Integrand size = 32, antiderivative size = 441 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {b^2 x \sqrt {a+b x^2}}{d f \sqrt {c+d x^2}}-\frac {\left (2 a b c d f-a^2 d^2 f+b^2 c (d e-2 c f)\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} d^{3/2} f (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \left (b^3 c d e^2-a^3 d^2 f^2-a b^2 c f (4 d e-c f)+a^2 b d f (2 d e+c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a d^{3/2} f (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} (b e-a f)^3 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
b^2*x*(b*x^2+a)^(1/2)/d/f/(d*x^2+c)^(1/2)-(2*a*b*c*d*f-a^2*d^2*f+b^2*c*(-2 *c*f+d*e))*(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))/c^(1/2)/d^(3/2)/f/(-c*f+d*e)/(c*(b*x^2+a)/a/(d*x^2+c))^( 1/2)/(d*x^2+c)^(1/2)+c^(1/2)*(b^3*c*d*e^2-a^3*d^2*f^2-a*b^2*c*f*(-c*f+4*d* e)+a^2*b*d*f*(c*f+2*d*e))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x /c^(1/2)),(1-b*c/a/d)^(1/2))/a/d^(3/2)/f/(-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^ 2+c))^(1/2)/(d*x^2+c)^(1/2)-c^(3/2)*(-a*f+b*e)^3*(b*x^2+a)^(1/2)*EllipticP i(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/f/(-c*f+d*e)^2/(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 = 7.64 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {-i b c e f \left (-2 a b c d f+a^2 d^2 f+b^2 c (-d e+2 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b^2 c e (-d e+c f) (3 a d f-b (d e+2 c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-d \left (\sqrt {\frac {b}{a}} (b c-a d)^2 e f^2 x \left (a+b x^2\right )+i c d (-b e+a f)^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} c d^2 e f^2 (-d e+c f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*(e + f*x^2)),x]
Output:
((-I)*b*c*e*f*(-2*a*b*c*d*f + a^2*d^2*f + b^2*c*(-(d*e) + 2*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b *c)] - I*b^2*c*e*(-(d*e) + c*f)*(3*a*d*f - b*(d*e + 2*c*f))*Sqrt[1 + (b*x^ 2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - d*(Sqrt[b/a]*(b*c - a*d)^2*e*f^2*x*(a + b*x^2) + I*c*d*(-(b*e) + a*f)^3*S qrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[S qrt[b/a]*x], (a*d)/(b*c)]))/(Sqrt[b/a]*c*d^2*e*f^2*(-(d*e) + c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 1.38 (sec) , antiderivative size = 822, normalized size of antiderivative = 1.86, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {419, 25, 401, 25, 27, 403, 25, 406, 320, 388, 313, 418, 25, 403, 27, 406, 320, 388, 313, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 419 |
| \(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {-\frac {\int -\frac {d \sqrt {b x^2+a} \left (b (b c (4 d e-3 c f)-a d (3 d e-2 c f)) x^2+a c d (b e-a f)\right )}{\sqrt {d x^2+c}}dx}{c d}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {d \sqrt {b x^2+a} \left (b (b c (4 d e-3 c f)-a d (3 d e-2 c f)) x^2+a c d (b e-a f)\right )}{\sqrt {d x^2+c}}dx}{c d}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {b x^2+a} \left (b (b c (4 d e-3 c f)-a d (3 d e-2 c f)) x^2+a c d (b e-a f)\right )}{\sqrt {d x^2+c}}dx}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {a c \left (c (4 d e-3 c f) b^2-2 a d (3 d e-c f) b+3 a^2 d^2 f\right )-b \left (-2 b^2 (4 d e-3 c f) c^2+a b d (13 d e-7 c f) c-a^2 d^2 (3 d e+c f)\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\int \frac {a c \left (c (4 d e-3 c f) b^2-2 a d (3 d e-c f) b+3 a^2 d^2 f\right )-b \left (-2 b^2 (4 d e-3 c f) c^2+a b d (13 d e-7 c f) c-a^2 d^2 (3 d e+c f)\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {a c \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}{f x^2+e}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 418 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f)^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f^2}+\frac {b \int -\frac {\sqrt {d x^2+c} \left (-b f x^2+b e-2 a f\right )}{\sqrt {b x^2+a}}dx}{f^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f)^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f^2}-\frac {b \int \frac {\sqrt {d x^2+c} \left (-b f x^2+b e-2 a f\right )}{\sqrt {b x^2+a}}dx}{f^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f)^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f^2}-\frac {b \left (\frac {\int \frac {b \left ((3 b d e-b c f-4 a d f) x^2+c (3 b e-5 a f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f)^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f^2}-\frac {b \left (\frac {1}{3} \int \frac {(3 b d e-b c f-4 a d f) x^2+c (3 b e-5 a f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f)^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f^2}-\frac {b \left (\frac {1}{3} \left (c (3 b e-5 a f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+(-4 a d f-b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f)^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f^2}-\frac {b \left (\frac {1}{3} \left ((-4 a d f-b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (3 b e-5 a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f)^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f^2}-\frac {b \left (\frac {1}{3} \left ((-4 a d f-b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (3 b e-5 a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c (4 d e-3 c f)-a d (3 d e-2 c f))}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (3 a^2 d^2 f-2 a b d (3 d e-c f)+b^2 c (4 d e-3 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b \left (-a^2 d^2 (c f+3 d e)+a b c d (13 d e-7 c f)-2 b^2 c^2 (4 d e-3 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{c \sqrt {c+d x^2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f)^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f^2}-\frac {b \left (\frac {1}{3} \left (\frac {c^{3/2} \sqrt {a+b x^2} (3 b e-5 a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(-4 a d f-b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {\frac {\frac {b (b c (4 d e-3 c f)-a d (3 d e-2 c f)) x \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 d}-\frac {\frac {c^{3/2} \left (c (4 d e-3 c f) b^2-2 a d (3 d e-c f) b+3 a^2 d^2 f\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-b \left (-2 b^2 (4 d e-3 c f) c^2+a b d (13 d e-7 c f) c-a^2 d^2 (3 d e+c f)\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{3 d}}{c}-\frac {(b c-a d) (d e-c f) x \left (b x^2+a\right )^{3/2}}{c \sqrt {d x^2+c}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {c^{3/2} (b e-a f)^2 \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \left (\frac {1}{3} \left (\frac {(3 b e-5 a f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+(3 b d e-b c f-4 a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{(d e-c f)^2}\) |
Input:
Int[(a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*(e + f*x^2)),x]
Output:
(-(((b*c - a*d)*(d*e - c*f)*x*(a + b*x^2)^(3/2))/(c*Sqrt[c + d*x^2])) + (( b*(b*c*(4*d*e - 3*c*f) - a*d*(3*d*e - 2*c*f))*x*Sqrt[a + b*x^2]*Sqrt[c + d *x^2])/(3*d) - (-(b*(a*b*c*d*(13*d*e - 7*c*f) - 2*b^2*c^2*(4*d*e - 3*c*f) - a^2*d^2*(3*d*e + c*f))*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[ c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)] )/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))) + (c ^(3/2)*(3*a^2*d^2*f + b^2*c*(4*d*e - 3*c*f) - 2*a*b*d*(3*d*e - c*f))*Sqrt[ a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[ d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*d))/c)/(d*e - c*f)^2 - (f*(b*e - a*f)*(-((b*(-1/3*(f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2] ) + ((3*b*d*e - b*c*f - 4*a*d*f)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c )/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2] )) + (c^(3/2)*(3*b*e - 5*a*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x) /Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2 ))]*Sqrt[c + d*x^2]))/3))/f^2) + (c^(3/2)*(b*e - a*f)^2*Sqrt[a + b*x^2]*El lipticPi[1 - (c*f)/(d*e), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/( a*Sqrt[d]*e*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/( d*e - c*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[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, 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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
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)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[(b*c - a*d)^2/b^2 Int[Sqrt[e + f*x^2]/((a + b*x ^2)*Sqrt[c + d*x^2]), x], x] + Simp[d/b^2 Int[(2*b*c - a*d + b*d*x^2)*(Sq rt[e + f*x^2]/Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && P osQ[d/c] && PosQ[f/e]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b*((b*e - a*f)/(b*c - a*d)^2) Int[(c + d*x^2)^( q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2 I nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt Q[r, 1]
Leaf count of result is larger than twice the leaf count of optimal. \(1064\) vs. \(2(423)=846\).
