\(\int \frac {x^2 (A+B x+C x^2)}{(c+d x)^{3/2} (a-b x^2)^{3/2}} \, dx\) [217]

Optimal result

Integrand size = 35, antiderivative size = 556 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a (b B c-A b d-a C d)+b (A b c+a c C-a B d) x}{b^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\left (a^2 C d^4+a b d^2 \left (c^2 C-2 B c d+A d^2\right )+b^2 c^2 \left (2 c^2 C-2 B c d+3 A d^2\right )\right ) \sqrt {a-b x^2}}{b^2 d \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {\sqrt {a} \left (3 a^2 C d^4-a b d^2 \left (3 c^2 C+2 B c d-A d^2\right )+b^2 c^2 \left (4 c^2 C-2 B c d+3 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{b^{3/2} d^2 \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \left (a d^2 (3 c C-B d)-b c \left (4 c^2 C-2 B c d+A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{b^{3/2} d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.77 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.32 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (3 a^2 C d^4+a b d^2 \left (-3 c^2 C-2 B c d+A d^2\right )-2 b^2 c^2 \left (c^2 C-B c d+A d^2\right )+b^2 c^2 \left (4 c^2 C-2 B c d+3 A d^2\right )-\frac {b d (c+d x) \left (A b^2 c^2 x+a^2 d (-2 c C+d (B+C x))+a b \left (c^2 C x+B c (c-2 d x)+A d (-2 c+d x)\right )\right )}{-a+b x^2}-\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 C d^4+a b d^2 \left (-3 c^2 C-2 B c d+A d^2\right )+b^2 c^2 \left (4 c^2 C-2 B c d+3 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (2 A b^2 c^2 d+3 a^2 C d^3+a b d^2 (-3 B c+A d)+a^{3/2} \sqrt {b} d^2 (3 c C-B d)-\sqrt {a} b^{3/2} c \left (4 c^2 C-2 B c d+A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{b^2 d \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(3*a^2*C*d^4 + a*b*d^2*(-3*c^2*C - 2*B*c*d + A*d^2) - 2*b 
^2*c^2*(c^2*C - B*c*d + A*d^2) + b^2*c^2*(4*c^2*C - 2*B*c*d + 3*A*d^2) - ( 
b*d*(c + d*x)*(A*b^2*c^2*x + a^2*d*(-2*c*C + d*(B + C*x)) + a*b*(c^2*C*x + 
 B*c*(c - 2*d*x) + A*d*(-2*c + d*x))))/(-a + b*x^2) - (I*Sqrt[b]*(Sqrt[b]* 
c - Sqrt[a]*d)*(3*a^2*C*d^4 + a*b*d^2*(-3*c^2*C - 2*B*c*d + A*d^2) + b^2*c 
^2*(4*c^2*C - 2*B*c*d + 3*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x) 
]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE 
[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqr 
t[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a 
+ b*x^2)) + (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(2*A*b^2*c^2*d + 3*a^2*C*d^ 
3 + a*b*d^2*(-3*B*c + A*d) + a^(3/2)*Sqrt[b]*d^2*(3*c*C - B*d) - Sqrt[a]*b 
^(3/2)*c*(4*c^2*C - 2*B*c*d + A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + 
d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*Ellip 
ticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + 
 Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(- 
a + b*x^2))))/(b^2*d*(b*c^2 - a*d^2)^2*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2180, 27, 2182, 27, 600, 509, 508, 327, 512, 511, 321}

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.

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

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 C d^4-a b d^2 \left (-A d^2+2 B c d+3 c^2 C\right )+b^2 c^2 \left (3 A d^2-2 B c d+4 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (3 c C-B d)-b c \left (A d^2-2 B c d+4 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )}{b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (a^2 C d^4+a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (3 A d^2-2 B c d+2 c^2 C\right )\right )}{b^2 d \sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(a*(b*B*c - A*b*d - a*C*d) + b*(A*b*c + a*c*C - a*B*d)*x)/(b^2*(b*c^2 - a* 
d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2]) - ((2*a*(a^2*C*d^4 + a*b*d^2*(c^2*C - 
2*B*c*d + A*d^2) + b^2*c^2*(2*c^2*C - 2*B*c*d + 3*A*d^2))*Sqrt[a - b*x^2]) 
/(b^2*d*(b*c^2 - a*d^2)*Sqrt[c + d*x]) + (a*((-2*Sqrt[a]*(3*a^2*C*d^4 - a* 
b*d^2*(3*c^2*C + 2*B*c*d - A*d^2) + b^2*c^2*(4*c^2*C - 2*B*c*d + 3*A*d^2)) 
*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/S 
qrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[ 
b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*(b*c^ 
2 - a*d^2)*(a*d^2*(3*c*C - B*d) - b*c*(4*c^2*C - 2*B*c*d + A*d^2))*Sqrt[(S 
qrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[A 
rcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + 
 d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(b*d*(b*c^2 - a*d^2)))/( 
2*a*(b*c^2 - a*d^2))
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 2180

