\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx\) [152]

Optimal result

Integrand size = 37, antiderivative size = 568 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {2 \left (3 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (2 c^2 C d+B c d^2-4 A d^3-5 c^3 D\right )\right ) \sqrt {a-b x^2}}{3 d^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}-\frac {2 \sqrt {a} \left (3 a^2 d^4 D+3 a b d^2 \left (2 c C d-B d^2-5 c^2 D\right )-b^2 c \left (2 c^2 C d+B c d^2-4 A d^3-8 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^3 \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (3 a d^2 (C d-3 c D)-b \left (2 c^2 C d+B c d^2-A d^3-8 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^3 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2/3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-b*x^2+a)^(1/2)/d^2/(-a*d^2+b*c^2)/(d*x 
+c)^(3/2)+2/3*(3*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(-4*A*d^3+B*c*d^2+2*C* 
c^2*d-5*D*c^3))*(-b*x^2+a)^(1/2)/d^2/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)-2/3*a^ 
(1/2)*(3*a^2*d^4*D+3*a*b*d^2*(-B*d^2+2*C*c*d-5*D*c^2)-b^2*c*(-4*A*d^3+B*c* 
d^2+2*C*c^2*d-8*D*c^3))*(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^3/(-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)+2/3*a^(1/2)*(3*a*d^2*(C*d-3*D*c)-b*(-A*d^3+B*c*d^2+2*C 
*c^2*d-8*D*c^3))*((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^3/(-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 = 30.54 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.35 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-2 b c^3 C d-b B c^2 d^2+4 A b c d^3+6 a c C d^3-3 a B d^4+5 b c^4 D-9 a c^2 d^2 D-\frac {3 a^2 d^4 D}{b}+3 a d^2 \left (-2 c C d+B d^2+5 c^2 D\right )+b c \left (2 c^2 C d+B c d^2-4 A d^3-8 c^3 D\right )+\frac {\left (b c^2-a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{c+d x}+\frac {i \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 d^4 D-3 a b d^2 \left (-2 c C d+B d^2+5 c^2 D\right )+b^2 c \left (-2 c^2 C d-B c d^2+4 A d^3+8 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}-\frac {i \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 A b^2 c d^2+3 a^2 d^3 D-3 a^{3/2} \sqrt {b} d^2 (C d-3 c D)-3 a b d \left (-c C d+B d^2+2 c^2 D\right )+\sqrt {a} b^{3/2} \left (2 c^2 C d+B c d^2-A d^3-8 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 \left (b c^2 d-a d^3\right )^2 \sqrt {c+d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2182, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[a - b*x^2])/(3*d^2*(b*c^2 - a* 
d^2)*(c + d*x)^(3/2)) + ((2*(3*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) - b*c*(2* 
c^2*C*d + B*c*d^2 - 4*A*d^3 - 5*c^3*D))*Sqrt[a - b*x^2])/(d^2*(b*c^2 - a*d 
^2)*Sqrt[c + d*x]) + ((-2*Sqrt[a]*(3*a^2*d^4*D + 3*a*b*d^2*(2*c*C*d - B*d^ 
2 - 5*c^2*D) - b^2*c*(2*c^2*C*d + B*c*d^2 - 4*A*d^3 - 8*c^3*D))*Sqrt[c + d 
*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqr 
t[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*(C*d - 3*c*D) - b*(2*c^2*C*d + B*c*d^2 - A*d^3 - 8*c^3*D))*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]))/(d^2*(b*c^2 - a*d^2)))/( 
3*(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 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 [A] (verified)

Time = 7.19 (sec) , antiderivative size = 947, normalized size of antiderivative = 1.67