Time = 6.86 (sec) , antiderivative size = 1065, normalized size of antiderivative = 2.41
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1065\) |
| elliptic | \(\text {Expression too large to display}\) | \(1254\) |
Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
(-(-b/a)^(1/2)*a^2*b*d^3*e*f^2*x^3+2*(-b/a)^(1/2)*a*b^2*c*d^2*e*f^2*x^3-(- b/a)^(1/2)*b^3*c^2*d*e*f^2*x^3+3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*E llipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^2*c^2*d*e*f^2-3*((b*x^2+a)/a) ^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^2 *c*d^2*e^2*f-2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^ (1/2),(a*d/b/c)^(1/2))*b^3*c^3*e*f^2+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/ 2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^3*c^2*d*e^2*f+((b*x^2+a)/a) ^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^3*c *d^2*e^3+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2), (a*d/b/c)^(1/2))*a^2*b*c*d^2*e*f^2-2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/ 2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^2*c^2*d*e*f^2+2*((b*x^2+a )/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b ^3*c^3*e*f^2-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1 /2),(a*d/b/c)^(1/2))*b^3*c^2*d*e^2*f+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/ 2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a^3*c*d^ 2*f^3-3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2), a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a^2*b*c*d^2*e*f^2+3*((b*x^2+a)/a)^(1/ 2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(- b/a)^(1/2))*a*b^2*c*d^2*e^2*f-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elli pticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b^3*c*d^2*e^...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(3/2)/(f*x**2+e),x)
Output:
Integral((a + b*x**2)**(5/2)/((c + d*x**2)**(3/2)*(e + f*x**2)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(3/2)*(f*x^2 + e)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(3/2)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*(e + f*x^2)),x)
Output:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*(e + f*x^2)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x)
Output:
(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*x - 6*int((sqrt(c + d*x**2)*sqrt( a + b*x**2)*x**6)/(2*a*c**3*e*f + 2*a*c**3*f**2*x**2 + a*c**2*d*e**2 + 5*a *c**2*d*e*f*x**2 + 4*a*c**2*d*f**2*x**4 + 2*a*c*d**2*e**2*x**2 + 4*a*c*d** 2*e*f*x**4 + 2*a*c*d**2*f**2*x**6 + a*d**3*e**2*x**4 + a*d**3*e*f*x**6 + 2 *b*c**3*e*f*x**2 + 2*b*c**3*f**2*x**4 + b*c**2*d*e**2*x**2 + 5*b*c**2*d*e* f*x**4 + 4*b*c**2*d*f**2*x**6 + 2*b*c*d**2*e**2*x**4 + 4*b*c*d**2*e*f*x**6 + 2*b*c*d**2*f**2*x**8 + b*d**3*e**2*x**6 + b*d**3*e*f*x**8),x)*a*b**2*c* *2*d*f**2 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a*c**3*e*f + 2*a*c**3*f**2*x**2 + a*c**2*d*e**2 + 5*a*c**2*d*e*f*x**2 + 4*a*c**2*d*f** 2*x**4 + 2*a*c*d**2*e**2*x**2 + 4*a*c*d**2*e*f*x**4 + 2*a*c*d**2*f**2*x**6 + a*d**3*e**2*x**4 + a*d**3*e*f*x**6 + 2*b*c**3*e*f*x**2 + 2*b*c**3*f**2* x**4 + b*c**2*d*e**2*x**2 + 5*b*c**2*d*e*f*x**4 + 4*b*c**2*d*f**2*x**6 + 2 *b*c*d**2*e**2*x**4 + 4*b*c*d**2*e*f*x**6 + 2*b*c*d**2*f**2*x**8 + b*d**3* e**2*x**6 + b*d**3*e*f*x**8),x)*a*b**2*c*d**2*e*f - 6*int((sqrt(c + d*x**2 )*sqrt(a + b*x**2)*x**6)/(2*a*c**3*e*f + 2*a*c**3*f**2*x**2 + a*c**2*d*e** 2 + 5*a*c**2*d*e*f*x**2 + 4*a*c**2*d*f**2*x**4 + 2*a*c*d**2*e**2*x**2 + 4* a*c*d**2*e*f*x**4 + 2*a*c*d**2*f**2*x**6 + a*d**3*e**2*x**4 + a*d**3*e*f*x **6 + 2*b*c**3*e*f*x**2 + 2*b*c**3*f**2*x**4 + b*c**2*d*e**2*x**2 + 5*b*c* *2*d*e*f*x**4 + 4*b*c**2*d*f**2*x**6 + 2*b*c*d**2*e**2*x**4 + 4*b*c*d**2*e *f*x**6 + 2*b*c*d**2*f**2*x**8 + b*d**3*e**2*x**6 + b*d**3*e*f*x**8),x)...