\(\int \frac {x^2 (A+B x+C x^2)}{(c+d x)^{5/2} (a-b x^2)^{3/2}} \, dx\) [225]

Optimal result

Integrand size = 35, antiderivative size = 714 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a (b B c-A b d-a C d)+b (A b c+a c C-a B d) x}{b^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {\left (3 a^2 C d^4+3 a b d^2 \left (c^2 C-2 B c d+A d^2\right )+b^2 c^2 \left (2 c^2 C-2 B c d+5 A d^2\right )\right ) \sqrt {a-b x^2}}{3 b^2 d \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}-\frac {\left (3 a^2 d^4 (3 c C-B d)-b^2 c^3 \left (4 c^2 C+2 B c d-11 A d^2\right )+3 a b c d^2 \left (9 c^2 C-9 B c d+7 A d^2\right )\right ) \sqrt {a-b x^2}}{3 b d \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}+\frac {\sqrt {a} \left (3 a^2 d^4 (3 c C-B d)-b^2 c^3 \left (4 c^2 C+2 B c d-11 A d^2\right )+3 a b c d^2 \left (9 c^2 C-9 B c d+7 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^2 \left (b c^2-a d^2\right )^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \left (3 a^2 C d^4+b^2 c^2 \left (4 c^2 C+2 B c d-5 A d^2\right )-3 a b d^2 \left (5 c^2 C-2 B c d+A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(a*(-A*b*d+B*b*c-C*a*d)+b*(A*b*c-B*a*d+C*a*c)*x)/b^2/(-a*d^2+b*c^2)/(d*x+c 
)^(3/2)/(-b*x^2+a)^(1/2)-1/3*(3*a^2*C*d^4+3*a*b*d^2*(A*d^2-2*B*c*d+C*c^2)+ 
b^2*c^2*(5*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)^(3/2)-1/3*(3*a^2*d^4*(-B*d+3*C*c)-b^2*c^3*(-11*A*d^2+2*B*c*d+4*C* 
c^2)+3*a*b*c*d^2*(7*A*d^2-9*B*c*d+9*C*c^2))*(-b*x^2+a)^(1/2)/b/d/(-a*d^2+b 
*c^2)^3/(d*x+c)^(1/2)+1/3*a^(1/2)*(3*a^2*d^4*(-B*d+3*C*c)-b^2*c^3*(-11*A*d 
^2+2*B*c*d+4*C*c^2)+3*a*b*c*d^2*(7*A*d^2-9*B*c*d+9*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^(1/2)/d^2/(-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)+1/3*a^(1/2)*(3*a^2* 
C*d^4+b^2*c^2*(-5*A*d^2+2*B*c*d+4*C*c^2)-3*a*b*d^2*(A*d^2-2*B*c*d+5*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)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 31.21 (sec) , antiderivative size = 891, normalized size of antiderivative = 1.25 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (-3 a^2 d^4 (-3 c C+B d)-b^2 c^3 \left (4 c^2 C+2 B c d-11 A d^2\right )+3 a b c d^2 \left (9 c^2 C-9 B c d+7 A d^2\right )+2 b c \left (b c^2 \left (2 c^2 C+B c d-4 A d^2\right )-3 a d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )\right )-\frac {2 b c^2 \left (b c^2-a d^2\right ) \left (c^2 C-B c d+A d^2\right )}{c+d x}+\frac {3 d (c+d x) \left (a^3 C d^3-A b^3 c^3 x+a^2 b d \left (3 c^2 C+d^2 (A+B x)-3 c d (B+C x)\right )-a b^2 c \left (c^2 C x+B c (c-3 d x)+3 A d (-c+d x)\right )\right )}{-a+b x^2}+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 d^4 (-3 c C+B d)+b^2 c^3 \left (4 c^2 C+2 B c d-11 A d^2\right )-3 a b c d^2 \left (9 c^2 C-9 B c d+7 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}+\frac {i \left (\sqrt {b} c-\sqrt {a} d\right ) \left (6 A b^{5/2} c^3 d+3 a^{5/2} C d^4-3 a^2 \sqrt {b} d^3 (-4 c C+B d)+\sqrt {a} b^2 c^2 \left (4 c^2 C+2 B c d-5 A d^2\right )-3 a^{3/2} b d^2 \left (5 c^2 C-2 B c d+A d^2\right )+3 a b^{3/2} c d \left (4 c^2 C-7 B c d+6 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )^3 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(-3*a^2*d^4*(-3*c*C + B*d) - b^2*c^3*(4*c^2*C + 2*B*c*d - 
 11*A*d^2) + 3*a*b*c*d^2*(9*c^2*C - 9*B*c*d + 7*A*d^2) + 2*b*c*(b*c^2*(2*c 
^2*C + B*c*d - 4*A*d^2) - 3*a*d^2*(4*c^2*C - 3*B*c*d + 2*A*d^2)) - (2*b*c^ 
2*(b*c^2 - a*d^2)*(c^2*C - B*c*d + A*d^2))/(c + d*x) + (3*d*(c + d*x)*(a^3 
*C*d^3 - A*b^3*c^3*x + a^2*b*d*(3*c^2*C + d^2*(A + B*x) - 3*c*d*(B + C*x)) 
 - a*b^2*c*(c^2*C*x + B*c*(c - 3*d*x) + 3*A*d*(-c + d*x))))/(-a + b*x^2) + 
 (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(3*a^2*d^4*(-3*c*C + B*d) + b^2*c^3*(4 
*c^2*C + 2*B*c*d - 11*A*d^2) - 3*a*b*c*d^2*(9*c^2*C - 9*B*c*d + 7*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)/S 
qrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/ 
(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2)) + (I*(Sqrt[b]*c - Sqrt[a 
]*d)*(6*A*b^(5/2)*c^3*d + 3*a^(5/2)*C*d^4 - 3*a^2*Sqrt[b]*d^3*(-4*c*C + B* 
d) + Sqrt[a]*b^2*c^2*(4*c^2*C + 2*B*c*d - 5*A*d^2) - 3*a^(3/2)*b*d^2*(5*c^ 
2*C - 2*B*c*d + A*d^2) + 3*a*b^(3/2)*c*d*(4*c^2*C - 7*B*c*d + 6*A*d^2))*Sq 
rt[(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)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqr 
t[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))))/(3*b*d*(b*c^2 - a*d^2)^3*S 
qrt[c + d*x])
 

