Integrand size = 32, antiderivative size = 658 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {b \left (a b d^2 e-a^2 d^2 f+3 b^2 c (d e-c f)\right ) x}{3 a c (b c-a d)^2 (b e-a f) (d e-c f) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (a b^2 c d^2 e (7 d e-10 c f)+a^3 d^3 f (2 d e-5 c f)+3 b^3 c^2 (d e-c f)^2-2 a^2 b d^2 \left (d^2 e^2+c d e f-5 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} (b c-a d)^3 (b e-a f) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {d^{3/2} \left (3 a^2 c d^2 f^2+3 b^2 c \left (3 d^2 e^2-8 c d e f+6 c^2 f^2\right )-a b d \left (d^2 e^2-8 c d e f+13 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} (b c-a d)^3 (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} f^4 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (b e-a f) (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
-1/3*d^2*x/c/(-a*d+b*c)/(-c*f+d*e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)+1/3*b*( a*b*d^2*e-a^2*d^2*f+3*b^2*c*(-c*f+d*e))*x/a/c/(-a*d+b*c)^2/(-a*f+b*e)/(-c* f+d*e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+1/3*d^(1/2)*(a*b^2*c*d^2*e*(-10*c*f +7*d*e)+a^3*d^3*f*(-5*c*f+2*d*e)+3*b^3*c^2*(-c*f+d*e)^2-2*a^2*b*d^2*(-5*c^ 2*f^2+c*d*e*f+d^2*e^2))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x ^2/c)^(1/2),(1-b*c/a/d)^(1/2))/a/c^(3/2)/(-a*d+b*c)^3/(-a*f+b*e)/(-c*f+d*e )^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*d^(3/2)*(3*a^2*c*d ^2*f^2+3*b^2*c*(6*c^2*f^2-8*c*d*e*f+3*d^2*e^2)-a*b*d*(13*c^2*f^2-8*c*d*e*f +d^2*e^2))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b* c/a/d)^(1/2))/a/c^(1/2)/(-a*d+b*c)^3/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c) )^(1/2)/(d*x^2+c)^(1/2)+c^(3/2)*f^4*(b*x^2+a)^(1/2)*EllipticPi(d^(1/2)*x/c ^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e/(-a*f+b* e)/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 13.37 (sec) , antiderivative size = 1660, normalized size of antiderivative = 2.52 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x]
Output:
(Sqrt[b/a]*((-I)*b*c*e*(-3*b^3*c^2*(d*e - c*f)^2 + a^3*d^3*f*(-2*d*e + 5*c *f) + a*b^2*c*d^2*e*(-7*d*e + 10*c*f) + 2*a^2*b*d^2*(d^2*e^2 + c*d*e*f - 5 *c^2*f^2))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*EllipticE[I *ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (Sqrt[b/a]*(3*b^5*c^4*d^2*e^3*x + 8* a^2*b^3*c^2*d^4*e^3*x - 3*a^3*b^2*c*d^5*e^3*x - 6*b^5*c^5*d*e^2*f*x - 11*a ^2*b^3*c^3*d^3*e^2*f*x - 2*a^3*b^2*c^2*d^4*e^2*f*x + 3*a^4*b*c*d^5*e^2*f*x + 3*b^5*c^6*e*f^2*x + 11*a^3*b^2*c^3*d^3*e*f^2*x - 6*a^4*b*c^2*d^4*e*f^2* x + 6*b^5*c^3*d^3*e^3*x^3 + 8*a*b^4*c^2*d^4*e^3*x^3 + 4*a^2*b^3*c*d^5*e^3* x^3 - 2*a^3*b^2*d^6*e^3*x^3 - 12*b^5*c^4*d^2*e^2*f*x^3 - 11*a*b^4*c^3*d^3* e^2*f*x^3 - 12*a^2*b^3*c^2*d^4*e^2*f*x^3 + a^3*b^2*c*d^5*e^2*f*x^3 + 2*a^4 *b*d^6*e^2*f*x^3 + 6*b^5*c^5*d*e*f^2*x^3 + 11*a^2*b^3*c^3*d^3*e*f^2*x^3 + 4*a^3*b^2*c^2*d^4*e*f^2*x^3 - 5*a^4*b*c*d^5*e*f^2*x^3 + 3*b^5*c^2*d^4*e^3* x^5 + 7*a*b^4*c*d^5*e^3*x^5 - 2*a^2*b^3*d^6*e^3*x^5 - 6*b^5*c^3*d^3*e^2*f* x^5 - 10*a*b^4*c^2*d^4*e^2*f*x^5 - 2*a^2*b^3*c*d^5*e^2*f*x^5 + 2*a^3*b^2*d ^6*e^2*f*x^5 + 3*b^5*c^4*d^2*e*f^2*x^5 + 10*a^2*b^3*c^2*d^4*e*f^2*x^5 - 5* a^3*b^2*c*d^5*e*f^2*x^5 + I*a*b*Sqrt[b/a]*c*(-(b*c) + a*d)*e*(-(d*e) + c*f )*(-(a*b*d^2*e) + a^2*d^2*f + 3*b^2*c*(-(d*e) + c*f))*Sqrt[1 + (b*x^2)/a]* (c + d*x^2)*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b *c)] + ((3*I)*b^5*c^6*f^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic Pi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(b/a)^(3/2) - (9*...