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{4} \left (a \,d^{2}-b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {2 \left (-b d \,x^{2}+a d \right ) \left (4 A \,d^{3} b c -3 B a \,d^{4}-B b \,c^{2} d^{2}+6 a c C \,d^{3}-2 b d C \,c^{3}-9 a \,c^{2} D d^{2}+5 c^{4} D b \right )}{3 d^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}+\frac {2 \left (\frac {C d -2 D c}{d^{3}}+\frac {b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{3 d^{3} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {b c \left (4 A \,d^{3} b c -3 B a \,d^{4}-B b \,c^{2} d^{2}+6 a c C \,d^{3}-2 b d C \,c^{3}-9 a \,c^{2} D d^{2}+5 c^{4} D b \right )}{3 d^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {D}{d^{2}}+\frac {b \left (4 A \,d^{3} b c -3 B a \,d^{4}-B b \,c^{2} d^{2}+6 a c C \,d^{3}-2 b d C \,c^{3}-9 a \,c^{2} D d^{2}+5 c^{4} D b \right )}{3 d^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(947\)
default	\(\text {Expression too large to display}\)	\(9472\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((d*x+c)*(-b*x^2+a))^(1/2)*(-2/3/d^4/(a*d 
^2-b*c^2)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2) 
/(x+c/d)^2+2/3*(-b*d*x^2+a*d)/d^3/(a*d^2-b*c^2)^2*(4*A*b*c*d^3-3*B*a*d^4-B 
*b*c^2*d^2+6*C*a*c*d^3-2*C*b*c^3*d-9*D*a*c^2*d^2+5*D*b*c^4)/((x+c/d)*(-b*d 
*x^2+a*d))^(1/2)+2*((C*d-2*D*c)/d^3+1/3*b/d^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3 
)/(a*d^2-b*c^2)+1/3*b*c/d^3*(4*A*b*c*d^3-3*B*a*d^4-B*b*c^2*d^2+6*C*a*c*d^3 
-2*C*b*c^3*d-9*D*a*c^2*d^2+5*D*b*c^4)/(a*d^2-b*c^2)^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*(D/d^2+1/3*b/ 
d^2*(4*A*b*c*d^3-3*B*a*d^4-B*b*c^2*d^2+6*C*a*c*d^3-2*C*b*c^3*d-9*D*a*c^2*d 
^2+5*D*b*c^4)/(a*d^2-b*c^2)^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)/(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)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.74 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm= 
"fricas")
 

Output:

2/9*((8*D*b^2*c^7 - 2*C*b^2*c^6*d - (21*D*a*b + B*b^2)*c^5*d^2 + (3*C*a*b 
- 5*A*b^2)*c^4*d^3 + 3*(7*D*a^2 + 3*B*a*b)*c^3*d^4 - 3*(3*C*a^2 + A*a*b)*c 
^2*d^5 + (8*D*b^2*c^5*d^2 - 2*C*b^2*c^4*d^3 - (21*D*a*b + B*b^2)*c^3*d^4 + 
 (3*C*a*b - 5*A*b^2)*c^2*d^5 + 3*(7*D*a^2 + 3*B*a*b)*c*d^6 - 3*(3*C*a^2 + 
A*a*b)*d^7)*x^2 + 2*(8*D*b^2*c^6*d - 2*C*b^2*c^5*d^2 - (21*D*a*b + B*b^2)* 
c^4*d^3 + (3*C*a*b - 5*A*b^2)*c^3*d^4 + 3*(7*D*a^2 + 3*B*a*b)*c^2*d^5 - 3* 
(3*C*a^2 + A*a*b)*c*d^6)*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* 
(8*D*b^2*c^6*d - 2*C*b^2*c^5*d^2 - (15*D*a*b + B*b^2)*c^4*d^3 + 2*(3*C*a*b 
 + 2*A*b^2)*c^3*d^4 + 3*(D*a^2 - B*a*b)*c^2*d^5 + (8*D*b^2*c^4*d^3 - 2*C*b 
^2*c^3*d^4 - (15*D*a*b + B*b^2)*c^2*d^5 + 2*(3*C*a*b + 2*A*b^2)*c*d^6 + 3* 
(D*a^2 - B*a*b)*d^7)*x^2 + 2*(8*D*b^2*c^5*d^2 - 2*C*b^2*c^4*d^3 - (15*D*a* 
b + B*b^2)*c^3*d^4 + 2*(3*C*a*b + 2*A*b^2)*c^2*d^5 + 3*(D*a^2 - B*a*b)*c*d 
^6)*x)*sqrt(-b*d)*weierstrassZeta(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), 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*D*b^2*c^5 
*d^2 - C*b^2*c^4*d^3 - 2*B*a*b*c*d^6 - A*a*b*d^7 - 2*(4*D*a*b + B*b^2)*c^3 
*d^4 + 5*(C*a*b + A*b^2)*c^2*d^5 + (5*D*b^2*c^4*d^3 - 2*C*b^2*c^3*d^4 - 3* 
B*a*b*d^7 - (9*D*a*b + B*b^2)*c^2*d^5 + 2*(3*C*a*b + 2*A*b^2)*c*d^6)*x)*sq 
rt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*c^6*d^4 - 2*a*b^2*c^4*d^6 + a^2*b*c^...
 

Sympy [F]

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

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2)/(-b*x**2+a)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm= 
"maxima")
 

Output:

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

Giac [F]

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

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm= 
"giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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