\(\int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} (a-b x^2)^{3/2}} \, dx\) [230]

Optimal result

Integrand size = 35, antiderivative size = 1012 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {b (A b c+a c C-a B d+(b B c-A b d-a C d) x)}{a^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {d \left (3 b^2 c^4 (B c-2 A d)-3 a b c^3 d (2 c C-B d)-2 a^2 d^3 \left (c^2 C-B c d+A d^2\right )\right ) \sqrt {a-b x^2}}{3 a^2 c^3 \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}-\frac {d \left (3 b^3 c^6 (B c-3 A d)-3 a b^2 c^4 d \left (3 c^2 C-3 B c d+A d^2\right )+6 a^3 d^5 \left (c^2 C-2 B c d+3 A d^2\right )-a^2 b c^2 d^3 \left (29 c^2 C-32 B c d+38 A d^2\right )\right ) \sqrt {a-b x^2}}{3 a^2 c^4 \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{2 a^2 c^3 x^2}-\frac {(4 B c-11 A d) \sqrt {c+d x} \sqrt {a-b x^2}}{4 a^2 c^4 x}+\frac {\sqrt {b} \left (3 b^3 c^6 (8 B c-23 A d)+3 a^3 d^5 \left (8 c^2 C-20 B c d+35 A d^2\right )-a^2 b c^2 d^3 \left (116 c^2 C-164 B c d+251 A d^2\right )-a b^2 \left (36 c^6 C d-87 A c^4 d^3\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{12 a^{3/2} c^4 \left (b c^2-a d^2\right )^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \left (3 b^2 c^4 (8 B c-17 A d)-6 a b c^2 d \left (4 c^2 C+2 B c d-9 A d^2\right )-a^2 d^3 \left (8 c^2 C-20 B c d+35 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{12 a^{3/2} c^3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\left (12 A b c^2+8 a c^2 C-20 a B c d+35 a A d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a^2 c^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

b*(A*b*c+C*a*c-B*a*d+(-A*b*d+B*b*c-C*a*d)*x)/a^2/(-a*d^2+b*c^2)/(d*x+c)^(3 
/2)/(-b*x^2+a)^(1/2)-1/3*d*(3*b^2*c^4*(-2*A*d+B*c)-3*a*b*c^3*d*(-B*d+2*C*c 
)-2*a^2*d^3*(A*d^2-B*c*d+C*c^2))*(-b*x^2+a)^(1/2)/a^2/c^3/(-a*d^2+b*c^2)^2 
/(d*x+c)^(3/2)-1/3*d*(3*b^3*c^6*(-3*A*d+B*c)-3*a*b^2*c^4*d*(A*d^2-3*B*c*d+ 
3*C*c^2)+6*a^3*d^5*(3*A*d^2-2*B*c*d+C*c^2)-a^2*b*c^2*d^3*(38*A*d^2-32*B*c* 
d+29*C*c^2))*(-b*x^2+a)^(1/2)/a^2/c^4/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)-1/2*A 
*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/c^3/x^2-1/4*(-11*A*d+4*B*c)*(d*x+c)^(1 
/2)*(-b*x^2+a)^(1/2)/a^2/c^4/x+1/12*b^(1/2)*(3*b^3*c^6*(-23*A*d+8*B*c)+3*a 
^3*d^5*(35*A*d^2-20*B*c*d+8*C*c^2)-a^2*b*c^2*d^3*(251*A*d^2-164*B*c*d+116* 
C*c^2)-a*b^2*(-87*A*c^4*d^3+36*C*c^6*d))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2 
)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^ 
(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2)/c^4/(-a*d^2+b*c^2)^3/((d*x+c)/(c+a^(1/2 
)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/12*b^(1/2)*(3*b^2*c^4*(-17*A*d+8*B* 
c)-6*a*b*c^2*d*(-9*A*d^2+2*B*c*d+4*C*c^2)-a^2*d^3*(35*A*d^2-20*B*c*d+8*C*c 
^2))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF( 
1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1 
/2)*d))^(1/2))/a^(3/2)/c^3/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2) 
-1/4*(35*A*a*d^2+12*A*b*c^2-20*B*a*c*d+8*C*a*c^2)*((d*x+c)/(c+a^(1/2)*d/b^ 
(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1 
/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^2/c^4/...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 36.56 (sec) , antiderivative size = 12354, normalized size of antiderivative = 12.21 \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x + C*x^2)/(x^3*(c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
 

Output:

Result too large to show
 

Rubi [F]

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.

