Integrand size = 32, antiderivative size = 655 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d)^2 x \sqrt {a+b x^2}}{5 c d (d e-c f) \left (c+d x^2\right )^{5/2}}-\frac {(b c-a d) (a d (4 d e-9 c f)+b c (7 d e-2 c f)) x \sqrt {a+b x^2}}{15 c^2 d (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (a b c d \left (7 d^2 e^2-29 c d e f-8 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2+9 c d e f-2 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-26 c d e f+33 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 c^{5/2} d^{3/2} (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (15 b^3 c^3 d e^2 f-15 a^3 c^2 d^2 f^3-a^2 b d \left (4 d^3 e^3-17 c d^2 e^2 f-8 c^2 d e f^2-24 c^3 f^3\right )-a b^2 c \left (4 d^3 e^3+8 c d^2 e^2 f+32 c^2 d e f^2+c^3 f^3\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a c^{3/2} d^{3/2} (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} f (b e-a f)^3 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/5*(-a*d+b*c)^2*x*(b*x^2+a)^(1/2)/c/d/(-c*f+d*e)/(d*x^2+c)^(5/2)-1/15*(-a *d+b*c)*(a*d*(-9*c*f+4*d*e)+b*c*(-2*c*f+7*d*e))*x*(b*x^2+a)^(1/2)/c^2/d/(- c*f+d*e)^2/(d*x^2+c)^(3/2)+1/15*(a*b*c*d*(-8*c^2*f^2-29*c*d*e*f+7*d^2*e^2) +b^2*c^2*(-2*c^2*f^2+9*c*d*e*f+8*d^2*e^2)+a^2*d^2*(33*c^2*f^2-26*c*d*e*f+8 *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))/c^(5/2)/d^(3/2)/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^( 1/2)/(d*x^2+c)^(1/2)+1/15*(15*b^3*c^3*d*e^2*f-15*a^3*c^2*d^2*f^3-a^2*b*d*( -24*c^3*f^3-8*c^2*d*e*f^2-17*c*d^2*e^2*f+4*d^3*e^3)-a*b^2*c*(c^3*f^3+32*c^ 2*d*e*f^2+8*c*d^2*e^2*f+4*d^3*e^3))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan (d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/c^(3/2)/d^(3/2)/(-c*f+d*e)^4/(c*( b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-c^(3/2)*f*(-a*f+b*e)^3*(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/(-c*f+d*e)^4/(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 = 8.87 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} d e x \left (a+b x^2\right ) \left (3 c^2 (b c-a d)^2 (d e-c f)^2-c (b c-a d) (-d e+c f) (b c (-7 d e+2 c f)+a d (-4 d e+9 c f)) \left (c+d x^2\right )+\left (a b c d \left (7 d^2 e^2-29 c d e f-8 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2+9 c d e f-2 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-26 c d e f+33 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )+i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \left (b e \left (a b c d \left (7 d^2 e^2-29 c d e f-8 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2+9 c d e f-2 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-26 c d e f+33 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+b (b c-a d) e (-d e+c f) (b c (-7 d e+2 c f)+a d (-4 d e+9 c f)) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-15 c^2 d^2 (b e-a f)^3 \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 \sqrt {\frac {b}{a}} c^3 d^2 e (d e-c f)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \] Input:
Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)),x]
Output:
(Sqrt[b/a]*d*e*x*(a + b*x^2)*(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2 - c*(b*c - a*d)*(-(d*e) + c*f)*(b*c*(-7*d*e + 2*c*f) + a*d*(-4*d*e + 9*c*f))*(c + d* x^2) + (a*b*c*d*(7*d^2*e^2 - 29*c*d*e*f - 8*c^2*f^2) + b^2*c^2*(8*d^2*e^2 + 9*c*d*e*f - 2*c^2*f^2) + a^2*d^2*(8*d^2*e^2 - 26*c*d*e*f + 33*c^2*f^2))* (c + d*x^2)^2) + I*c*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c] *(b*e*(a*b*c*d*(7*d^2*e^2 - 29*c*d*e*f - 8*c^2*f^2) + b^2*c^2*(8*d^2*e^2 + 9*c*d*e*f - 2*c^2*f^2) + a^2*d^2*(8*d^2*e^2 - 26*c*d*e*f + 33*c^2*f^2))*E llipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + b*(b*c - a*d)*e*(-(d*e) + c*f)*(b*c*(-7*d*e + 2*c*f) + a*d*(-4*d*e + 9*c*f))*EllipticF[I*ArcSinh[Sqr t[b/a]*x], (a*d)/(b*c)] - 15*c^2*d^2*(b*e - a*f)^3*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*Sqrt[b/a]*c^3*d^2*e*(d*e - c*f )^3*Sqrt[a + b*x^2]*(c + d*x^2)^(5/2))
Time = 1.00 (sec) , antiderivative size = 630, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {419, 25, 401, 25, 27, 401, 25, 400, 313, 320, 417, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 419 |
