Integrand size = 30, antiderivative size = 653 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {b \left (2 b^3 c e (d e-c f)^2+3 a^3 d^2 f^2 (d e+c f)-6 a^2 b d f \left (d^2 e^2+c^2 f^2\right )+3 a b^2 \left (d^3 e^3+c^3 f^3\right )\right ) x}{6 a c (b c-a d)^2 e (b e-a f)^2 (d e-c f)^2 \left (a+b x^2\right )^{3/2}}+\frac {b \left (4 b^5 c^2 e^2 (d e-c f)^2+3 a^5 d^3 f^3 (d e+c f)-16 a b^4 c e (d e-c f)^2 (d e+c f)-9 a^4 b d^2 f^2 \left (d^2 e^2+c^2 f^2\right )+9 a^3 b^2 d f \left (d^3 e^3+c^3 f^3\right )-a^2 b^3 \left (3 d^4 e^4-28 c d^3 e^3 f+56 c^2 d^2 e^2 f^2-28 c^3 d e f^3+3 c^4 f^4\right )\right ) x}{6 a^2 c (b c-a d)^3 e (b e-a f)^3 (d e-c f)^2 \sqrt {a+b x^2}}-\frac {d^3 x}{2 c (b c-a d) (d e-c f)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}-\frac {f^3 x}{2 e (b e-a f) (d e-c f)^2 \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {d^4 (a d (d e-5 c f)-2 b c (3 d e-5 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{7/2} (d e-c f)^3}+\frac {f^4 (2 b e (5 d e-3 c f)-a f (5 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{7/2} (d e-c f)^3} \] Output:
1/6*b*(2*b^3*c*e*(-c*f+d*e)^2+3*a^3*d^2*f^2*(c*f+d*e)-6*a^2*b*d*f*(c^2*f^2 +d^2*e^2)+3*a*b^2*(c^3*f^3+d^3*e^3))*x/a/c/(-a*d+b*c)^2/e/(-a*f+b*e)^2/(-c *f+d*e)^2/(b*x^2+a)^(3/2)+1/6*b*(4*b^5*c^2*e^2*(-c*f+d*e)^2+3*a^5*d^3*f^3* (c*f+d*e)-16*a*b^4*c*e*(-c*f+d*e)^2*(c*f+d*e)-9*a^4*b*d^2*f^2*(c^2*f^2+d^2 *e^2)+9*a^3*b^2*d*f*(c^3*f^3+d^3*e^3)-a^2*b^3*(3*c^4*f^4-28*c^3*d*e*f^3+56 *c^2*d^2*e^2*f^2-28*c*d^3*e^3*f+3*d^4*e^4))*x/a^2/c/(-a*d+b*c)^3/e/(-a*f+b *e)^3/(-c*f+d*e)^2/(b*x^2+a)^(1/2)-1/2*d^3*x/c/(-a*d+b*c)/(-c*f+d*e)^2/(b* x^2+a)^(3/2)/(d*x^2+c)-1/2*f^3*x/e/(-a*f+b*e)/(-c*f+d*e)^2/(b*x^2+a)^(3/2) /(f*x^2+e)-1/2*d^4*(a*d*(-5*c*f+d*e)-2*b*c*(-5*c*f+3*d*e))*arctanh((-a*d+b *c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(3/2)/(-a*d+b*c)^(7/2)/(-c*f+d*e)^3 +1/2*f^4*(2*b*e*(-3*c*f+5*d*e)-a*f*(-c*f+5*d*e))*arctanh((-a*f+b*e)^(1/2)* x/e^(1/2)/(b*x^2+a)^(1/2))/e^(3/2)/(-a*f+b*e)^(7/2)/(-c*f+d*e)^3
Time = 19.56 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {1}{6} \left (x \sqrt {a+b x^2} \left (\frac {2 b^4}{a (b c-a d)^2 (b e-a f)^2 \left (a+b x^2\right )^2}+\frac {4 b^4 \left (b^2 c e+7 a^2 d f-4 a b (d e+c f)\right )}{a^2 (b c-a d)^3 (b e-a f)^3 \left (a+b x^2\right )}-\frac {3 d^5}{c (b c-a d)^3 (d e-c f)^2 \left (c+d x^2\right )}-\frac {3 f^5}{e (b e-a f)^3 (d e-c f)^2 \left (e+f x^2\right )}\right )-\frac {3 d^4 (a d (d e-5 c f)+2 b c (-3 d e+5 c f)) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{3/2} (-b c+a d)^{7/2} (-d e+c f)^3}-\frac {3 f^4 (2 b e (5 d e-3 c f)+a f (-5 d e+c f)) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{3/2} (-b e+a f)^{7/2} (d e-c f)^3}\right ) \] Input:
Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^2*(e + f*x^2)^2),x]
Output:
(x*Sqrt[a + b*x^2]*((2*b^4)/(a*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x^2)^2) + (4*b^4*(b^2*c*e + 7*a^2*d*f - 4*a*b*(d*e + c*f)))/(a^2*(b*c - a*d)^3*(b* e - a*f)^3*(a + b*x^2)) - (3*d^5)/(c*(b*c - a*d)^3*(d*e - c*f)^2*(c + d*x^ 2)) - (3*f^5)/(e*(b*e - a*f)^3*(d*e - c*f)^2*(e + f*x^2))) - (3*d^4*(a*d*( d*e - 5*c*f) + 2*b*c*(-3*d*e + 5*c*f))*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt [c]*Sqrt[a + b*x^2])])/(c^(3/2)*(-(b*c) + a*d)^(7/2)*(-(d*e) + c*f)^3) - ( 3*f^4*(2*b*e*(5*d*e - 3*c*f) + a*f*(-5*d*e + c*f))*ArcTan[(Sqrt[-(b*e) + a *f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(e^(3/2)*(-(b*e) + a*f)^(7/2)*(d*e - c* f)^3))/6
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{5/2} \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{5/2} \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int -\frac {6 d f a^2-b (5 d e+3 c f) a+4 b (b c-a d) f x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{3 a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {6 d f a^2-5 b d e a-3 b c f a+4 b (b c-a d) f x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}-\frac {\int -\frac {f \left (2 b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (d e+3 c f) a+2 b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\int \frac {f \left (2 b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (d e+3 c f) a+2 b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {f \int \frac {2 