\(\int \frac {x (A+B x+C x^2)}{(c+d x)^{5/2} (a-b x^2)^{3/2}} \, dx\) [226]

Optimal result

Integrand size = 33, antiderivative size = 642 \[ \int \frac {x \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {A b c+a c C-a B d+b \left (B c-\left (A+\frac {a C}{b}\right ) d\right ) x}{b \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}+\frac {\left (3 a d^2 (2 c C-B d)+b c \left (2 c^2 C-5 B c d+8 A d^2\right )\right ) \sqrt {a-b x^2}}{3 b \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}+\frac {\left (3 a^2 C d^4+3 a b d^2 \left (9 c^2 C-7 B c d+3 A d^2\right )+b^2 c^2 \left (2 c^2 C-11 B c d+23 A d^2\right )\right ) \sqrt {a-b x^2}}{3 b \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {\sqrt {a} \left (3 a^2 C d^4+3 a b d^2 \left (9 c^2 C-7 B c d+3 A d^2\right )+b^2 c^2 \left (2 c^2 C-11 B c d+23 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d \left (b c^2-a d^2\right )^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \left (3 a d^2 (2 c C-B d)+b c \left (2 c^2 C-5 B c d+8 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(A*b*c+C*a*c-B*a*d+b*(B*c-(A+a*C/b)*d)*x)/b/(-a*d^2+b*c^2)/(d*x+c)^(3/2)/( 
-b*x^2+a)^(1/2)+1/3*(3*a*d^2*(-B*d+2*C*c)+b*c*(8*A*d^2-5*B*c*d+2*C*c^2))*( 
-b*x^2+a)^(1/2)/b/(-a*d^2+b*c^2)^2/(d*x+c)^(3/2)+1/3*(3*a^2*C*d^4+3*a*b*d^ 
2*(3*A*d^2-7*B*c*d+9*C*c^2)+b^2*c^2*(23*A*d^2-11*B*c*d+2*C*c^2))*(-b*x^2+a 
)^(1/2)/b/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)-1/3*a^(1/2)*(3*a^2*C*d^4+3*a*b*d^ 
2*(3*A*d^2-7*B*c*d+9*C*c^2)+b^2*c^2*(23*A*d^2-11*B*c*d+2*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/(-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*d^2*(-B*d+2*C*c)+b*c*(8*A*d^2-5*B*c*d+2*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^(1/2)/d/(-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 = 30.06 (sec) , antiderivative size = 866, normalized size of antiderivative = 1.35 \[ \int \frac {x \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (\frac {2 c \left (-b c^2+a d^2\right ) \left (c^2 C-B c d+A d^2\right )}{(c+d x)^2}-\frac {2 \left (3 a d^2 \left (3 c^2 C-2 B c d+A d^2\right )+b c^2 \left (c^2 C-4 B c d+7 A d^2\right )\right )}{c+d x}+\frac {3 \left (-b^2 B c^3 x+a^2 d^2 (-3 c C+d (B+C x))+a b c \left (-c^2 C-3 B d^2 x+3 c d (B+C x)\right )+A b \left (-b c^2 (c-3 d x)+a d^2 (-3 c+d x)\right )\right )}{a-b x^2}\right )}{\left (-b c^2+a d^2\right )^3}+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (3 a^2 C d^4+3 a b d^2 \left (9 c^2 C-7 B c d+3 A d^2\right )+b^2 c^2 \left (2 c^2 C-11 B c d+23 A d^2\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 C d^4+3 a b d^2 \left (9 c^2 C-7 B c d+3 A d^2\right )+b^2 c^2 \left (2 c^2 C-11 B c d+23 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {b} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 C d^3+3 b^2 c^2 (-2 B c+5 A d)+3 a^{3/2} \sqrt {b} d^2 (-2 c C+B d)+\sqrt {a} b^{3/2} c \left (-2 c^2 C+5 B c d-8 A d^2\right )+3 a b d \left (7 c^2 C-6 B c d+3 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{b d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right )^3 \left (-a+b x^2\right )}\right )}{3 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*((2*c*(-(b*c^2) + a*d^2)*(c^2*C - B*c*d + A*d 
^2))/(c + d*x)^2 - (2*(3*a*d^2*(3*c^2*C - 2*B*c*d + A*d^2) + b*c^2*(c^2*C 
- 4*B*c*d + 7*A*d^2)))/(c + d*x) + (3*(-(b^2*B*c^3*x) + a^2*d^2*(-3*c*C + 
d*(B + C*x)) + a*b*c*(-(c^2*C) - 3*B*d^2*x + 3*c*d*(B + C*x)) + A*b*(-(b*c 
^2*(c - 3*d*x)) + a*d^2*(-3*c + d*x))))/(a - b*x^2)))/(-(b*c^2) + a*d^2)^3 
 + (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(3*a^2*C*d^4 + 3*a*b*d^2*(9*c^2*C - 
 7*B*c*d + 3*A*d^2) + b^2*c^2*(2*c^2*C - 11*B*c*d + 23*A*d^2))*(a - b*x^2) 
 + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(3*a^2*C*d^4 + 3*a*b*d^2*(9*c^2*C - 7 
*B*c*d + 3*A*d^2) + b^2*c^2*(2*c^2*C - 11*B*c*d + 23*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 + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[b]*d* 
(Sqrt[b]*c - Sqrt[a]*d)*(3*a^2*C*d^3 + 3*b^2*c^2*(-2*B*c + 5*A*d) + 3*a^(3 
/2)*Sqrt[b]*d^2*(-2*c*C + B*d) + Sqrt[a]*b^(3/2)*c*(-2*c^2*C + 5*B*c*d - 8 
*A*d^2) + 3*a*b*d*(7*c^2*C - 6*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)*EllipticF[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)])/(b*d^2*Sqrt[-c + (Sqrt[a] 
*d)/Sqrt[b]]*(b*c^2 - a*d^2)^3*(-a + b*x^2))))/(3*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {2180, 27, 688, 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.