\(\displaystyle \int \frac {A+B x+C x^2}{x^3 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 2355

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx+\int \frac {\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 2355

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\int \frac {C}{d^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 638

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 7239

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {1}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}dx-\frac {(2 c C-B d) \int \frac {1}{x^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

\(\Big \downarrow \) 637

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \left (-\frac {3 d^3}{c^4 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}-\frac {d^3}{c^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}}+\frac {3 d^2}{c^4 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {2 d}{c^3 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx-\frac {(2 c C-B d) \int \left (-\frac {d^3}{c^3 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}}+\frac {d^2}{c^3 x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {1}{c x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}\right )dx}{d^2}+\frac {C \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{d^2}\)

Input:

Int[(A + B*x + C*x^2)/(x^3*(c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

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 637

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 
/2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n 
 + 1/2] && IntegerQ[m]
 

rule 638

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Unintegrable[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x] /; FreeQ 
[{a, b, c, d, e, m, n, p}, x]
 

rule 2355

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(e*x)^m*(c + d* 
x)^(n + 1)*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x]   I 
nt[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p} 
, x] && PolynomialQ[Px, x] && LtQ[n, 0]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

Maple [A] (verified)

Time = 16.14 (sec) , antiderivative size = 1732, normalized size of antiderivative = 1.71

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1732\)
risch	\(\text {Expression too large to display}\)	\(2916\)
default	\(\text {Expression too large to display}\)	\(21237\)

Input:

int((C*x^2+B*x+A)/x^3/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBO 
SE)
 

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(2/3*d^2/(a*d^2- 
b*c^2)^2*(A*d^2-B*c*d+C*c^2)/c^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d 
)^2+2/3*(-b*d*x^2+a*d)*d^3/(a*d^2-b*c^2)^3*(9*A*a*d^4-19*A*b*c^2*d^2-6*B*a 
*c*d^3+16*B*b*c^3*d+3*C*a*c^2*d^2-13*C*b*c^4)/c^4/((x+c/d)*(-b*d*x^2+a*d)) 
^(1/2)-2*(-b*d*x-b*c)*(1/2/a^2*b*(A*a*b*d^3+3*A*b^2*c^2*d-3*B*a*b*c*d^2-B* 
b^2*c^3+C*a^2*d^3+3*C*a*b*c^2*d)/(a*d^2-b*c^2)^3*x-1/2*b*(3*A*a*b*c*d^2+A* 
b^2*c^3-B*a^2*d^3-3*B*a*b*c^2*d+3*C*a^2*c*d^2+C*a*b*c^3)/(a*d^2-b*c^2)^3/a 
^2)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-1/2*A/c^3/a^2/x^2*(-b*d*x^3-b*c*x^2+a*d 
*x+a*c)^(1/2)+1/4/a^2*(11*A*d-4*B*c)/c^4*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2 
)/x+2*(-1/3*b*d^3*(A*d^2-B*c*d+C*c^2)/(a*d^2-b*c^2)^2/c^3+1/3/c^3*d^3*b*(9 
*A*a*d^4-19*A*b*c^2*d^2-6*B*a*c*d^3+16*B*b*c^3*d+3*C*a*c^2*d^2-13*C*b*c^4) 
/(a*d^2-b*c^2)^3-b^2/a^2*(2*A*b*c*d-B*a*d^2-B*b*c^2+2*C*a*c*d)/(a*d^2-b*c^ 
2)^2+1/2*b^2*d*(3*A*a*b*c*d^2+A*b^2*c^3-B*a^2*d^3-3*B*a*b*c^2*d+3*C*a^2*c* 
d^2+C*a*b*c^3)/(a*d^2-b*c^2)^3/a^2-b^2*c/a^2*(A*a*b*d^3+3*A*b^2*c^2*d-3*B* 
a*b*c*d^2-B*b^2*c^3+C*a^2*d^3+3*C*a*b*c^2*d)/(a*d^2-b*c^2)^3+1/4*A*b*d/a^2 
/c^3)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b* 
(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b* 
(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/ 
(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2) 
))^(1/2))+2*(1/3*b*d^4*(9*A*a*d^4-19*A*b*c^2*d^2-6*B*a*c*d^3+16*B*b*c^3...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="f 
ricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((C*x**2+B*x+A)/x**3/(d*x+c)**(5/2)/(-b*x**2+a)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="m 
axima")
 

Output:

integrate((C*x^2 + B*x + A)/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)*x^3), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="g 
iac")
 

Output:

integrate((C*x^2 + B*x + A)/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {C\,x^2+B\,x+A}{x^3\,{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int((A + B*x + C*x^2)/(x^3*(a - b*x^2)^(3/2)*(c + d*x)^(5/2)),x)
 

Output:

int((A + B*x + C*x^2)/(x^3*(a - b*x^2)^(3/2)*(c + d*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{x^3 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {C \,x^{2}+B x +A}{x^{3} \left (d x +c \right )^{\frac {5}{2}} \left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:

int((C*x^2+B*x+A)/x^3/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)
 

Output:

int((C*x^2+B*x+A)/x^3/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)