b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (d e+3 c f) a+2 b^2 c e\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}+\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {f \left (\frac {\int \frac {3 a^2 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {f \left (\frac {3 a^2 \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {f \left (\frac {3 a^2 \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b e (2 d e-c f)-a f (3 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b e (2 d e-c f)-a f (3 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b e (2 d e-c f)-a f (3 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b e (2 d e-c f)-a f (3 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {b x \left (10 a^2 d f-7 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right ) (b e-a f)}+\frac {f \left (\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (2 a^2 d f^2-a b f (c f+11 d e)+2 b^2 e (3 c f+2 d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (19 d e-3 c f)-2 a b^2 e (8 c f+5 d e)+4 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \left (\frac {d^2 \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (2 b e (2 d e-c f)-a f (3 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {3 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) a^2}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {\left (6 d f^2 a^3+b f (19 d e-3 c f) a^2-2 b^2 e (5 d e+8 c f) a+4 b^3 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}-\frac {\int \frac {a (b d e+b c f-2 a d f)-2 b (b c-a d) f x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {3 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) a^2}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {\left (6 d f^2 a^3+b f (19 d e-3 c f) a^2-2 b^2 e (5 d e+8 c f) a+4 b^3 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}-\frac {\frac {\int \frac {a \left (2 e (d e+2 c f) b^2-a f (7 d e+c f) b+2 a^2 d f^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (d e-c f) a+2 b^2 c e\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {3 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) a^2}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {\left (6 d f^2 a^3+b f (19 d e-3 c f) a^2-2 b^2 e (5 d e+8 c f) a+4 b^3 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}-\frac {\frac {a \left (2 e (d e+2 c f) b^2-a f (7 d e+c f) b+2 a^2 d f^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (d e-c f) a+2 b^2 c e\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {3 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) a^2}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {\left (6 d f^2 a^3+b f (19 d e-3 c f) a^2-2 b^2 e (5 d e+8 c f) a+4 b^3 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}-\frac {\frac {a \left (2 e (d e+2 c f) b^2-a f (7 d e+c f) b+2 a^2 d f^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (d e-c f) a+2 b^2 c e\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {3 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) a^2}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {\left (6 d f^2 a^3+b f (19 d e-3 c f) a^2-2 b^2 e (5 d e+8 c f) a+4 b^3 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}-\frac {\frac {a \left (2 e (d e+2 c f) b^2-a f (7 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f \left (-2 d f a^2-b (d e-c f) a+2 b^2 c e\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^2}-\frac {f \left (\frac {(2 b e (2 d e-c f)-a f (3 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {3 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) a^2}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {\left (6 d f^2 a^3+b f (19 d e-3 c f) a^2-2 b^2 e (5 d e+8 c f) a+4 b^3 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}-\frac {\frac {a \left (2 e (d e+2 c f) b^2-a f (7 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f \left (-2 d f a^2-b (d e-c f) a+2 b^2 c e\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 301 |