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

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \int -\frac {b \left (3 b^2 (2 B c-5 A d) c^3-a b d \left (23 C c^2-23 B d c+17 A d^2\right ) c-3 a^2 d^3 (3 c C-B d)-\left (3 a^2 C d^4+3 a b \left (9 C c^2-7 B d c+3 A d^2\right ) d^2+b^2 c^2 \left (2 C c^2-11 B d c+23 A d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {b \int \frac {3 b^2 (2 B c-5 A d) c^3-a b d \left (23 C c^2-23 B d c+17 A d^2\right ) c-3 a^2 d^3 (3 c C-B d)-\left (3 a^2 C d^4+3 a b \left (9 C c^2-7 B d c+3 A d^2\right ) d^2+b^2 c^2 \left (2 C c^2-11 B d c+23 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {b \left (\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {b \left (\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {b \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {b \left (\frac {\left (b c^2-a d^2\right ) \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {b \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {b \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {b x \left (B c-d \left (\frac {a C}{b}+A\right )\right )-a B d+a c C+A b c}{b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {-\frac {\frac {2 \sqrt {a-b x^2} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {b \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 C d^4+3 a b d^2 \left (3 A d^2-7 B c d+9 c^2 C\right )+b^2 c^2 \left (23 A d^2-11 B c d+2 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2 (2 c C-B d)+b c \left (8 A d^2-5 B c d+2 c^2 C\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 b \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(A*b*c + a*c*C - a*B*d + b*(B*c - (A + (a*C)/b)*d)*x)/(b*(b*c^2 - a*d^2)*( 
c + d*x)^(3/2)*Sqrt[a - b*x^2]) - ((-2*(3*a*d^2*(2*c*C - B*d) + b*c*(2*c^2 
*C - 5*B*c*d + 8*A*d^2))*Sqrt[a - b*x^2])/(3*(b*c^2 - a*d^2)*(c + d*x)^(3/ 
2)) - ((2*(3*a^2*C*d^4 + 3*a*b*d^2*(9*c^2*C - 7*B*c*d + 3*A*d^2) + b^2*c^2 
*(2*c^2*C - 11*B*c*d + 23*A*d^2))*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c 
 + d*x]) - (b*((2*Sqrt[a]*(3*a^2*C*d^4 + 3*a*b*d^2*(9*c^2*C - 7*B*c*d + 3* 
A*d^2) + b^2*c^2*(2*c^2*C - 11*B*c*d + 23*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)])/(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)*(3*a*d^2*(2*c* 
C - B*d) + b*c*(2*c^2*C - 5*B*c*d + 8*A*d^2))*Sqrt[(Sqrt[b]*(c + d*x))/(Sq 
rt[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*c^2 - a*d^2)))/(2*b*(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]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1317\) vs. \(2(572)=1144\).