| \(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{7/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{7/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {-\frac {\int -\frac {d \sqrt {b x^2+a} \left (b (a d (d e-6 c f)+b c (4 d e+c f)) x^2+a (a d (4 d e-9 c f)+b c (d e+4 c f))\right )}{\left (d x^2+c\right )^{5/2}}dx}{5 c d}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {d \sqrt {b x^2+a} \left (b (a d (d e-6 c f)+b c (4 d e+c f)) x^2+a (a d (4 d e-9 c f)+b c (d e+4 c f))\right )}{\left (d x^2+c\right )^{5/2}}dx}{5 c d}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {b x^2+a} \left (b (a d (d e-6 c f)+b c (4 d e+c f)) x^2+a (a d (4 d e-9 c f)+b c (d e+4 c f))\right )}{\left (d x^2+c\right )^{5/2}}dx}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {b \left (2 b^2 (4 d e+c f) c^2+a b d (3 d e-8 c f) c+a^2 d^2 (4 d e-9 c f)\right ) x^2+a \left (b^2 (4 d e+c f) c^2+a b d (3 d e+2 c f) c+2 a^2 d^2 (4 d e-9 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (4 d e-9 c f)+b c (c f+4 d e))}{3 c d \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b \left (2 b^2 (4 d e+c f) c^2+a b d (3 d e-8 c f) c+a^2 d^2 (4 d e-9 c f)\right ) x^2+a \left (b^2 (4 d e+c f) c^2+a b d (3 d e+2 c f) c+2 a^2 d^2 (4 d e-9 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (4 d e-9 c f)+b c (c f+4 d e))}{3 c d \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^2 d^2 (4 d e-9 c f)+7 a b c d (d e-c f)+2 b^2 c^2 (c f+4 d e)\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx-a b (a d (4 d e-9 c f)+b c (c f+4 d e)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (4 d e-9 c f)+b c (c f+4 d e))}{3 c d \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)+7 a b c d (d e-c f)+2 b^2 c^2 (c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-a b (a d (4 d e-9 c f)+b c (c f+4 d e)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (4 d e-9 c f)+b c (c f+4 d e))}{3 c d \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)+7 a b c d (d e-c f)+2 b^2 c^2 (c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (c f+4 d e)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (4 d e-9 c f)+b c (c f+4 d e))}{3 c d \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 417 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)+7 a b c d (d e-c f)+2 b^2 c^2 (c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (c f+4 d e)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (4 d e-9 c f)+b c (c f+4 d e))}{3 c d \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{d e-c f}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{d e-c f}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)+7 a b c d (d e-c f)+2 b^2 c^2 (c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (c f+4 d e)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (4 d e-9 c f)+b c (c f+4 d e))}{3 c d \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{d e-c f}-\frac {\sqrt {a+b x^2} (b c-a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)+7 a b c d (d e-c f)+2 b^2 c^2 (c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (c f+4 d e)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (4 d e-9 c f)+b c (c f+4 d e))}{3 c d \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {a^{3/2} \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a+b x^2} (b c-a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\) |
Input:
Int[(a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)),x]
Output:
(-1/5*((b*c - a*d)*(d*e - c*f)*x*(a + b*x^2)^(3/2))/(c*(c + d*x^2)^(5/2)) + (-1/3*((b*c - a*d)*(a*d*(4*d*e - 9*c*f) + b*c*(4*d*e + c*f))*x*Sqrt[a + b*x^2])/(c*d*(c + d*x^2)^(3/2)) + (((2*a^2*d^2*(4*d*e - 9*c*f) + 7*a*b*c*d *(d*e - c*f) + 2*b^2*c^2*(4*d*e + c*f))*Sqrt[a + b*x^2]*EllipticE[ArcTan[( Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*Sqrt[(c*(a + b*x^2 ))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*(a*d*(4*d*e - 9*c*f) + b *c*(4*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*c*d))/(5*c))/(d*e - c*f)^2 - (f*(b*e - a*f)*(-(((b*c - a*d)*Sqr t[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqr t[c]*Sqrt[d]*(d*e - c*f)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d* x^2])) + (a^(3/2)*(b*e - a*f)*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)*Sq rt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*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[((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[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[((e_) + (f_.)*(x_)^2)^(3/2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2 )^(3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[Sqrt[e + f*x^2]/( (a + b*x^2)*Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[S qrt[e + f*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[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. \(2416\) vs. \(2(625)=1250\).
