\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^{7/2} (e+f x^2)} \, dx\) [103]

Optimal result

Integrand size = 32, antiderivative size = 659 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=-\frac {(b c-a d) x \sqrt {a+b x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-9 c f)+b c (2 d e+3 c f)) x \sqrt {a+b x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2 \left (2 d^2 e^2-14 c d e f-3 c^2 f^2\right )+3 a b c d \left (d^2 e^2-2 c d e f+11 c^2 f^2\right )-a^2 d^2 \left (8 d^2 e^2-26 c d e f+33 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 c^{5/2} \sqrt {d} (b c-a d) (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (15 b^3 c^4 e f^2-15 a^3 c^2 d^2 f^3-a^2 b d \left (4 d^3 e^3-17 c d^2 e^2 f+7 c^2 d e f^2-39 c^3 f^3\right )+a b^2 c \left (d^3 e^3-8 c d^2 e^2 f-17 c^2 d e f^2-21 c^3 f^3\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a c^{3/2} \sqrt {d} (b c-a d) (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} f^2 (b e-a f)^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-1/5*(-a*d+b*c)*x*(b*x^2+a)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^(5/2)+1/15*(a*d*( 
-9*c*f+4*d*e)+b*c*(3*c*f+2*d*e))*x*(b*x^2+a)^(1/2)/c^2/(-c*f+d*e)^2/(d*x^2 
+c)^(3/2)+1/15*(b^2*c^2*(-3*c^2*f^2-14*c*d*e*f+2*d^2*e^2)+3*a*b*c*d*(11*c^ 
2*f^2-2*c*d*e*f+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^(1/2)/(-a*d+b*c)/(-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^4*e*f^2-15*a^3*c^2*d^2*f^3-a^2*b*d*(-39*c^3* 
f^3+7*c^2*d*e*f^2-17*c*d^2*e^2*f+4*d^3*e^3)+a*b^2*c*(-21*c^3*f^3-17*c^2*d* 
e*f^2-8*c*d^2*e^2*f+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^(1/2)/(-a*d+b*c)/(-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^2*(-a*f+b*e)^2* 
(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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.75 (sec) , antiderivative size = 547, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {\frac {b}{a}} d e x \left (a+b x^2\right ) \left (3 c^2 (b c-a d)^2 (d e-c f)^2+c (b c-a d) (-d e+c f) (a d (4 d e-9 c f)+b c (2 d e+3 c f)) \left (c+d x^2\right )+\left (b^2 c^2 \left (-2 d^2 e^2+14 c d e f+3 c^2 f^2\right )-3 a b c d \left (d^2 e^2-2 c d e f+11 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-26 c d e f+33 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )-i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \left (b e \left (b^2 c^2 \left (-2 d^2 e^2+14 c d e f+3 c^2 f^2\right )-3 a b c d \left (d^2 e^2-2 c d e f+11 c^2 f^2\right )+a^2 d^2 \left (8 d^2 e^2-26 c d e f+33 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (b e (-d e+c f) (a d (4 d e-9 c f)+b c (2 d e+3 c f)) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+15 c^2 d f (b e-a f)^2 \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{15 \sqrt {\frac {b}{a}} c^3 d (b c-a d) e (d e-c f)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \] Input:

Integrate[(a + b*x^2)^(3/2)/((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)*(a*d*(4*d*e - 9*c*f) + b*c*(2*d*e + 3*c*f))*(c + d* 
x^2) + (b^2*c^2*(-2*d^2*e^2 + 14*c*d*e*f + 3*c^2*f^2) - 3*a*b*c*d*(d^2*e^2 
 - 2*c*d*e*f + 11*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*(b^2*c^2*(-2*d^2*e^2 + 14*c*d*e*f + 3*c^2*f^2) - 3*a*b*c*d*(d^2*e 
^2 - 2*c*d*e*f + 11*c^2*f^2) + a^2*d^2*(8*d^2*e^2 - 26*c*d*e*f + 33*c^2*f^ 
2))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (b*c - a*d)*(b*e*(-(d 
*e) + c*f)*(a*d*(4*d*e - 9*c*f) + b*c*(2*d*e + 3*c*f))*EllipticF[I*ArcSinh 
[Sqrt[b/a]*x], (a*d)/(b*c)] + 15*c^2*d*f*(b*e - a*f)^2*EllipticPi[(a*f)/(b 
*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(15*Sqrt[b/a]*c^3*d*(b*c - a* 
d)*e*(d*e - c*f)^3*Sqrt[a + b*x^2]*(c + d*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 624, normalized size of antiderivative = 0.95, 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, 402, 27, 400, 313, 320, 416, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 419

