\(\int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} (a-b x^2)^{3/2}} \, dx\) [221]

Optimal result

Integrand size = 35, antiderivative size = 719 \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a (b B c-A b d-a C d)+b (A b c+a c C-a B d) x}{a^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {d \left (A \left (b^2 c^4+a b c^2 d^2+2 a^2 d^4\right )+a c \left (b c^2 (c C-2 B d)+a d^2 (3 c C-2 B d)\right )\right ) \sqrt {a-b x^2}}{a^2 c^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{a^2 c^2 x}+\frac {\sqrt {b} \left (A \left (2 b^2 c^4-a b c^2 d^2+3 a^2 d^4\right )+a c \left (b c^2 (c C-2 B d)+a d^2 (3 c C-2 B d)\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} c^2 \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \left (a c (c C-B d)+A \left (2 b c^2-a d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^{3/2} c \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {(2 B c-3 A d) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(a*(-A*b*d+B*b*c-C*a*d)+b*(A*b*c-B*a*d+C*a*c)*x)/a^2/(-a*d^2+b*c^2)/(d*x+c 
)^(1/2)/(-b*x^2+a)^(1/2)-d*(A*(2*a^2*d^4+a*b*c^2*d^2+b^2*c^4)+a*c*(b*c^2*( 
-2*B*d+C*c)+a*d^2*(-2*B*d+3*C*c)))*(-b*x^2+a)^(1/2)/a^2/c^2/(-a*d^2+b*c^2) 
^2/(d*x+c)^(1/2)-A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/c^2/x+b^(1/2)*(A*(3* 
a^2*d^4-a*b*c^2*d^2+2*b^2*c^4)+a*c*(b*c^2*(-2*B*d+C*c)+a*d^2*(-2*B*d+3*C*c 
)))*(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^ 
2/(-a*d^2+b*c^2)^2/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)- 
b^(1/2)*(a*c*(-B*d+C*c)+A*(-a*d^2+2*b*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/(-a*d^2+b* 
c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-(-3*A*d+2*B*c)*((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/c^2/(d*x 
+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.77 (sec) , antiderivative size = 2824, normalized size of antiderivative = 3.93 \[ \int \frac {A+B x+C x^2}{x^2 (c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Sqrt[c + d*x]*Sqrt[a - b*x^2]*(-(A/(a^2*c^2*x)) - (2*d^3*(c^2*C - B*c*d + 
A*d^2))/(c^2*(b*c^2 - a*d^2)^2*(c + d*x)) + (-(a*b^2*B*c^2) + 2*a*A*b^2*c* 
d + 2*a^2*b*c*C*d - a^2*b*B*d^2 - A*b^3*c^2*x - a*b^2*c^2*C*x + 2*a*b^2*B* 
c*d*x - a*A*b^2*d^2*x - a^2*b*C*d^2*x)/(a^2*(-(b*c^2) + a*d^2)^2*(-a + b*x 
^2))) - (d*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(-2*A*b^3*c^ 
5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - a*b^2*c^5*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[ 
b]] + 2*a*b^2*B*c^4*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + a*A*b^2*c^3*d^2*Sqr 
t[-c + (Sqrt[a]*d)/Sqrt[b]] - 3*a^2*b*c^3*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt 
[b]] + 2*a^2*b*B*c^2*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 3*a^2*A*b*c*d^4* 
Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - (2*A*b^3*c^7*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b 
]])/(c + d*x)^2 - (a*b^2*c^7*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 
 + (2*a*b^2*B*c^6*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (3*a*A*b 
^2*c^5*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (2*a^2*b*c^5*C*d^ 
2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (4*a^2*A*b*c^3*d^4*Sqrt[-c 
 + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (3*a^3*c^3*C*d^4*Sqrt[-c + (Sqrt[a] 
*d)/Sqrt[b]])/(c + d*x)^2 - (2*a^3*B*c^2*d^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b] 
])/(c + d*x)^2 + (3*a^3*A*c*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^ 
2 + (4*A*b^3*c^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) + (2*a*b^2*c^6* 
C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (4*a*b^2*B*c^5*d*Sqrt[-c + ( 
Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (2*a*A*b^2*c^4*d^2*Sqrt[-c + (Sqrt[a]*...
 

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.

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

\(\Big \downarrow \) 2355

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 2355

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 638

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 637

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

Input:

Int[(A + B*x + C*x^2)/(x^2*(c + d*x)^(3/2)*(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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1435\) vs. \(2(636)=1272\).

Time = 11.00 (sec) , antiderivative size = 1436, normalized size of antiderivative = 2.00

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1436\)
risch	\(\text {Expression too large to display}\)	\(1743\)
default	\(\text {Expression too large to display}\)	\(5829\)

Input:

int((C*x^2+B*x+A)/x^2/(d*x+c)^(3/2)/(-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)*(-A/c^2/a^2/x*(- 
b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*b*d*(1/2*(2*A*a^2*d^4+A*a*b*c^2*d^2+A*b 
^2*c^4-2*B*a^2*c*d^3-2*B*a*b*c^3*d+3*C*a^2*c^2*d^2+C*a*b*c^4)/(a*d^2-b*c^2 
)^2/c^2/a^2*x^2-1/2*(A*b*c-B*a*d+C*a*c)/d/(a*d^2-b*c^2)/a^2*x-1/2*(2*A*a^2 
*d^5+2*A*b^2*c^4*d-2*B*a^2*c*d^4-B*a*b*c^3*d^2-B*b^2*c^5+2*C*a^2*c^2*d^3+2 
*C*a*b*c^4*d)/a/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/d/b/c^2)/(-(x^3+c/d*x^2-a* 
x/b-a*c/b/d)*b*d)^(1/2)+2*(-1/2*b*(2*A*a^2*d^4+4*A*a*b*c^2*d^2-2*A*b^2*c^4 
-5*B*a^2*c*d^3+B*a*b*c^3*d+6*C*a^2*c^2*d^2-2*C*a*b*c^4)/a^2/(a^2*d^4-2*a*b 
*c^2*d^2+b^2*c^4)/c+b*(A*b*c-B*a*d+C*a*c)/(a*d^2-b*c^2)/a^2)*(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*(-1/2*A 
*b*d/a^2/c^2+3/2*b*d*(2*A*a^2*d^4+A*a*b*c^2*d^2+A*b^2*c^4-2*B*a^2*c*d^3-2* 
B*a*b*c^3*d+3*C*a^2*c^2*d^2+C*a*b*c^4)/a^2/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4) 
/c^2-2*b*d*(2*A*a^2*d^4+A*a*b*c^2*d^2+A*b^2*c^4-2*B*a^2*c*d^3-2*B*a*b*c^3* 
d+3*C*a^2*c^2*d^2+C*a*b*c^4)/(a*d^2-b*c^2)^2/c^2/a^2)*(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)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)...
 

Fricas [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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