}{9}+a \,d^{2} f \left (a f +\frac {20 b e}{63}\right ) c^{2}-\frac {2 a \left (a f +\frac {2 b e}{9}\right ) d^{3} e c}{7}+\frac {a^{2} e^{2} d^{4}}{21}\right ) \left (a f -b e \right ) \left (f \,x^{2}+e \right )^{2} e^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\sqrt {\left (a d -b c \right ) c}\, \left (\left (a d -b c \right ) c^{2} \left (x^{2} d +c \right )^{2} \left (f \,x^{2}+e \right )^{2} f^{2} \left (\left (21 a^{2} e^{2} f^{2}-\frac {112}{3} a b \,e^{3} f +16 b^{2} e^{4}\right ) d^{2}-6 a d \,f^{2} e \left (a f -\frac {10 b e}{9}\right ) c +a \,f^{3} \left (a f -\frac {4 b e}{3}\right ) c^{2}\right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {5 \sqrt {\left (a f -b e \right ) e}\, \left (-\left (-\frac {4 b \,e^{2}}{5}+f \left (-\frac {2 b \,x^{2}}{5}+a \right ) e +\frac {3 a \,f^{2} x^{2}}{5}\right ) b \,f^{4} c^{6}+d \left (-\frac {16 e^{3} b^{2}}{5}+\frac {13 \left (-\frac {6 b \,x^{2}}{13}+a \right ) b f \,e^{2}}{5}+f^{2} \left (\frac {4}{5} b^{2} x^{4}+\frac {3}{5} a b \,x^{2}+a^{2}\right ) e +\frac {3 a \,f^{3} x^{2} \left (-2 b \,x^{2}+a \right )}{5}\right ) f^{3} c^{5}-\frac {17 \left (-\frac {16 b \left (-2 b \,x^{2}+a \right ) e^{3}}{17}+f \left (\frac {24}{17} b^{2} x^{4}-\frac {40}{17} a b \,x^{2}+a^{2}\right ) e^{2}+\frac {5 \left (-\frac {2}{5} b^{2} x^{4}-\frac {21}{5} a b \,x^{2}+a^{2}\right ) x^{2} f^{2} e}{17}-\frac {6 a \left (-\frac {b \,x^{2}}{2}+a \right ) x^{4} f^{3}}{17}\right ) d^{2} f^{3} c^{4}}{5}-\frac {34 d^{3} \left (\frac {8 b^{2} e^{5}}{17}-\frac {8 b f \left (-2 b \,x^{2}+a \right ) e^{4}}{17}-\frac {32 \left (-\frac {b \,x^{2}}{2}+a \right ) b \,x^{2} f^{2} e^{3}}{17}+f^{3} x^{2} \left (\frac {7}{17} b^{2} x^{4}-\frac {57}{34} a b \,x^{2}+a^{2}\right ) e^{2}+\frac {25 a \,x^{4} f^{4} \left (-\frac {13 b \,x^{2}}{25}+a \right ) e}{34}-\frac {3 a^{2} f^{5} x^{6}}{34}\right ) f \,c^{3}}{5}-\frac {17 d^{4} \left (-\frac {4 b^{2} e^{5}}{17}-\frac {13 \left (-\frac {6 b \,x^{2}}{13}+a \right ) b f \,e^{4}}{17}+f^{2} \left (\frac {24}{17} b^{2} x^{4}-\frac {40}{17} a b \,x^{2}+a^{2}\right ) e^{3}+2 f^{3} x^{2} \left (\frac {7}{17} b^{2} x^{4}-\frac {57}{34} a b \,x^{2}+a^{2}\right ) e^{2}+2 a \left (-\frac {14 b \,x^{2}}{17}+a \right ) x^{4} f^{4} e +\frac {15 a^{2} f^{5} x^{6}}{17}\right ) e \,c^{2}}{5}+\left (\left (-\frac {2 b \,x^{2}}{5}+a \right ) e -3 a f \,x^{2}\right ) d^{5} \left (a f -b e \right ) \left (f \,x^{2}+e \right )^{2} e^{2} c +\frac {3 a \,d^{6} e^{3} x^{2} \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right )}{5}\right ) \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x}{3}\right )\right )}{8 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right )^{5} \left (a d -b c \right ) c^{2} \left (a f -b e \right ) e^{2} \left (x^{2} d +c \right )^{2} \left (f \,x^{2}+e \right )^{2}}\) | \(900\) |
| default | \(\text {Expression too large to display}\) | \(8862\) |
Input:
int((b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
-3/8/((a*d-b*c)*c)^(1/2)/((a*f-b*e)*e)^(1/2)*(-21*((a*f-b*e)*e)^(1/2)*d^2* (d*x^2+c)^2*(16/21*b^2*c^4*f^2-16/9*a*b*c^3*d*f^2+a*d^2*f*(a*f+20/63*b*e)* c^2-2/7*a*(a*f+2/9*b*e)*d^3*e*c+1/21*a^2*e^2*d^4)*(a*f-b*e)*(f*x^2+e)^2*e^ 2*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+((a*d-b*c)*c)^(1/2)*((a* d-b*c)*c^2*(d*x^2+c)^2*(f*x^2+e)^2*f^2*((21*a^2*e^2*f^2-112/3*a*b*e^3*f+16 *b^2*e^4)*d^2-6*a*d*f^2*e*(a*f-10/9*b*e)*c+a*f^3*(a*f-4/3*b*e)*c^2)*arctan (e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))-5/3*((a*f-b*e)*e)^(1/2)*(-(-4/5* b*e^2+f*(-2/5*b*x^2+a)*e+3/5*a*f^2*x^2)*b*f^4*c^6+d*(-16/5*e^3*b^2+13/5*(- 