\(\displaystyle -\frac {\int -\frac {\sqrt {b x^2+a} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{7/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{7/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {-\frac {\int -\frac {d \left (b (a d (3 d e-8 c f)+b c (2 d e+3 c f)) x^2+a (a d (4 d e-9 c f)+b c (d e+4 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {d \left (b (a d (3 d e-8 c f)+b c (2 d e+3 c f)) x^2+a (a d (4 d e-9 c f)+b c (d e+4 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {b (a d (3 d e-8 c f)+b c (2 d e+3 c f)) x^2+a (a d (4 d e-9 c f)+b c (d e+4 c f))}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {\int \frac {(b c-a d) \left (b (a d (4 d e-9 c f)+b c (2 d e+3 c f)) x^2+a (2 a d (4 d e-9 c f)+b c (d e+9 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (3 c f+2 d e))}{3 c \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {b (a d (4 d e-9 c f)+b c (2 d e+3 c f)) x^2+a (2 a d (4 d e-9 c f)+b c (d e+9 c f))}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (3 c f+2 d e))}{3 c \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 400

\(\displaystyle \frac {\frac {\frac {\frac {a b (a d (4 d e-9 c f)-b c (d e-6 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\left (2 a^2 d^2 (4 d e-9 c f)-3 a b c d (d e-6 c f)-b^2 c^2 (3 c f+2 d e)\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}}{3 c}+\frac {x \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (3 c f+2 d e))}{3 c \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {\frac {\frac {a b (a d (4 d e-9 c f)-b c (d e-6 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)-3 a b c d (d e-6 c f)-b^2 c^2 (3 c f+2 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (3 c f+2 d e))}{3 c \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (4 d e-9 c f)-b c (d e-6 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)-3 a b c d (d e-6 c f)-b^2 c^2 (3 c f+2 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (3 c f+2 d e))}{3 c \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\)

\(\Big \downarrow \) 416

\(\displaystyle \frac {\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (4 d e-9 c f)-b c (d e-6 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)-3 a b c d (d e-6 c f)-b^2 c^2 (3 c f+2 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (3 c f+2 d e))}{3 c \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {d \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (4 d e-9 c f)-b c (d e-6 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)-3 a b c d (d e-6 c f)-b^2 c^2 (3 c f+2 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (3 c f+2 d e))}{3 c \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {f \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 414

\(\displaystyle \frac {\frac {\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (4 d e-9 c f)-b c (d e-6 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (4 d e-9 c f)-3 a b c d (d e-6 c f)-b^2 c^2 (3 c f+2 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (a d (4 d e-9 c f)+b c (3 c f+2 d e))}{3 c \left (c+d x^2\right )^{3/2}}}{5 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{5 c \left (c+d x^2\right )^{5/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a^{3/2} f \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{(d e-c f)^2}\)

Input:

Int[(a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)),x]
 

Output:

(-1/5*((b*c - a*d)*(d*e - c*f)*x*Sqrt[a + b*x^2])/(c*(c + d*x^2)^(5/2)) + 
(((a*d*(4*d*e - 9*c*f) + b*c*(2*d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(3*c*(c + 
 d*x^2)^(3/2)) + (-(((2*a^2*d^2*(4*d*e - 9*c*f) - 3*a*b*c*d*(d*e - 6*c*f) 
- b^2*c^2*(2*d*e + 3*c*f))*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sq 
rt[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]*(a*d*(4*d*e - 9*c*f) - b 
*c*(d*e - 6*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))/(5*c))/(d*e - c*f)^2 - (f*(b*e - a*f)*((Sqrt[d] 
*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/ 
(Sqrt[c]*(d*e - c*f)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2] 
) - (a^(3/2)*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)*Sqrt[a + b*x^2]*S 
qrt[(a*(c + d*x^2))/(c*(a + b*x^2))])))/(d*e - c*f)^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 313