| \(\displaystyle \frac {b \left (\frac {\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^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {3 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) a^2}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {\left (6 d f^2 a^3+b f (19 d e-3 c f) a^2-2 b^2 e (5 d e+8 c f) a+4 b^3 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}-\frac {\frac {a \left (2 e (d e+2 c f) b^2-a f (7 d e+c f) b+2 a^2 d f^2\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}}-\frac {f \left (-2 d f a^2-b (d e-c f) a+2 b^2 c e\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\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}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {\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^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}+\frac {\frac {b \left (10 d f a^2-5 b d e a-7 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}+\frac {f \left (\frac {3 \left (2 e (2 d e+3 c f) b^2-a f (11 d e+c f) b+2 a^2 d f^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ) a^2}{2 e^{3/2} (b e-a f)^{3/2}}+\frac {\left (6 d f^2 a^3+b f (19 d e-3 c f) a^2-2 b^2 e (5 d e+8 c f) a+4 b^3 c e^2\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^2}dx}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )}-\frac {\frac {a \left (2 e (d e+2 c f) b^2-a f (7 d e+c f) b+2 a^2 d f^2\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}}-\frac {f \left (-2 d f a^2-b (d e-c f) a+2 b^2 c e\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\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}{b c-a d}\right )}{b c-a d}\) |
Input:
Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^2*(e + f*x^2)^2),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], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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), x_Symbol] :> Simp[b/ d Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b*x^2)^ (p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E qQ[b*c + 3*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[(((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[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 = 5.54 (sec) , antiderivative size = 1048, normalized size of antiderivative = 1.60
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1048\) |
| default | \(\text {Expression too large to display}\) | \(7278\) |
Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^2/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
-5/2/((a*f-b*e)*e)^(1/2)*(((a*f-b*e)*e)^(1/2)*a^2*(b*x^2+a)^(3/2)*d^4*(a*f -b*e)^3*(d*x^2+c)*((-2*c^2*f+6/5*c*e*d)*b+d*(c*f-1/5*d*e)*a)*(f*x^2+e)*e*a rctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+1/5*((a*d-b*c)*c)^(1/2)*((a *d-b*c)^3*a^2*(b*x^2+a)^(3/2)*c*((-6*c*e*f+10*d*e^2)*b+f*a*(c*f-5*d*e))*(d *x^2+c)*(f*x^2+e)*f^4*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))-((a* f-b*e)*e)^(1/2)*(d^3*(c^2*f^2+c*d*f^2*x^2+d^2*e*(f*x^2+e))*f^3*a^7-3*d^2*( c^3*f^3+1/3*c^2*d*f^3*x^2-2/3*c*d^2*f^3*x^4+d^3*(-2/3*f*x^2+e)*(f*x^2+e)*e )*b*f^2*a^6+3*(c^4*f^4-c^3*d*f^4*x^2-5/3*c^2*d^2*f^4*x^4+1/3*c*d^3*f^4*x^6 +d^4*(1/3*f^2*x^4-2*e*f*x^2+e^2)*(f*x^2+e)*e)*d*b^2*f*a^5-(c^5*f^5-5*c^4*d *f^5*x^2-3*c^3*d^2*f^5*x^4+3*c^2*d^3*f^5*x^6+d^5*e^2*(f*x^2+e)*(3*f^2*x^4- 6*e*f*x^2+e^2))*b^3*a^4-2*b^4*(c^5*f^5*x^2-5*(1/10*f^2*x^4+e*f*x^2+e^2)*d* f^3*c^4+10*(-3/20*f^3*x^6-1/2*e*f^2*x^4+1/2*e^2*f*x^2+e^3)*d^2*f^2*c^3-5*f *e^2*d^3*(f*x^2+e)*(-2*f*x^2+e)*c^2-5*d^4*e^3*f*x^2*(f*x^2+e)*c+d^5*(-3/2* f*x^2+e)*x^2*(f*x^2+e)*e^3)*a^3-6*(f^3*(1/6*f^2*x^4+e*f*x^2+e^2)*c^5-d*(-1 /6*f^3*x^6+5/9*e*f^2*x^4+14/9*e^2*f*x^2+e^3)*f^2*c^4-(14/9*f^2*x^4-19/9*e* f*x^2+e^2)*d^2*(f*x^2+e)*f*e*c^3+d^3*(28/9*f^2*x^4-23/9*e*f*x^2+e^2)*(f*x^ 2+e)*e^2*c^2+d^4*(-14/9*f*x^2+e)*x^2*(f*x^2+e)*e^3*c+1/6*d^5*e^4*x^4*(f*x^ 2+e))*b^5*a^2+2*c*((-8/3*f*x^2+e)*c-8/3*d*e*x^2)*(c*f-d*e)^2*(d*x^2+c)*b^6 *(f*x^2+e)*e*a+4/3*b^7*c^2*e^2*x^2*(f*x^2+e)*(d*x^2+c)*(c*f-d*e)^2)*(c*f-d *e)*x))/((a*d-b*c)*c)^(1/2)/(b*x^2+a)^(3/2)/(d*x^2+c)/(c*f-d*e)^3/(a*d-...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^2/(f*x^2+e)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{2} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**2/(f*x**2+e)**2,x)
Output:
Integral(1/((a + b*x**2)**(5/2)*(c + d*x**2)**2*(e + f*x**2)**2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{2} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^2/(f*x^2+e)^2,x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^2*(f*x^2 + e)^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 5859 vs. \(2 (614) = 1228\).
