Integrand size = 30, antiderivative size = 566 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=-\frac {d^4 x \sqrt {a+b x^2}}{4 c (b c-a d) (d e-c f)^3 \left (c+d x^2\right )^2}+\frac {3 d^4 (a d (d e-5 c f)-2 b c (d e-3 c f)) x \sqrt {a+b x^2}}{8 c^2 (b c-a d)^2 (d e-c f)^4 \left (c+d x^2\right )}+\frac {f^4 x \sqrt {a+b x^2}}{4 e (b e-a f) (d e-c f)^3 \left (e+f x^2\right )^2}+\frac {3 f^4 (2 b e (3 d e-c f)-a f (5 d e-c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f)^2 (d e-c f)^4 \left (e+f x^2\right )}+\frac {d^3 \left (8 b^2 c^2 \left (d^2 e^2-5 c d e f+10 c^2 f^2\right )+3 a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )-4 a b c d \left (2 d^2 e^2-13 c d e f+35 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2} (d e-c f)^5}-\frac {f^3 \left (3 a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )+8 b^2 e^2 \left (10 d^2 e^2-5 c d e f+c^2 f^2\right )-4 a b e f \left (35 d^2 e^2-13 c d e f+2 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{5/2} (d e-c f)^5} \] Output:
-1/4*d^4*x*(b*x^2+a)^(1/2)/c/(-a*d+b*c)/(-c*f+d*e)^3/(d*x^2+c)^2+3/8*d^4*( a*d*(-5*c*f+d*e)-2*b*c*(-3*c*f+d*e))*x*(b*x^2+a)^(1/2)/c^2/(-a*d+b*c)^2/(- c*f+d*e)^4/(d*x^2+c)+1/4*f^4*x*(b*x^2+a)^(1/2)/e/(-a*f+b*e)/(-c*f+d*e)^3/( f*x^2+e)^2+3/8*f^4*(2*b*e*(-c*f+3*d*e)-a*f*(-c*f+5*d*e))*x*(b*x^2+a)^(1/2) /e^2/(-a*f+b*e)^2/(-c*f+d*e)^4/(f*x^2+e)+1/8*d^3*(8*b^2*c^2*(10*c^2*f^2-5* c*d*e*f+d^2*e^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*(35*c^ 2*f^2-13*c*d*e*f+2*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)^(5/2)/(-c*f+d*e)^5-1/8*f^3*(3*a^2*f^2*(c^2*f^2-6 *c*d*e*f+21*d^2*e^2)+8*b^2*e^2*(c^2*f^2-5*c*d*e*f+10*d^2*e^2)-4*a*b*e*f*(2 *c^2*f^2-13*c*d*e*f+35*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)^(5/2)/(-c*f+d*e)^5
Time = 17.27 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\frac {3 d^3 f x \left (d \left (a+b x^2\right )-\frac {(2 b c-a d) \left (c+d x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c \sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}\right )}{2 c (b c-a d) (d e-c f)^4 \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {d^3 x \left (c d \left (a+b x^2\right ) \left (2 b c \left (4 c+3 d x^2\right )-a d \left (5 c+3 d x^2\right )\right )-\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \left (c+d x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}\right )}{8 c^3 (b c-a d)^2 (d e-c f)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {3 d f^3 x \left (f \left (a+b x^2\right )-\frac {(2 b e-a f) \left (e+f x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{e \sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{2 e (b e-a f) (d e-c f)^4 \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {f^3 x \left (e f \left (a+b x^2\right ) \left (2 b e \left (4 e+3 f x^2\right )-a f \left (5 e+3 f x^2\right )\right )-\frac {\left (8 b^2 e^2-8 a b e f+3 a^2 f^2\right ) \left (e+f x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{8 e^3 (b e-a f)^2 (d e-c f)^3 \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {6 d^3 f^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)^5}-\frac {6 d^2 f^3 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)^5} \] Input:
Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^3*(e + f*x^2)^3),x]
Output:
(3*d^3*f*x*(d*(a + b*x^2) - ((2*b*c - a*d)*(c + d*x^2)*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]) ))/(2*c*(b*c - a*d)*(d*e - c*f)^4*Sqrt[a + b*x^2]*(c + d*x^2)) - (d^3*x*(c *d*(a + b*x^2)*(2*b*c*(4*c + 3*d*x^2) - a*d*(5*c + 3*d*x^2)) - ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*(c + d*x^2)^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c* (a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(8*c^3*(b*c - a* d)^2*(d*e - c*f)^3*Sqrt[a + b*x^2]*(c + d*x^2)^2) + (3*d*f^3*x*(f*(a + b*x ^2) - ((2*b*e - a*f)*(e + f*x^2)*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b* x^2))]])/(e*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))])))/(2*e*(b*e - a*f)*(d *e - c*f)^4*Sqrt[a + b*x^2]*(e + f*x^2)) + (f^3*x*(e*f*(a + b*x^2)*(2*b*e* (4*e + 3*f*x^2) - a*f*(5*e + 3*f*x^2)) - ((8*b^2*e^2 - 8*a*b*e*f + 3*a^2*f ^2)*(e + f*x^2)^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/Sqrt[( (b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(8*e^3*(b*e - a*f)^2*(d*e - c*f)^3*Sqr t[a + b*x^2]*(e + f*x^2)^2) + (6*d^3*f^2*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt [c]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt[b*c - a*d]*(d*e - c*f)^5) - (6*d^2*f^ 3*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*Sqrt[b* e - a*f]*(d*e - c*f)^5)
Leaf count is larger than twice the leaf count of optimal. \(1697\) vs. \(2(566)=1132\).
