\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} (a-b x^2)^{3/2}} \, dx\) [159]

Optimal result

Integrand size = 37, antiderivative size = 754 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a (b B c-A b d-a C d+a c D)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x}{a b \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {\left (A b d \left (3 b c^2+5 a d^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (5 c C d-8 B d^2-2 c^2 D\right )\right )\right ) \sqrt {a-b x^2}}{3 a b \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}-\frac {\left (A b^2 c d \left (3 b c^2+29 a d^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (7 c C d-3 B d^2-9 c^2 D\right )-b^2 c^2 \left (11 c C d-23 B d^2-2 c^2 D\right )\right )\right ) \sqrt {a-b x^2}}{3 a b \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}+\frac {\left (A b^2 c d \left (3 b c^2+29 a d^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (7 c C d-3 B d^2-9 c^2 D\right )-b^2 c^2 \left (11 c C d-23 B d^2-2 c^2 D\right )\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} \sqrt {b} d \left (b c^2-a d^2\right )^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (A b d \left (3 b c^2+5 a d^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (5 c C d-8 B d^2-2 c^2 D\right )\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} \sqrt {b} d \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(a*(-A*b*d+B*b*c-C*a*d+D*a*c)+b*(c*(A*b+C*a)-a*d*(B+a*D/b))*x)/a/b/(-a*d^2 
+b*c^2)/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2)-1/3*(A*b*d*(5*a*d^2+3*b*c^2)+a*(3*a 
*d^2*(C*d-2*D*c)+b*c*(-8*B*d^2+5*C*c*d-2*D*c^2)))*(-b*x^2+a)^(1/2)/a/b/(-a 
*d^2+b*c^2)^2/(d*x+c)^(3/2)-1/3*(A*b^2*c*d*(29*a*d^2+3*b*c^2)-a*(3*a^2*d^4 
*D-3*a*b*d^2*(-3*B*d^2+7*C*c*d-9*D*c^2)-b^2*c^2*(-23*B*d^2+11*C*c*d-2*D*c^ 
2)))*(-b*x^2+a)^(1/2)/a/b/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)+1/3*(A*b^2*c*d*(2 
9*a*d^2+3*b*c^2)-a*(3*a^2*d^4*D-3*a*b*d^2*(-3*B*d^2+7*C*c*d-9*D*c^2)-b^2*c 
^2*(-23*B*d^2+11*C*c*d-2*D*c^2)))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellip 
ticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c 
+a^(1/2)*d))^(1/2))/a^(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*b*d*(5*a*d^2+3*b*c^2)+a*(3*a*d^ 
2*(C*d-2*D*c)+b*c*(-8*B*d^2+5*C*c*d-2*D*c^2)))*((d*x+c)/(c+a^(1/2)*d/b^(1/ 
2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)* 
2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(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 = 33.07 (sec) , antiderivative size = 1181, normalized size of antiderivative = 1.57 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3 
*(-(b*c^2) + a*d^2)^2*(c + d*x)^2) - (2*(-4*b*c^3*C*d + 7*b*B*c^2*d^2 - 10 
*A*b*c*d^3 - 6*a*c*C*d^3 + 3*a*B*d^4 + b*c^4*D + 9*a*c^2*d^2*D))/(3*(-(b*c 
^2) + a*d^2)^3*(c + d*x)) + (a*b^2*B*c^3 - 3*a*A*b^2*c^2*d - 3*a^2*b*c^2*C 
*d + 3*a^2*b*B*c*d^2 - a^2*A*b*d^3 - a^3*C*d^3 + a^2*b*c^3*D + 3*a^3*c*d^2 
*D + A*b^3*c^3*x + a*b^2*c^3*C*x - 3*a*b^2*B*c^2*d*x + 3*a*A*b^2*c*d^2*x + 
 3*a^2*b*c*C*d^2*x - a^2*b*B*d^3*x - 3*a^2*b*c^2*d*D*x - a^3*d^3*D*x)/(a*( 
-(b*c^2) + a*d^2)^3*(-a + b*x^2))) + (Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + 
 d*x))^2)/d^2]*(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(A*b^2*c*d*(3*b*c^2 + 29*a* 
d^2) - a*(3*a^2*d^4*D + b^2*c^2*(-11*c*C*d + 23*B*d^2 + 2*c^2*D) + 3*a*b*d 
^2*(-7*c*C*d + 3*B*d^2 + 9*c^2*D)))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c 
 + d*x))^2) - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(A*b^2*c*d*(3*b*c^2 + 29* 
a*d^2) - a*(3*a^2*d^4*D + b^2*c^2*(-11*c*C*d + 23*B*d^2 + 2*c^2*D) + 3*a*b 
*d^2*(-7*c*C*d + 3*B*d^2 + 9*c^2*D)))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/( 
Sqrt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (Sqrt[a]*d)/(Sqrt[b]*(c + d*x)) 
]*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[c + d*x] - (I*Sqrt[a]*Sq 
rt[b]*d*(Sqrt[b]*c - Sqrt[a]*d)*(3*A*b^(5/2)*c^2*d - 3*Sqrt[a]*b^2*c*(2*c^ 
2*C - 5*B*c*d + 8*A*d^2) + 3*a^(5/2)*d^3*D + 3*a^2*Sqrt[b]*d^2*(C*d - 2*c* 
D) + 3*a^(3/2)*b*d*(-6*c*C*d + 3*B*d^2 + 7*c^2*D) + a*b^(3/2)*(5*c^2*C*...
 

Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 779, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, 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.

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

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

\(\displaystyle \frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{a b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {2 \int \frac {b \left (a \left (3 a^2 (C d-3 c D) d^3+a b \left (-23 D c^3+23 C d c^2-17 B d^2 c+5 A d^3\right ) d+3 b^2 c^2 \left (2 C c^2-5 B d c+9 A d^2\right )\right )+\left (A b^2 c d \left (3 b c^2+29 a d^2\right )-a \left (3 a^2 D d^4-3 a b \left (-9 D c^2+7 C d c-3 B d^2\right ) d^2-b^2 c^2 \left (-2 D c^2+11 C d c-23 B d^2\right )\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{a b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {b \int \frac {a \left (3 a^2 (C d-3 c D) d^3+a b \left (-23 D c^3+23 C d c^2-17 B d^2 c+5 A d^3\right ) d+3 b^2 c^2 \left (2 C c^2-5 B d c+9 A d^2\right )\right )+\left (A b^2 c d \left (3 b c^2+29 a d^2\right )-a \left (3 a^2 D d^4-3 a b \left (-9 D c^2+7 C d c-3 B d^2\right ) d^2-b^2 c^2 \left (-2 D c^2+11 C d c-23 B d^2\right )\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{a b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {b \left (\frac {\left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{a b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {b \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{a b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {b \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{a b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {b \left (-\frac {\left (b c^2-a d^2\right ) \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{a b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {b \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {a (b B c+a D c-A b d-a C d)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x}{a b \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {\frac {2 \sqrt {a-b x^2} \left (A b d \left (3 b c^2+5 a d^2\right )+a \left (3 a (C d-2 c D) d^2+b c \left (-2 D c^2+5 C d c-8 B d^2\right )\right )\right )}{3 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (3 b c^2+29 a d^2\right )-a \left (3 a^2 D d^4-3 a b \left (-9 D c^2+7 C d c-3 B d^2\right ) d^2-b^2 c^2 \left (-2 D c^2+11 C d c-23 B d^2\right )\right )\right )}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {b \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (A b d \left (3 b c^2+5 a d^2\right )+a \left (3 a (C d-2 c D) d^2+b c \left (-2 D c^2+5 C d c-8 B d^2\right )\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (A b^2 c d \left (3 b c^2+29 a d^2\right )-a \left (3 a^2 D d^4-3 a b \left (-9 D c^2+7 C d c-3 B d^2\right ) d^2-b^2 c^2 \left (-2 D c^2+11 C d c-23 B d^2\right )\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{a b \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {b \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A b^2 c d \left (29 a d^2+3 b c^2\right )-a \left (3 a^2 d^4 D-3 a b d^2 \left (-3 B d^2-9 c^2 D+7 c C d\right )-b^2 c^2 \left (-23 B d^2-2 c^2 D+11 c C d\right )\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A b d \left (5 a d^2+3 b c^2\right )+a \left (3 a d^2 (C d-2 c D)+b c \left (-8 B d^2-2 c^2 D+5 c C d\right )\right )\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a b \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(a*(b*B*c - A*b*d - a*C*d + a*c*D) + b*(c*(A*b + a*C) - a*d*(B + (a*D)/b)) 
*x)/(a*b*(b*c^2 - a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2]) - ((2*(A*b*d*(3* 
b*c^2 + 5*a*d^2) + a*(3*a*d^2*(C*d - 2*c*D) + b*c*(5*c*C*d - 8*B*d^2 - 2*c 
^2*D)))*Sqrt[a - b*x^2])/(3*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) + ((2*(A*b^2* 
c*d*(3*b*c^2 + 29*a*d^2) - a*(3*a^2*d^4*D - 3*a*b*d^2*(7*c*C*d - 3*B*d^2 - 
 9*c^2*D) - b^2*c^2*(11*c*C*d - 23*B*d^2 - 2*c^2*D)))*Sqrt[a - b*x^2])/((b 
*c^2 - a*d^2)*Sqrt[c + d*x]) + (b*((-2*Sqrt[a]*(A*b^2*c*d*(3*b*c^2 + 29*a* 
d^2) - a*(3*a^2*d^4*D - 3*a*b*d^2*(7*c*C*d - 3*B*d^2 - 9*c^2*D) - b^2*c^2* 
(11*c*C*d - 23*B*d^2 - 2*c^2*D)))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*Ellipt 
icE[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)]*Sq 
rt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(A*b*d*(3*b*c^2 + 5*a*d^2) + a 
*(3*a*d^2*(C*d - 2*c*D) + b*c*(5*c*C*d - 8*B*d^2 - 2*c^2*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])))/(b*c^2 - a*d^2))/(3*(b*c^2 - 
a*d^2)))/(2*a*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. \(1509\) vs. \(2(684)=1368\).

Time = 9.79 (sec) , antiderivative size = 1510, normalized size of antiderivative = 2.00

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1510\)
default	\(\text {Expression too large to display}\)	\(10903\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),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^2/(a*d 
^2-b*c^2)^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/(a*d^2-b*c^2)^3*(10*A*b*c*d^3-3*B*a*d^4- 
7*B*b*c^2*d^2+6*C*a*c*d^3+4*C*b*c^3*d-9*D*a*c^2*d^2-D*b*c^4)/((x+c/d)*(-b* 
d*x^2+a*d))^(1/2)-2*(-b*d*x-b*c)*(-1/2*(3*A*a*b^2*c*d^2+A*b^3*c^3-B*a^2*b* 
d^3-3*B*a*b^2*c^2*d+3*C*a^2*b*c*d^2+C*a*b^2*c^3-D*a^3*d^3-3*D*a^2*b*c^2*d) 
/(a*d^2-b*c^2)^3/a/b*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-3*D*a^2*c*d^2-D*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/d*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/(a*d^2 
-b*c^2)^2+1/3*b*c/d*(10*A*b*c*d^3-3*B*a*d^4-7*B*b*c^2*d^2+6*C*a*c*d^3+4*C* 
b*c^3*d-9*D*a*c^2*d^2-D*b*c^4)/(a*d^2-b*c^2)^3+(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^2-2*D*a^2*c*d)/(a*d^2-b*c^2)^2/a-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-3*D*a^2*c*d^ 
2-D*a*b*c^3)/(a*d^2-b*c^2)^3+c*(3*A*a*b^2*c*d^2+A*b^3*c^3-B*a^2*b*d^3-3*B* 
a*b^2*c^2*d+3*C*a^2*b*c*d^2+C*a*b^2*c^3-D*a^3*d^3-3*D*a^2*b*c^2*d)/(a*d^2- 
b*c^2)^3/a)*(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*(10*A*b*c*d^3-3*B*a*d^4-7*B*b*c^2*d^2+6*C*a*c*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2266 vs. \(2 (692) = 1384\).