Time = 1.17 (sec) , antiderivative size = 849, normalized size of antiderivative = 1.29, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {421, 25, 402, 402, 400, 313, 320, 421, 25, 401, 25, 27, 400, 313, 320, 414}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\int \frac {a (b d e+b c f-2 a d f)-3 b d (b e-a f) x^2}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\int \frac {a \left (3 c (2 d e+c f) b^2-a d (2 d e+11 c f) b+4 a^2 d^2 f\right )-b d \left (-2 d f a^2+b (d e-2 c f) a+3 b^2 c e\right ) x^2}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {a b \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {d \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {a b \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}-\frac {\int -\frac {d \left (b (d e-4 c f) x^2+a (2 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int \frac {d \left (b (d e-4 c f) x^2+a (2 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int \frac {b (d e-4 c f) x^2+a (2 d e-5 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\frac {a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {f^2 \left (\frac {d \left (\frac {\frac {a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)-b c (d e-4 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)-b c (d e-4 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {b x (b e-a f)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (2 a^2 d^2 f-a b d (13 c f+d e)+3 b^2 c (c f+3 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} \left (4 a^3 d^2 f-a^2 b d (13 c f+2 d e)+a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} \left (-2 a^2 d f+a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {\left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}-\frac {\frac {\frac {b \sqrt {c} \left (3 c (3 d e+c f) b^2-a d (d e+13 c f) b+2 a^2 d^2 f\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (4 d^2 f a^3-b d (2 d e+13 c f) a^2+b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}-\frac {d \left (-2 d f a^2+b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {b x^2+a}}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}}{a (b c-a d)}\right )}{(b e-a f)^2}\) |
Input:
Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x]
Output:
(b*((b*(b*e - a*f)*x)/(a*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) - (-1/3*(d*(3*b^2*c*e - 2*a^2*d*f + a*b*(d*e - 2*c*f))*x*Sqrt[a + b*x^2])/(c *(b*c - a*d)*(c + d*x^2)^(3/2)) + (-((Sqrt[d]*(3*b^3*c^2*e + 4*a^3*d^2*f + a*b^2*c*(7*d*e + c*f) - a^2*b*d*(2*d*e + 13*c*f))*Sqrt[a + b*x^2]*Ellipti cE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*(b*c - a*d)*Sqr t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (b*Sqrt[c]*(2*a^2*d ^2*f + 3*b^2*c*(3*d*e + c*f) - a*b*d*(d*e + 13*c*f))*Sqrt[a + b*x^2]*Ellip ticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*S qrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*c*(b*c - a*d)))/ (a*(b*c - a*d))))/(b*e - a*f)^2 + (f^2*((d*(((d*e - c*f)*x*Sqrt[a + b*x^2] )/(3*c*(c + d*x^2)^(3/2)) + (-(((a*d*(2*d*e - 5*c*f) - b*c*(d*e - 4*c*f))* Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/( Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (b*Sqrt[c]*(d*e - c*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[ d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2) )/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*c)))/(d*e - c*f)^2 + (a^(3/2)*f^2* Sqrt[c + d*x^2]*EllipticPi[1 - (a*f)/(b*e), ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[b]*c*e*(d*e - c*f)^2*Sqrt[a + b*x^2]*Sqrt[(a*(c + d *x^2))/(c*(a + b*x^2))])))/(b*e - a*f)^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 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[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(2754\) vs. \(2(628)=1256\).