Time = 2.32 (sec) , antiderivative size = 1697, normalized size of antiderivative = 3.00, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {426, 426, 421, 25, 402, 25, 402, 25, 27, 291, 221, 407, 291, 221, 426, 421, 25, 291, 221, 402, 25, 27, 291, 221}
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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {2 b d (d e-c f) x^2+a d (3 d e-7 c f)-4 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{4 c \left (c+d x^2\right )^2 (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int -\frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (-\frac {\int \frac {2 b d (d e-c f) x^2+a d (3 d e-7 c f)-4 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{4 c \left (c+d x^2\right )^2 (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (-\frac {\frac {\int -\frac {8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (a d (3 d e-7 c f)-2 b c (3 d e-5 c f))}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{4 c \left (c+d x^2\right )^2 (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {\int -\frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (-\frac {-\frac {\int \frac {8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (a d (3 d e-7 c f)-2 b c (3 d e-5 c f))}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{4 c \left (c+d x^2\right )^2 (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\int \frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (-\frac {-\frac {\left (a^2 d^2 (3 d e-7 c f)-4 a b c d (2 d e-5 c f)+8 b^2 c^2 (d e-2 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (a d (3 d e-7 c f)-2 b c (3 d e-5 c f))}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{4 c \left (c+d x^2\right )^2 (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (-\frac {-\frac {\left (a^2 d^2 (3 d e-7 c f)-4 a b c d (2 d e-5 c f)+8 b^2 c^2 (d e-2 c f)\right ) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (a d (3 d e-7 c f)-2 b c (3 d e-5 c f))}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{4 c \left (c+d x^2\right )^2 (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (-\frac {-\frac {\text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right ) \left (a^2 d^2 (3 d e-7 c f)-4 a b c d (2 d e-5 c f)+8 b^2 c^2 (d e-2 c f)\right )}{2 c^{3/2} (b c-a d)^{3/2}}-\frac {d x \sqrt {a+b x^2} (a d (3 d e-7 c f)-2 b c (3 d e-5 c f))}{2 c \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{4 c \left (c+d x^2\right )^2 (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 407 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right ) f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\left (8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right ) f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\left (8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right ) f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{4 c (b c-a d) \left (d x^2+c\right )^2}-\frac {-\frac {d (a d (3 d e-7 c f)-2 b c (3 d e-5 c f)) \sqrt {b x^2+a} x}{2 c (b c-a d) \left (d x^2+c\right )}-\frac {\left (8 b^2 (d e-2 c f) c^2-4 a b d (2 d e-5 c f) c+a^2 d^2 (3 d e-7 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}}{4 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\right )}{d e-c f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 {\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 e-c f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\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 ) 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 \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}\right )}{d e-c f}-\frac {f \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 e-c f}\) |