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]
 

rule 320

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]
 

rule 400

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]
 

rule 401

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]
 

rule 402

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]
 

rule 414

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]
 

rule 416

Int[Sqrt[(e_) + (f_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)^ 
(3/2)), x_Symbol] :> Simp[b/(b*c - a*d)   Int[Sqrt[e + f*x^2]/((a + b*x^2)* 
Sqrt[c + d*x^2]), x], x] - Simp[d/(b*c - a*d)   Int[Sqrt[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 
]
 

rule 419

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4504\) vs. \(2(629)=1258\).

Time = 19.06 (sec) , antiderivative size = 4505, normalized size of antiderivative = 6.84

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(4505\)
default	\(\text {Expression too large to display}\)	\(6065\)

Input:

int((b*x^2+a)^(3/2)/(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)*(-2/5/(-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/c/(c*f-d*e)^3*a*b* 
e*f-26/15/(-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))*a^ 
3/c^2/(a*d-b*c)/(c*f-d*e)^3*e*f*d^3-1/5*(a*d-b*c)/d^3/(c*f-d*e)/c*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*c*d*f-4*a*d^2*e-3*b*c^ 
2*f-2*b*c*d*e)/(c*f-d*e)^2/c^2/d^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/( 
x^2+c/d)^2-11/5*b^2/(a*d-b*c)*c/(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+a*c)^(1/2)*EllipticE(x*(-b/a)^ 
(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*a*f^2-1/5*b^2/(a*d-b*c)/c/(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+a*c) 
^(1/2)*d^2*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*a*e^2-14/15* 
b^3/(a*d-b*c)*c/(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+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b 
*c)/c/b)^(1/2))*e*f+11/5*b/(a*d-b*c)/(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+a*c)^(1/2)*d*EllipticE(x* 
(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*a^2*f^2+8/15*b/(a*d-b*c)/c^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+a*c)^(1/2)*d^3*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**(7/2)/(f*x**2+e),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(7/2)*(f*x^2 + e)), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(7/2)*(f*x^2 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{7/2}\,\left (f\,x^2+e\right )} \,d x \] Input:

int((a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)),x)
 

Output:

int((a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \left (e+f x^2\right )} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{d^{4} f \,x^{10}+4 c \,d^{3} f \,x^{8}+d^{4} e \,x^{8}+6 c^{2} d^{2} f \,x^{6}+4 c \,d^{3} e \,x^{6}+4 c^{3} d f \,x^{4}+6 c^{2} d^{2} e \,x^{4}+c^{4} f \,x^{2}+4 c^{3} d e \,x^{2}+c^{4} e}d x \right ) b +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d^{4} f \,x^{10}+4 c \,d^{3} f \,x^{8}+d^{4} e \,x^{8}+6 c^{2} d^{2} f \,x^{6}+4 c \,d^{3} e \,x^{6}+4 c^{3} d f \,x^{4}+6 c^{2} d^{2} e \,x^{4}+c^{4} f \,x^{2}+4 c^{3} d e \,x^{2}+c^{4} e}d x \right ) a \] Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e),x)
 

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(c**4*e + c**4*f*x**2 + 4*c** 
3*d*e*x**2 + 4*c**3*d*f*x**4 + 6*c**2*d**2*e*x**4 + 6*c**2*d**2*f*x**6 + 4 
*c*d**3*e*x**6 + 4*c*d**3*f*x**8 + d**4*e*x**8 + d**4*f*x**10),x)*b + int( 
(sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c**4*e + c**4*f*x**2 + 4*c**3*d*e*x** 
2 + 4*c**3*d*f*x**4 + 6*c**2*d**2*e*x**4 + 6*c**2*d**2*f*x**6 + 4*c*d**3*e 
*x**6 + 4*c*d**3*f*x**8 + d**4*e*x**8 + d**4*f*x**10),x)*a