Time = 12.03 (sec) , antiderivative size = 2417, normalized size of antiderivative = 3.69
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2417\) |
| default | \(\text {Expression too large to display}\) | \(5577\) |
Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(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)*(-1/5*(a^2*d^2 -2*a*b*c*d+b^2*c^2)/c/d^4/(c*f-d*e)*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/ (x^2+c/d)^3-1/15*(9*a^2*c*d^2*f-4*a^2*d^3*e-7*a*b*c^2*d*f-3*a*b*c*d^2*e-2* b^2*c^3*f+7*b^2*c^2*d*e)/d^3/(c*f-d*e)^2/c^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a* c)^(1/2)/(x^2+c/d)^2-1/15*(b*d*x^2+a*d)*(33*a^2*c^2*d^2*f^2-26*a^2*c*d^3*e *f+8*a^2*d^4*e^2-8*a*b*c^3*d*f^2-29*a*b*c^2*d^2*e*f+7*a*b*c*d^3*e^2-2*b^2* c^4*f^2+9*b^2*c^3*d*e*f+8*b^2*c^2*d^2*e^2)/d^2/(c*f-d*e)^3/c^3*x/((x^2+c/d )*(b*d*x^2+a*d))^(1/2)-7/15/(-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))*b^3/d/(c*f-d*e)^2*e+1/(c*f-d*e)^3/e*f^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)*Ellipti cPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a^3-3/(c*f-d*e)^3* f^2/(-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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/ 2))*a^2*b+4/15/(-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 ))*b/c^2*d/(c*f-d*e)^2*a^2*e-26/15*d*b/(c*f-d*e)^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)*EllipticE( x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*a^2*e*f-29/15*b^2/(c*f-d*e)^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+...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(7/2)/(f*x**2+e),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(7/2)*(f*x^2 + e)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(7/2)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{7/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)),x)
Output:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(7/2)*(e + f*x^2)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(7/2)/(f*x^2+e),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*x - 20*int((sqrt(c + d*x**2 )*sqrt(a + b*x**2)*x**6)/(4*a**2*c**4*d*e*f + 4*a**2*c**4*d*f**2*x**2 + 16 *a**2*c**3*d**2*e*f*x**2 + 16*a**2*c**3*d**2*f**2*x**4 + 24*a**2*c**2*d**3 *e*f*x**4 + 24*a**2*c**2*d**3*f**2*x**6 + 16*a**2*c*d**4*e*f*x**6 + 16*a** 2*c*d**4*f**2*x**8 + 4*a**2*d**5*e*f*x**8 + 4*a**2*d**5*f**2*x**10 - 2*a*b *c**5*e*f - 2*a*b*c**5*f**2*x**2 + 3*a*b*c**4*d*e**2 - a*b*c**4*d*e*f*x**2 - 4*a*b*c**4*d*f**2*x**4 + 12*a*b*c**3*d**2*e**2*x**2 + 16*a*b*c**3*d**2* e*f*x**4 + 4*a*b*c**3*d**2*f**2*x**6 + 18*a*b*c**2*d**3*e**2*x**4 + 34*a*b *c**2*d**3*e*f*x**6 + 16*a*b*c**2*d**3*f**2*x**8 + 12*a*b*c*d**4*e**2*x**6 + 26*a*b*c*d**4*e*f*x**8 + 14*a*b*c*d**4*f**2*x**10 + 3*a*b*d**5*e**2*x** 8 + 7*a*b*d**5*e*f*x**10 + 4*a*b*d**5*f**2*x**12 - 2*b**2*c**5*e*f*x**2 - 2*b**2*c**5*f**2*x**4 + 3*b**2*c**4*d*e**2*x**2 - 5*b**2*c**4*d*e*f*x**4 - 8*b**2*c**4*d*f**2*x**6 + 12*b**2*c**3*d**2*e**2*x**4 - 12*b**2*c**3*d**2 *f**2*x**8 + 18*b**2*c**2*d**3*e**2*x**6 + 10*b**2*c**2*d**3*e*f*x**8 - 8* b**2*c**2*d**3*f**2*x**10 + 12*b**2*c*d**4*e**2*x**8 + 10*b**2*c*d**4*e*f* x**10 - 2*b**2*c*d**4*f**2*x**12 + 3*b**2*d**5*e**2*x**10 + 3*b**2*d**5*e* f*x**12),x)*a**2*b**3*c**3*d**2*f**2 - 60*int((sqrt(c + d*x**2)*sqrt(a + b *x**2)*x**6)/(4*a**2*c**4*d*e*f + 4*a**2*c**4*d*f**2*x**2 + 16*a**2*c**3*d **2*e*f*x**2 + 16*a**2*c**3*d**2*f**2*x**4 + 24*a**2*c**2*d**3*e*f*x**4 + 24*a**2*c**2*d**3*f**2*x**6 + 16*a**2*c*d**4*e*f*x**6 + 16*a**2*c*d**4*...