Input:
Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)^3*(e + f*x^2)^3),x]
Output:
-((f*(-((f*((d^2*((d*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2]) ])/(Sqrt[c]*Sqrt[b*c - a*d]*(d*e - c*f)) - (f*ArcTanh[(Sqrt[b*e - a*f]*x)/ (Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*Sqrt[b*e - a*f]*(d*e - c*f))))/(d*e - c*f)^2 - (f*(-1/4*(f*(d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f *x^2)^2) - ((f*(2*b*e*(5*d*e - 3*c*f) - a*f*(7*d*e - 3*c*f))*x*Sqrt[a + b* x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) - ((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))*ArcTanh[(Sqrt[b*e - a*f]*x)/( Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e - a*f) )))/(d*e - c*f)^2))/(d*e - c*f)) + (d*((d*((d*(-1/2*(d*(d*e - c*f)*x*Sqrt[ a + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)) - ((a*d*(d*e - 3*c*f) - 2*b*c*(d*e - 2*c*f))*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3 /2)*(b*c - a*d)^(3/2))))/(d*e - c*f)^2 + (f^2*ArcTanh[(Sqrt[b*e - a*f]*x)/ (Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*Sqrt[b*e - a*f]*(d*e - c*f)^2)))/(d*e - c*f) - (f*((d^2*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])]) /(Sqrt[c]*Sqrt[b*c - a*d]*(d*e - c*f)^2) - (f*(-1/2*(f*(d*e - c*f)*x*Sqrt[ a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) + ((2*b*e*(2*d*e - c*f) - a*f*(3*d *e - c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3 /2)*(b*e - a*f)^(3/2))))/(d*e - c*f)^2))/(d*e - c*f)))/(d*e - c*f)))/(d*e - c*f)) + (d*((d*((d*(-1/4*(d*(d*e - c*f)*x*Sqrt[a + b*x^2])/(c*(b*c - a*d )*(c + d*x^2)^2) - (-1/2*(d*(a*d*(3*d*e - 7*c*f) - 2*b*c*(3*d*e - 5*c*f...
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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1167\) vs. \(2(526)=1052\).
Time = 17.01 (sec) , antiderivative size = 1168, normalized size of antiderivative = 2.06
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1168\) |
| default | \(\text {Expression too large to display}\) | \(4322\) |
Input:
int(1/(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)*(-21*((a*f-b*e)*e)^(1/2)*d^3*(80/63*b^2*c^4*f^2-2 0/9*d*(a*f+2/7*b*e)*b*f*c^3+d^2*(a^2*f^2+52/63*a*b*f*e+8/63*b^2*e^2)*c^2-2 /7*a*(a*f+4/9*b*e)*d^3*e*c+1/21*a^2*e^2*d^4)*(d*x^2+c)^2*(a*f-b*e)^2*(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)^2*c^ 2*(d*x^2+c)^2*(f*x^2+e)^2*f^3*((21*a^2*e^2*f^2-140/3*a*b*e^3*f+80/3*b^2*e^ 4)*d^2-6*d*(a^2*f^2-26/9*a*b*f*e+20/9*b^2*e^2)*f*e*c+f^2*(a^2*f^2-8/3*a*b* f*e+8/3*b^2*e^2)*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)*(c*f-d*e)*(b*x^2+a)^(1/2)*(3/5*a*d^7*e^3*x^2*(f*x^2+e) ^2*(a*f-b*e)^2+d^6*(a*f-b*e)^2*(f*x^2+e)^2*((-6/5*b*x^2+a)*e-3*a*f*x^2)*e^ 2*c-17/5*d^5*(8/17*b^3*e^6+1/17*b^2*f*(-2*b*x^2+a)*e^5-26/17*b*f^2*(14/13* b^2*x^4-19/13*a*b*x^2+a^2)*e^4+f^3*(-18/17*b^3*x^6+73/17*a*b^2*x^4-70/17*a ^2*b*x^2+a^3)*e^3+2*a*f^4*x^2*(18/17*b^2*x^4-41/17*a*b*x^2+a^2)*e^2+2*a^2* (-18/17*b*x^2+a)*x^4*f^5*e+15/17*a^3*f^6*x^6)*e*c^2-34/5*d^4*(-10/17*b^3*e ^6+20/17*b^2*f*(-b*x^2+a)*e^5-10/17*b*f^2*(b^2*x^4-4*a*b*x^2+a^2)*e^4-40/1 7*a*b*f^3*x^2*(-b*x^2+a)*e^3+a*f^4*x^2*(18/17*b^2*x^4-41/17*a*b*x^2+a^2)*e ^2+25/34*a^2*(-24/25*b*x^2+a)*x^4*f^5*e-3/34*a^3*f^6*x^6)*f*c^3-17/5*d^3*f ^4*(-20/17*b*(b^2*x^4-4*a*b*x^2+a^2)*e^3+f*(-18/17*b^3*x^6+73/17*a*b^2*x^4 -70/17*a^2*b*x^2+a^3)*e^2+5/17*a*(3/5*b^2*x^4-38/5*a*b*x^2+a^2)*x^2*f^2*e- 6/17*a^2*x^4*f^3*(-b*x^2+a))*c^4+c^5*f^4*(3/5*a*f^3*x^2*(b^2*x^4-4*a*b*x^2 +a^2)+f^2*(4/5*a^2*b*x^2-6/5*b^3*x^6-1/5*a*b^2*x^4+a^3)*e+26/5*b*f*(14/...