Rubi [A] (verified)

Time = 2.81 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2180, 27, 2182, 27, 688, 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.

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

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {2 \int \frac {a \left (3 d \left (2 A b c \left (b c^2+3 a d^2\right )+a \left (b (4 c C-7 B d) c^2+a d^2 (4 c C-B d)\right )\right )+\left (3 a^2 C d^4-3 a b \left (5 C c^2-2 B d c+A d^2\right ) d^2+b^2 c^2 \left (4 C c^2+2 B d c-5 A d^2\right )\right ) x\right )}{2 b d (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \int \frac {3 d \left (2 A b c \left (b c^2+3 a d^2\right )+a \left (b (4 c C-7 B d) c^2+a d^2 (4 c C-B d)\right )\right )+\left (3 a^2 C d^4-3 a b \left (5 C c^2-2 B d c+A d^2\right ) d^2+b^2 c^2 \left (4 C c^2+2 B d c-5 A d^2\right )\right ) x}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {2 \int \frac {d \left (A b \left (6 b^2 c^4+23 a b d^2 c^2+3 a^2 d^4\right )-a \left (3 a^2 C d^4-9 a b c (3 c C-B d) d^2-b^2 c^3 (8 c C-23 B d)\right )\right )+b \left (3 a^2 (3 c C-B d) d^4+3 a b c \left (9 C c^2-9 B d c+7 A d^2\right ) d^2-b^2 c^3 \left (4 C c^2+2 B d c-11 A d^2\right )\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\int \frac {d \left (A b \left (6 b^2 c^4+23 a b d^2 c^2+3 a^2 d^4\right )-a \left (3 a^2 C d^4-9 a b c (3 c C-B d) d^2-b^2 c^3 (8 c C-23 B d)\right )\right )+b \left (3 a^2 (3 c C-B d) d^4+3 a b c \left (9 C c^2-9 B d c+7 A d^2\right ) d^2-b^2 c^3 \left (4 C c^2+2 B d c-11 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\frac {\left (b c^2-a d^2\right ) \left (3 a^2 C d^4-3 a b d^2 \left (A d^2-2 B c d+5 c^2 C\right )+b^2 c^2 \left (-5 A d^2+2 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\frac {\left (b c^2-a d^2\right ) \left (3 a^2 C d^4-3 a b d^2 \left (A d^2-2 B c d+5 c^2 C\right )+b^2 c^2 \left (-5 A d^2+2 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\frac {\left (b c^2-a d^2\right ) \left (3 a^2 C d^4-3 a b d^2 \left (A d^2-2 B c d+5 c^2 C\right )+b^2 c^2 \left (-5 A d^2+2 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\frac {\left (b c^2-a d^2\right ) \left (3 a^2 C d^4-3 a b d^2 \left (A d^2-2 B c d+5 c^2 C\right )+b^2 c^2 \left (-5 A d^2+2 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 C d^4-3 a b d^2 \left (A d^2-2 B c d+5 c^2 C\right )+b^2 c^2 \left (-5 A d^2+2 B c d+4 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {a (b B c-A b d-a C d)+b (A b c+a C c-a B d) x}{b^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b \left (C c^2-2 B d c+A d^2\right ) d^2+b^2 c^2 \left (2 C c^2-2 B d c+5 A d^2\right )\right )}{3 b^2 d \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {a \left (\frac {2 \sqrt {a-b x^2} \left (3 a^2 (3 c C-B d) d^4+3 a b c \left (9 C c^2-9 B d c+7 A d^2\right ) d^2-b^2 c^3 \left (4 C c^2+2 B d c-11 A d^2\right )\right )}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2 (3 c C-B d) d^4+3 a b c \left (9 C c^2-9 B d c+7 A d^2\right ) d^2-b^2 c^3 \left (4 C c^2+2 B d c-11 A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a^2 C d^4-3 a b \left (5 C c^2-2 B d c+A d^2\right ) d^2+b^2 c^2 \left (4 C c^2+2 B d c-5 A d^2\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}}{b c^2-a d^2}\right )}{3 b d \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \left (\frac {-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (3 a^2 C d^4-3 a b d^2 \left (A d^2-2 B c d+5 c^2 C\right )+b^2 c^2 \left (-5 A d^2+2 B c d+4 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 d^4 (3 c C-B d)+3 a b c d^2 \left (7 A d^2-9 B c d+9 c^2 C\right )-b^2 c^3 \left (-11 A d^2+2 B c d+4 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (A d^2-2 B c d+c^2 C\right )+b^2 c^2 \left (5 A d^2-2 B c d+2 c^2 C\right )\right )}{3 b^2 d (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