Time = 20.36 (sec) , antiderivative size = 2755, normalized size of antiderivative = 4.19
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2755\) |
| default | \(\text {Expression too large to display}\) | \(4117\) |
Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/3/c*d/(a*d -b*c)*x/(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1 /2)/(x^2+c/d)^2-1/3*(b*d*x^2+a*d)/c^2*d^2/(a*d-b*c)^2*x*(5*a*c*d*f-2*a*d^2 *e-10*b*c^2*f+7*b*c*d*e)/(c*f-d*e)/(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e)/((x^2 +c/d)*(b*d*x^2+a*d))^(1/2)+(b*d*x^2+b*c)*b^3/a/(a*d-b*c)^3*x/(a*f-b*e)/((x ^2+a/b)*(b*d*x^2+b*c))^(1/2)+3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^ (1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a* d+b*c)/c/b)^(1/2))*a*d^4/c/(a*d-b*c)^2/(c*f-d*e)/(a*c*d*f-a*d^2*e-b*c^2*f+ b*c*d*e)*b*e+5/3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4 +a*d*x^2+b*c*x^2+a*c)^(1/2)*d^3*b/(a*d-b*c)^2/(c*f-d*e)/(a*c*d*f-a*d^2*e-b *c^2*f+b*c*d*e)*a*f*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-2/3 /c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x ^2+a*c)^(1/2)*d^4*b/(a*d-b*c)^2/(c*f-d*e)/(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e )*a*e*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/3/(-b/a)^(1/2)* (1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*El lipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))/c*d^2/(a*d-b*c)*b/(a*c*d* f-a*d^2*e-b*c^2*f+b*c*d*e)-5/3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^ (1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a* d+b*c)/c/b)^(1/2))*d^3/(a*d-b*c)/c/(c*f-d*e)/(a*c*d*f-a*d^2*e-b*c^2*f+b*c* d*e)*a*f+2/3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(5/2)/(f*x**2+e),x)
Output:
Integral(1/((a + b*x**2)**(3/2)*(c + d*x**2)**(5/2)*(e + f*x**2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="maxima ")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2)*(f*x^2 + e)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{5/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x)
Output:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)*(e + f*x^2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d^{3} f \,x^{12}+2 a b \,d^{3} f \,x^{10}+3 b^{2} c \,d^{2} f \,x^{10}+b^{2} d^{3} e \,x^{10}+a^{2} d^{3} f \,x^{8}+6 a b c \,d^{2} f \,x^{8}+2 a b \,d^{3} e \,x^{8}+3 b^{2} c^{2} d f \,x^{8}+3 b^{2} c \,d^{2} e \,x^{8}+3 a^{2} c \,d^{2} f \,x^{6}+a^{2} d^{3} e \,x^{6}+6 a b \,c^{2} d f \,x^{6}+6 a b c \,d^{2} e \,x^{6}+b^{2} c^{3} f \,x^{6}+3 b^{2} c^{2} d e \,x^{6}+3 a^{2} c^{2} d f \,x^{4}+3 a^{2} c \,d^{2} e \,x^{4}+2 a b \,c^{3} f \,x^{4}+6 a b \,c^{2} d e \,x^{4}+b^{2} c^{3} e \,x^{4}+a^{2} c^{3} f \,x^{2}+3 a^{2} c^{2} d e \,x^{2}+2 a b \,c^{3} e \,x^{2}+a^{2} c^{3} e}d x \] Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c**3*e + a**2*c**3*f*x**2 + 3*a**2*c**2*d*e*x**2 + 3*a**2*c**2*d*f*x**4 + 3*a**2*c*d**2*e*x**4 + 3*a** 2*c*d**2*f*x**6 + a**2*d**3*e*x**6 + a**2*d**3*f*x**8 + 2*a*b*c**3*e*x**2 + 2*a*b*c**3*f*x**4 + 6*a*b*c**2*d*e*x**4 + 6*a*b*c**2*d*f*x**6 + 6*a*b*c* d**2*e*x**6 + 6*a*b*c*d**2*f*x**8 + 2*a*b*d**3*e*x**8 + 2*a*b*d**3*f*x**10 + b**2*c**3*e*x**4 + b**2*c**3*f*x**6 + 3*b**2*c**2*d*e*x**6 + 3*b**2*c** 2*d*f*x**8 + 3*b**2*c*d**2*e*x**8 + 3*b**2*c*d**2*f*x**10 + b**2*d**3*e*x* *10 + b**2*d**3*f*x**12),x)