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema 
inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 
, x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((a*(e*R - d*S) 
 + (b*d*R + a*e*S)*x)/(2*a*(p + 1)*(b*d^2 + a*e^2))), x] + Simp[1/(2*a*(p + 
 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a* 
(p + 1)*(b*d^2 + a*e^2)*Qx + b*d^2*R*(2*p + 3) - a*e*(d*S*m - e*R*(m + 2*p 
+ 3)) + e*(b*d*R + a*e*S)*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, d, e 
, m}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] &&  !(IGtQ[ 
m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1057\) vs. \(2(498)=996\).

Time = 6.03 (sec) , antiderivative size = 1058, normalized size of antiderivative = 1.90

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1058\)
default	\(\text {Expression too large to display}\)	\(3766\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1220 vs. \(2 (502) = 1004\).

Time = 0.13 (sec) , antiderivative size = 1220, normalized size of antiderivative = 2.19 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*((4*C*a*b^2*c^6 - 2*B*a*b^2*c^5*d + 13*B*a^2*b*c^3*d^3 - 3*B*a^3*c*d^ 
5 - 3*(5*C*a^2*b + A*a*b^2)*c^4*d^2 + (3*C*a^3 - 5*A*a^2*b)*c^2*d^4 - (4*C 
*b^3*c^5*d - 2*B*b^3*c^4*d^2 + 13*B*a*b^2*c^2*d^4 - 3*B*a^2*b*d^6 - 3*(5*C 
*a*b^2 + A*b^3)*c^3*d^3 + (3*C*a^2*b - 5*A*a*b^2)*c*d^5)*x^3 - (4*C*b^3*c^ 
6 - 2*B*b^3*c^5*d + 13*B*a*b^2*c^3*d^3 - 3*B*a^2*b*c*d^5 - 3*(5*C*a*b^2 + 
A*b^3)*c^4*d^2 + (3*C*a^2*b - 5*A*a*b^2)*c^2*d^4)*x^2 + (4*C*a*b^2*c^5*d - 
 2*B*a*b^2*c^4*d^2 + 13*B*a^2*b*c^2*d^4 - 3*B*a^3*d^6 - 3*(5*C*a^2*b + A*a 
*b^2)*c^3*d^3 + (3*C*a^3 - 5*A*a^2*b)*c*d^5)*x)*sqrt(-b*d)*weierstrassPInv 
erse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3 
*(3*d*x + c)/d) + 3*(4*C*a*b^2*c^5*d - 2*B*a*b^2*c^4*d^2 - 2*B*a^2*b*c^2*d 
^4 - 3*(C*a^2*b - A*a*b^2)*c^3*d^3 + (3*C*a^3 + A*a^2*b)*c*d^5 - (4*C*b^3* 
c^4*d^2 - 2*B*b^3*c^3*d^3 - 2*B*a*b^2*c*d^5 - 3*(C*a*b^2 - A*b^3)*c^2*d^4 
+ (3*C*a^2*b + A*a*b^2)*d^6)*x^3 - (4*C*b^3*c^5*d - 2*B*b^3*c^4*d^2 - 2*B* 
a*b^2*c^2*d^4 - 3*(C*a*b^2 - A*b^3)*c^3*d^3 + (3*C*a^2*b + A*a*b^2)*c*d^5) 
*x^2 + (4*C*a*b^2*c^4*d^2 - 2*B*a*b^2*c^3*d^3 - 2*B*a^2*b*c*d^5 - 3*(C*a^2 
*b - A*a*b^2)*c^2*d^4 + (3*C*a^3 + A*a^2*b)*d^6)*x)*sqrt(-b*d)*weierstrass 
Zeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), wei 
erstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/ 
(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(2*C*a*b^2*c^4*d^2 - 3*B*a*b^2*c^3*d^3 - 
B*a^2*b*c*d^5 + 2*(C*a^2*b + 2*A*a*b^2)*c^2*d^4 - (2*C*b^3*c^4*d^2 - 2*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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