Time = 8.00 (sec) , antiderivative size = 1318, normalized size of antiderivative = 2.05

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1318\)
default	\(\text {Expression too large to display}\)	\(8308\)

Input:

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

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*c*(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/(a*d^2-b*c^2)^3*(3*A*a*d^4+7*A*b*c^2*d^2-6*B*a*c*d^ 
3-4*B*b*c^3*d+9*C*a*c^2*d^2+C*b*c^4)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-2*(-b* 
d*x-b*c)*(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*d^2-b*c^2)^3/b*x-1/2*(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/b)/((x^2-a/b)*(-b*d* 
x-b*c))^(1/2)+2*(-1/3*b*c/d*(A*d^2-B*c*d+C*c^2)/(a*d^2-b*c^2)^2-1/3*b*c/d* 
(3*A*a*d^4+7*A*b*c^2*d^2-6*B*a*c*d^3-4*B*b*c^3*d+9*C*a*c^2*d^2+C*b*c^4)/(a 
*d^2-b*c^2)^3-(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*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-c*(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)*(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*(3*A*a*d^4+7*A*b*c^2*d^2- 
6*B*a*c*d^3-4*B*b*c^3*d+9*C*a*c^2*d^2+C*b*c^4)/(a*d^2-b*c^2)^3-1/2*d*(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)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1759 vs. \(2 (576) = 1152\).

Time = 0.31 (sec) , antiderivative size = 1759, normalized size of antiderivative = 2.74 \[ \int \frac {x \left (A+B x+C x^2\right )}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/9*((2*C*a*b^2*c^7 + 7*B*a*b^2*c^6*d + 48*B*a^2*b*c^4*d^3 + 9*B*a^3*c^2*d 
^5 - 2*(21*C*a^2*b + 11*A*a*b^2)*c^5*d^2 - 6*(4*C*a^3 + 7*A*a^2*b)*c^3*d^4 
 - (2*C*b^3*c^5*d^2 + 7*B*b^3*c^4*d^3 + 48*B*a*b^2*c^2*d^5 + 9*B*a^2*b*d^7 
 - 2*(21*C*a*b^2 + 11*A*b^3)*c^3*d^4 - 6*(4*C*a^2*b + 7*A*a*b^2)*c*d^6)*x^ 
4 - 2*(2*C*b^3*c^6*d + 7*B*b^3*c^5*d^2 + 48*B*a*b^2*c^3*d^4 + 9*B*a^2*b*c* 
d^6 - 2*(21*C*a*b^2 + 11*A*b^3)*c^4*d^3 - 6*(4*C*a^2*b + 7*A*a*b^2)*c^2*d^ 
5)*x^3 - (2*C*b^3*c^7 + 7*B*b^3*c^6*d + 41*B*a*b^2*c^4*d^3 - 39*B*a^2*b*c^ 
2*d^5 - 9*B*a^3*d^7 - 22*(2*C*a*b^2 + A*b^3)*c^5*d^2 + 2*(9*C*a^2*b - 10*A 
*a*b^2)*c^3*d^4 + 6*(4*C*a^3 + 7*A*a^2*b)*c*d^6)*x^2 + 2*(2*C*a*b^2*c^6*d 
+ 7*B*a*b^2*c^5*d^2 + 48*B*a^2*b*c^3*d^4 + 9*B*a^3*c*d^6 - 2*(21*C*a^2*b + 
 11*A*a*b^2)*c^4*d^3 - 6*(4*C*a^3 + 7*A*a^2*b)*c^2*d^5)*x)*sqrt(-b*d)*weie 
rstrassPInverse(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^6*d - 11*B*a*b^2*c^5*d^2 - 21* 
B*a^2*b*c^3*d^4 + (27*C*a^2*b + 23*A*a*b^2)*c^4*d^3 + 3*(C*a^3 + 3*A*a^2*b 
)*c^2*d^5 - (2*C*b^3*c^4*d^3 - 11*B*b^3*c^3*d^4 - 21*B*a*b^2*c*d^6 + (27*C 
*a*b^2 + 23*A*b^3)*c^2*d^5 + 3*(C*a^2*b + 3*A*a*b^2)*d^7)*x^4 - 2*(2*C*b^3 
*c^5*d^2 - 11*B*b^3*c^4*d^3 - 21*B*a*b^2*c^2*d^5 + (27*C*a*b^2 + 23*A*b^3) 
*c^3*d^4 + 3*(C*a^2*b + 3*A*a*b^2)*c*d^6)*x^3 - (2*C*b^3*c^6*d - 11*B*b^3* 
c^5*d^2 - 10*B*a*b^2*c^3*d^4 + 21*B*a^2*b*c*d^6 + (25*C*a*b^2 + 23*A*b^3)* 
c^4*d^3 - 2*(12*C*a^2*b + 7*A*a*b^2)*c^2*d^5 - 3*(C*a^3 + 3*A*a^2*b)*d^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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