6/13*b*x^2+a)*b*f*e^2+f^2*(4/5*b^2*x^4+3/5*a*b*x^2+a^2)*e+3/5*a*f^3*x^2*(- 2*b*x^2+a))*f^3*c^5-17/5*(-16/17*b*(-2*b*x^2+a)*e^3+f*(24/17*b^2*x^4-40/17 *a*b*x^2+a^2)*e^2+5/17*(-2/5*b^2*x^4-21/5*a*b*x^2+a^2)*x^2*f^2*e-6/17*a*(- 1/2*b*x^2+a)*x^4*f^3)*d^2*f^3*c^4-34/5*d^3*(8/17*b^2*e^5-8/17*b*f*(-2*b*x^ 2+a)*e^4-32/17*(-1/2*b*x^2+a)*b*x^2*f^2*e^3+f^3*x^2*(7/17*b^2*x^4-57/34*a* b*x^2+a^2)*e^2+25/34*a*x^4*f^4*(-13/25*b*x^2+a)*e-3/34*a^2*f^5*x^6)*f*c^3- 17/5*d^4*(-4/17*b^2*e^5-13/17*(-6/13*b*x^2+a)*b*f*e^4+f^2*(24/17*b^2*x^4-4 0/17*a*b*x^2+a^2)*e^3+2*f^3*x^2*(7/17*b^2*x^4-57/34*a*b*x^2+a^2)*e^2+2*a*( -14/17*b*x^2+a)*x^4*f^4*e+15/17*a^2*f^5*x^6)*e*c^2+((-2/5*b*x^2+a)*e-3*a*f *x^2)*d^5*(a*f-b*e)*(f*x^2+e)^2*e^2*c+3/5*a*d^6*e^3*x^2*(f*x^2+e)^2*(a*f-b *e))*(c*f-d*e)*(b*x^2+a)^(1/2)*x))/(c*f-d*e)^5/(a*d-b*c)/c^2/(a*f-b*e)/e^2 /(d*x^2+c)^2/(f*x^2+e)^2
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(1/2)/(d*x**2+c)**3/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{3} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^3*(f*x^2 + e)^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 8143 vs. \(2 (467) = 934\).
Time = 8.58 (sec) , antiderivative size = 8143, normalized size of antiderivative = 16.06 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="giac")
Output:
-1/8*(4*a*b^(3/2)*c*d^5*e^2 - 3*a^2*sqrt(b)*d^6*e^2 - 20*a*b^(3/2)*c^2*d^4 *e*f + 18*a^2*sqrt(b)*c*d^5*e*f - 48*b^(5/2)*c^4*d^2*f^2 + 112*a*b^(3/2)*c ^3*d^3*f^2 - 63*a^2*sqrt(b)*c^2*d^4*f^2)*arctan(1/2*((sqrt(b)*x - sqrt(b*x ^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/((b*c^3*d^5*e^5 - a* c^2*d^6*e^5 - 5*b*c^4*d^4*e^4*f + 5*a*c^3*d^5*e^4*f + 10*b*c^5*d^3*e^3*f^2 - 10*a*c^4*d^4*e^3*f^2 - 10*b*c^6*d^2*e^2*f^3 + 10*a*c^5*d^3*e^2*f^3 + 5* b*c^7*d*e*f^4 - 5*a*c^6*d^2*e*f^4 - b*c^8*f^5 + a*c^7*d*f^5)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/8*(48*b^(5/2)*d^2*e^4*f^2 - 112*a*b^(3/2)*d^2*e^3*f^3 + 2 0*a*b^(3/2)*c*d*e^2*f^4 + 63*a^2*sqrt(b)*d^2*e^2*f^4 - 4*a*b^(3/2)*c^2*e*f ^5 - 18*a^2*sqrt(b)*c*d*e*f^5 + 3*a^2*sqrt(b)*c^2*f^6)*arctan(1/2*((sqrt(b )*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b*d^ 5*e^8 - 5*b*c*d^4*e^7*f - a*d^5*e^7*f + 10*b*c^2*d^3*e^6*f^2 + 5*a*c*d^4*e ^6*f^2 - 10*b*c^3*d^2*e^5*f^3 - 10*a*c^2*d^3*e^5*f^3 + 5*b*c^4*d*e^4*f^4 + 10*a*c^3*d^2*e^4*f^4 - b*c^5*e^3*f^5 - 5*a*c^4*d*e^3*f^5 + a*c^5*e^2*f^6) *sqrt(-b^2*e^2 + a*b*e*f)) - 1/4*(4*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a*b^( 5/2)*c*d^5*e^4*f^2 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(3/2)*d^6*e^ 4*f^2 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^14*b^(7/2)*c^3*d^3*e^3*f^3 - 64*( sqrt(b)*x - sqrt(b*x^2 + a))^14*a*b^(5/2)*c^2*d^4*e^3*f^3 + 11*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(3/2)*c*d^5*e^3*f^3 + 3*(sqrt(b)*x - sqrt(b*x^ 2 + a))^14*a^3*sqrt(b)*d^6*e^3*f^3 - 64*(sqrt(b)*x - sqrt(b*x^2 + a))^1...
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int \frac {\sqrt {b\,x^2+a}}{{\left (d\,x^2+c\right )}^3\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int((a + b*x^2)^(1/2)/((c + d*x^2)^3*(e + f*x^2)^3),x)
Output:
int((a + b*x^2)^(1/2)/((c + d*x^2)^3*(e + f*x^2)^3), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int \frac {\sqrt {b \,x^{2}+a}}{\left (d \,x^{2}+c \right )^{3} \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int((b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x)
Output:
int((b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x)