Time = 13.16 (sec) , antiderivative size = 5859, normalized size of antiderivative = 8.97 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^2/(f*x^2+e)^2,x, algorithm="giac")
Output:
1/3*(2*(b^14*c^4*e^4 - 7*a*b^13*c^3*d*e^4 + 15*a^2*b^12*c^2*d^2*e^4 - 13*a ^3*b^11*c*d^3*e^4 + 4*a^4*b^10*d^4*e^4 - 7*a*b^13*c^4*e^3*f + 40*a^2*b^12* c^3*d*e^3*f - 78*a^3*b^11*c^2*d^2*e^3*f + 64*a^4*b^10*c*d^3*e^3*f - 19*a^5 *b^9*d^4*e^3*f + 15*a^2*b^12*c^4*e^2*f^2 - 78*a^3*b^11*c^3*d*e^2*f^2 + 144 *a^4*b^10*c^2*d^2*e^2*f^2 - 114*a^5*b^9*c*d^3*e^2*f^2 + 33*a^6*b^8*d^4*e^2 *f^2 - 13*a^3*b^11*c^4*e*f^3 + 64*a^4*b^10*c^3*d*e*f^3 - 114*a^5*b^9*c^2*d ^2*e*f^3 + 88*a^6*b^8*c*d^3*e*f^3 - 25*a^7*b^7*d^4*e*f^3 + 4*a^4*b^10*c^4* f^4 - 19*a^5*b^9*c^3*d*f^4 + 33*a^6*b^8*c^2*d^2*f^4 - 25*a^7*b^7*c*d^3*f^4 + 7*a^8*b^6*d^4*f^4)*x^2/(a^2*b^13*c^6*e^6 - 6*a^3*b^12*c^5*d*e^6 + 15*a^ 4*b^11*c^4*d^2*e^6 - 20*a^5*b^10*c^3*d^3*e^6 + 15*a^6*b^9*c^2*d^4*e^6 - 6* a^7*b^8*c*d^5*e^6 + a^8*b^7*d^6*e^6 - 6*a^3*b^12*c^6*e^5*f + 36*a^4*b^11*c ^5*d*e^5*f - 90*a^5*b^10*c^4*d^2*e^5*f + 120*a^6*b^9*c^3*d^3*e^5*f - 90*a^ 7*b^8*c^2*d^4*e^5*f + 36*a^8*b^7*c*d^5*e^5*f - 6*a^9*b^6*d^6*e^5*f + 15*a^ 4*b^11*c^6*e^4*f^2 - 90*a^5*b^10*c^5*d*e^4*f^2 + 225*a^6*b^9*c^4*d^2*e^4*f ^2 - 300*a^7*b^8*c^3*d^3*e^4*f^2 + 225*a^8*b^7*c^2*d^4*e^4*f^2 - 90*a^9*b^ 6*c*d^5*e^4*f^2 + 15*a^10*b^5*d^6*e^4*f^2 - 20*a^5*b^10*c^6*e^3*f^3 + 120* a^6*b^9*c^5*d*e^3*f^3 - 300*a^7*b^8*c^4*d^2*e^3*f^3 + 400*a^8*b^7*c^3*d^3* e^3*f^3 - 300*a^9*b^6*c^2*d^4*e^3*f^3 + 120*a^10*b^5*c*d^5*e^3*f^3 - 20*a^ 11*b^4*d^6*e^3*f^3 + 15*a^6*b^9*c^6*e^2*f^4 - 90*a^7*b^8*c^5*d*e^2*f^4 + 2 25*a^8*b^7*c^4*d^2*e^2*f^4 - 300*a^9*b^6*c^3*d^3*e^2*f^4 + 225*a^10*b^5...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^2\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^2*(e + f*x^2)^2),x)
Output:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^2*(e + f*x^2)^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{2} \left (f \,x^{2}+e \right )^{2}}d x \] Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^2/(f*x^2+e)^2,x)
Output:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^2/(f*x^2+e)^2,x)