Input:

Int[(x^2*(A + B*x + C*x^2))/((c + d*x)^(5/2)*(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)*(c + d*x)^(3/2)*Sqrt[a - b*x^2]) - ((2*a*(3*a^2*C*d^4 + 3*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 + 5*A*d^2))*Sqrt[a - b 
*x^2])/(3*b^2*d*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) + (a*((2*(3*a^2*d^4*(3*c* 
C - B*d) - b^2*c^3*(4*c^2*C + 2*B*c*d - 11*A*d^2) + 3*a*b*c*d^2*(9*c^2*C - 
 9*B*c*d + 7*A*d^2))*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) + (( 
-2*Sqrt[a]*Sqrt[b]*(3*a^2*d^4*(3*c*C - B*d) - b^2*c^3*(4*c^2*C + 2*B*c*d - 
 11*A*d^2) + 3*a*b*c*d^2*(9*c^2*C - 9*B*c*d + 7*A*d^2))*Sqrt[c + d*x]*Sqrt 
[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], ( 
2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(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)*(3*a^2*C*d^4 + b 
^2*c^2*(4*c^2*C + 2*B*c*d - 5*A*d^2) - 3*a*b*d^2*(5*c^2*C - 2*B*c*d + A*d^ 
2))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]* 
EllipticF[ArcSin[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*c^2 - a*d^2 
)))/(3*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 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(1357\) vs. \(2(644)=1288\).

Time = 7.77 (sec) , antiderivative size = 1358, normalized size of antiderivative = 1.90

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1358\)
default	\(\text {Expression too large to display}\)	\(9032\)

Input:

int(x^2*(C*x^2+B*x+A)/(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^3/(a*d^2 
-b*c^2)^2*c^2*(A*d^2-B*c*d+C*c^2)*(-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^2/(a*d^2-b*c^2)^3*c*(6*A*a*d^4+4*A*b*c^2*d^2-9*B 
*a*c*d^3-B*b*c^3*d+12*C*a*c^2*d^2-2*C*b*c^4)/((x+c/d)*(-b*d*x^2+a*d))^(1/2 
)-2*(-b*d*x-b*c)*(-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*x+1/2*(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/(a*d^2-b*c^2)^3/b^2)/((x^ 
2-a/b)*(-b*d*x-b*c))^(1/2)+2*(-C/d^2/b+1/3*b*c^2/d^2*(A*d^2-B*c*d+C*c^2)/( 
a*d^2-b*c^2)^2+1/3*b*c^2/d^2*(6*A*a*d^4+4*A*b*c^2*d^2-9*B*a*c*d^3-B*b*c^3* 
d+12*C*a*c^2*d^2-2*C*b*c^4)/(a*d^2-b*c^2)^3+1/(a*d^2-b*c^2)^2/b*(A*a*b*d^2 
+A*b^2*c^2-2*B*a*b*c*d+C*a^2*d^2+C*a*b*c^2)-1/2*d*a*(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)/b/(a*d^2-b*c^2)^3+c*(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)*(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*c/d*(6*A*a*d^4+4*A*b*c^2*d^2-9*B*a*c*d^3-B*b*c 
^3*d+12*C*a*c^2*d^2-2*C*b*c^4)/(a*d^2-b*c^2)^3+1/2*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)*(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2078 vs. \(2 (648) = 1296\).

Time = 0.26 (sec) , antiderivative size = 2078, normalized size of antiderivative = 2.91 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/9*((4*C*a*b^3*c^8 + 2*B*a*b^3*c^7*d - 42*B*a^2*b^2*c^5*d^3 - 24*B*a^3*b* 
c^3*d^5 - (3*C*a^2*b^2 - 7*A*a*b^3)*c^6*d^2 + 24*(3*C*a^3*b + 2*A*a^2*b^2) 
*c^4*d^4 - 9*(C*a^4 - A*a^3*b)*c^2*d^6 - (4*C*b^4*c^6*d^2 + 2*B*b^4*c^5*d^ 
3 - 42*B*a*b^3*c^3*d^5 - 24*B*a^2*b^2*c*d^7 - (3*C*a*b^3 - 7*A*b^4)*c^4*d^ 
4 + 24*(3*C*a^2*b^2 + 2*A*a*b^3)*c^2*d^6 - 9*(C*a^3*b - A*a^2*b^2)*d^8)*x^ 
4 - 2*(4*C*b^4*c^7*d + 2*B*b^4*c^6*d^2 - 42*B*a*b^3*c^4*d^4 - 24*B*a^2*b^2 
*c^2*d^6 - (3*C*a*b^3 - 7*A*b^4)*c^5*d^3 + 24*(3*C*a^2*b^2 + 2*A*a*b^3)*c^ 
3*d^5 - 9*(C*a^3*b - A*a^2*b^2)*c*d^7)*x^3 - (4*C*b^4*c^8 + 2*B*b^4*c^7*d 
- 44*B*a*b^3*c^5*d^3 + 18*B*a^2*b^2*c^3*d^5 + 24*B*a^3*b*c*d^7 - 7*(C*a*b^ 
3 - A*b^4)*c^6*d^2 + (75*C*a^2*b^2 + 41*A*a*b^3)*c^4*d^4 - 3*(27*C*a^3*b + 
 13*A*a^2*b^2)*c^2*d^6 + 9*(C*a^4 - A*a^3*b)*d^8)*x^2 + 2*(4*C*a*b^3*c^7*d 
 + 2*B*a*b^3*c^6*d^2 - 42*B*a^2*b^2*c^4*d^4 - 24*B*a^3*b*c^2*d^6 - (3*C*a^ 
2*b^2 - 7*A*a*b^3)*c^5*d^3 + 24*(3*C*a^3*b + 2*A*a^2*b^2)*c^3*d^5 - 9*(C*a 
^4 - A*a^3*b)*c*d^7)*x)*sqrt(-b*d)*weierstrassPInverse(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^3*c^7*d + 2*B*a*b^3*c^6*d^2 + 27*B*a^2*b^2*c^4*d^4 + 3*B*a^3*b*c^2*d^ 
6 - (27*C*a^2*b^2 + 11*A*a*b^3)*c^5*d^3 - 3*(3*C*a^3*b + 7*A*a^2*b^2)*c^3* 
d^5 - (4*C*b^4*c^5*d^3 + 2*B*b^4*c^4*d^4 + 27*B*a*b^3*c^2*d^6 + 3*B*a^2*b^ 
2*d^8 - (27*C*a*b^3 + 11*A*b^4)*c^3*d^5 - 3*(3*C*a^2*b^2 + 7*A*a*b^3)*c*d^ 
7)*x^4 - 2*(4*C*b^4*c^6*d^2 + 2*B*b^4*c^5*d^3 + 27*B*a*b^3*c^3*d^5 + 3*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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