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x**2+a)**(1/2)/(d*x**2+c)**3/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{3} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate(1/(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. 11237 vs. \(2 (527) = 1054\).
Time = 7.70 (sec) , antiderivative size = 11237, normalized size of antiderivative = 19.85 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="giac")
Output:
-1/8*b^(11/2)*((8*b^2*c^2*d^5*e^2 - 8*a*b*c*d^6*e^2 + 3*a^2*d^7*e^2 - 40*b ^2*c^3*d^4*e*f + 52*a*b*c^2*d^5*e*f - 18*a^2*c*d^6*e*f + 80*b^2*c^4*d^3*f^ 2 - 140*a*b*c^3*d^4*f^2 + 63*a^2*c^2*d^5*f^2)*arctan(1/2*((sqrt(b)*x - sqr t(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/((b^7*c^4*d^5*e ^5 - 2*a*b^6*c^3*d^6*e^5 + a^2*b^5*c^2*d^7*e^5 - 5*b^7*c^5*d^4*e^4*f + 10* a*b^6*c^4*d^5*e^4*f - 5*a^2*b^5*c^3*d^6*e^4*f + 10*b^7*c^6*d^3*e^3*f^2 - 2 0*a*b^6*c^5*d^4*e^3*f^2 + 10*a^2*b^5*c^4*d^5*e^3*f^2 - 10*b^7*c^7*d^2*e^2* f^3 + 20*a*b^6*c^6*d^3*e^2*f^3 - 10*a^2*b^5*c^5*d^4*e^2*f^3 + 5*b^7*c^8*d* e*f^4 - 10*a*b^6*c^7*d^2*e*f^4 + 5*a^2*b^5*c^6*d^3*e*f^4 - b^7*c^9*f^5 + 2 *a*b^6*c^8*d*f^5 - a^2*b^5*c^7*d^2*f^5)*sqrt(-b^2*c^2 + a*b*c*d)) - (80*b^ 2*d^2*e^4*f^3 - 40*b^2*c*d*e^3*f^4 - 140*a*b*d^2*e^3*f^4 + 8*b^2*c^2*e^2*f ^5 + 52*a*b*c*d*e^2*f^5 + 63*a^2*d^2*e^2*f^5 - 8*a*b*c^2*e*f^6 - 18*a^2*c* d*e*f^6 + 3*a^2*c^2*f^7)*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^7*d^5*e^9 - 5*b^7*c*d^4*e^8*f - 2*a*b^6*d^5*e^8*f + 10*b^7*c^2*d^3*e^7*f^2 + 10*a*b^6*c*d^4*e^7*f^2 + a^2* b^5*d^5*e^7*f^2 - 10*b^7*c^3*d^2*e^6*f^3 - 20*a*b^6*c^2*d^3*e^6*f^3 - 5*a^ 2*b^5*c*d^4*e^6*f^3 + 5*b^7*c^4*d*e^5*f^4 + 20*a*b^6*c^3*d^2*e^5*f^4 + 10* a^2*b^5*c^2*d^3*e^5*f^4 - b^7*c^5*e^4*f^5 - 10*a*b^6*c^4*d*e^4*f^5 - 10*a^ 2*b^5*c^3*d^2*e^4*f^5 + 2*a*b^6*c^5*e^3*f^6 + 5*a^2*b^5*c^4*d*e^3*f^6 - a^ 2*b^5*c^5*e^2*f^7)*sqrt(-b^2*e^2 + a*b*e*f)) + 2*(8*(sqrt(b)*x - sqrt(b...
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^3\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^3*(e + f*x^2)^3),x)
Output:
int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^3*(e + f*x^2)^3), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\sqrt {b \,x^{2}+a}\, \left (d \,x^{2}+c \right )^{3} \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x)
Output:
int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^3/(f*x^2+e)^3,x)