Time = 0.22 (sec) , antiderivative size = 2266, normalized size of antiderivative = 3.01 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/9*((2*D*a^2*b^2*c^7 + (7*C*a^2*b^2 - 3*A*a*b^3)*c^6*d - 2*(21*D*a^3*b + 
11*B*a^2*b^2)*c^5*d^2 + 4*(12*C*a^3*b + 13*A*a^2*b^2)*c^4*d^3 - 6*(4*D*a^4 
 + 7*B*a^3*b)*c^3*d^4 + 3*(3*C*a^4 + 5*A*a^3*b)*c^2*d^5 - (2*D*a*b^3*c^5*d 
^2 + (7*C*a*b^3 - 3*A*b^4)*c^4*d^3 - 2*(21*D*a^2*b^2 + 11*B*a*b^3)*c^3*d^4 
 + 4*(12*C*a^2*b^2 + 13*A*a*b^3)*c^2*d^5 - 6*(4*D*a^3*b + 7*B*a^2*b^2)*c*d 
^6 + 3*(3*C*a^3*b + 5*A*a^2*b^2)*d^7)*x^4 - 2*(2*D*a*b^3*c^6*d + (7*C*a*b^ 
3 - 3*A*b^4)*c^5*d^2 - 2*(21*D*a^2*b^2 + 11*B*a*b^3)*c^4*d^3 + 4*(12*C*a^2 
*b^2 + 13*A*a*b^3)*c^3*d^4 - 6*(4*D*a^3*b + 7*B*a^2*b^2)*c^2*d^5 + 3*(3*C* 
a^3*b + 5*A*a^2*b^2)*c*d^6)*x^3 - (2*D*a*b^3*c^7 + (7*C*a*b^3 - 3*A*b^4)*c 
^6*d - 22*(2*D*a^2*b^2 + B*a*b^3)*c^5*d^2 + (41*C*a^2*b^2 + 55*A*a*b^3)*c^ 
4*d^3 + 2*(9*D*a^3*b - 10*B*a^2*b^2)*c^3*d^4 - (39*C*a^3*b + 37*A*a^2*b^2) 
*c^2*d^5 + 6*(4*D*a^4 + 7*B*a^3*b)*c*d^6 - 3*(3*C*a^4 + 5*A*a^3*b)*d^7)*x^ 
2 + 2*(2*D*a^2*b^2*c^6*d + (7*C*a^2*b^2 - 3*A*a*b^3)*c^5*d^2 - 2*(21*D*a^3 
*b + 11*B*a^2*b^2)*c^4*d^3 + 4*(12*C*a^3*b + 13*A*a^2*b^2)*c^3*d^4 - 6*(4* 
D*a^4 + 7*B*a^3*b)*c^2*d^5 + 3*(3*C*a^4 + 5*A*a^3*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*(2*D*a^2*b^2*c^6*d - (11*C*a^2*b^2 + 3* 
A*a*b^3)*c^5*d^2 + (27*D*a^3*b + 23*B*a^2*b^2)*c^4*d^3 - (21*C*a^3*b + 29* 
A*a^2*b^2)*c^3*d^4 + 3*(D*a^4 + 3*B*a^3*b)*c^2*d^5 - (2*D*a*b^3*c^4*d^3 - 
(11*C*a*b^3 + 3*A*b^4)*c^3*d^4 + (27*D*a^2*b^2 + 23*B*a*b^3)*c^2*d^5 - ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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