\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} (a-b x^2)^{5/2}} \, dx\) [165]

Optimal result

Integrand size = 37, antiderivative size = 849 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^{5/2}} \, dx=\frac {a (b B c-A b d-a C d+a c D)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x}{3 a b \left (b c^2-a d^2\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {a \left (A b d \left (b c^2+7 a d^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (7 c C d-8 B d^2-6 c^2 D\right )\right )\right )+\left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (b^2 c^2 (2 c C-3 B d)+a^2 d^3 D+a b d \left (6 c C d-5 B d^2-9 c^2 D\right )\right )\right ) x}{6 a^2 b \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {d \left (A b \left (4 b^2 c^4-15 a b c^2 d^2-21 a^2 d^4\right )-a \left (b^2 c^3 (2 c C-3 B d)+a^2 d^3 (3 C d-5 c D)+a b c d \left (27 c C d-29 B d^2-27 c^2 D\right )\right )\right ) \sqrt {a-b x^2}}{6 a^2 b \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}+\frac {\left (A b \left (4 b^2 c^4-15 a b c^2 d^2-21 a^2 d^4\right )-a \left (b^2 c^3 (2 c C-3 B d)+a^2 d^3 (3 C d-5 c D)+a b c d \left (27 c C d-29 B d^2-27 c^2 D\right )\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} \sqrt {b} \left (b c^2-a d^2\right )^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (b^2 c^2 (2 c C-3 B d)+a^2 d^3 D+a b d \left (6 c C d-5 B d^2-9 c^2 D\right )\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{3/2} \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

1/3*(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)^(1/2)/(-b*x^2+a)^(3/2)+1/6*(a*(A*b*d*(7*a*d^2+b*c^2)+a 
*(a*d^2*(C*d-2*D*c)+b*c*(-8*B*d^2+7*C*c*d-6*D*c^2)))+(4*A*b^2*c*(-3*a*d^2+ 
b*c^2)-a*(b^2*c^2*(-3*B*d+2*C*c)+a^2*d^3*D+a*b*d*(-5*B*d^2+6*C*c*d-9*D*c^2 
)))*x)/a^2/b/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-1/6*d*(A*b*(- 
21*a^2*d^4-15*a*b*c^2*d^2+4*b^2*c^4)-a*(b^2*c^3*(-3*B*d+2*C*c)+a^2*d^3*(3* 
C*d-5*D*c)+a*b*c*d*(-29*B*d^2+27*C*c*d-27*D*c^2)))*(-b*x^2+a)^(1/2)/a^2/b/ 
(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)+1/6*(A*b*(-21*a^2*d^4-15*a*b*c^2*d^2+4*b^2* 
c^4)-a*(b^2*c^3*(-3*B*d+2*C*c)+a^2*d^3*(3*C*d-5*D*c)+a*b*c*d*(-29*B*d^2+27 
*C*c*d-27*D*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))/a^(3/2)/b^(1/2)/(-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)-1/6*(4*A*b^2*c*(-3*a*d^2+b*c^2)-a*(b^2*c^2*(-3*B*d+2*C* 
c)+a^2*d^3*D+a*b*d*(-5*B*d^2+6*C*c*d-9*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^(3/2)/b^(3/2)/ 
(-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 = 33.38 (sec) , antiderivative size = 1346, normalized size of antiderivative = 1.59 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-2*d^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) 
)/((-(b*c^2) + a*d^2)^3*(c + d*x)) + (a*b^2*B*c^2 - 2*a*A*b^2*c*d - 2*a^2* 
b*c*C*d + a^2*b*B*d^2 + a^2*b*c^2*D + a^3*d^2*D + A*b^3*c^2*x + a*b^2*c^2* 
C*x - 2*a*b^2*B*c*d*x + a*A*b^2*d^2*x + a^2*b*C*d^2*x - 2*a^2*b*c*d*D*x)/( 
3*a*b*(-(b*c^2) + a*d^2)^2*(-a + b*x^2)^2) + (-(a*A*b^3*c^3*d) + 11*a^2*b^ 
2*c^3*C*d - 15*a^2*b^2*B*c^2*d^2 + 21*a^2*A*b^2*c*d^3 + 9*a^3*b*c*C*d^3 - 
5*a^3*b*B*d^4 - 6*a^2*b^2*c^4*D - 15*a^3*b*c^2*d^2*D + a^4*d^4*D + 4*A*b^4 
*c^4*x - 2*a*b^3*c^4*C*x + 3*a*b^3*B*c^3*d*x - 15*a*A*b^3*c^2*d^2*x - 15*a 
^2*b^2*c^2*C*d^2*x + 17*a^2*b^2*B*c*d^3*x - 9*a^2*A*b^2*d^4*x - 3*a^3*b*C* 
d^4*x + 15*a^2*b^2*c^3*d*D*x + 5*a^3*b*c*d^3*D*x)/(6*a^2*b*(-(b*c^2) + a*d 
^2)^3*(-a + b*x^2))) + (d*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^ 
2]*(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(A*b*(4*b^2*c^4 - 15*a*b*c^2*d^2 - 21*a 
^2*d^4) + a*(b^2*c^3*(-2*c*C + 3*B*d) + a^2*d^3*(-3*C*d + 5*c*D) + a*b*c*d 
*(-27*c*C*d + 29*B*d^2 + 27*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*(4*b^2*c^4 - 15*a*b 
*c^2*d^2 - 21*a^2*d^4) + a*(b^2*c^3*(-2*c*C + 3*B*d) + a^2*d^3*(-3*C*d + 5 
*c*D) + a*b*c*d*(-27*c*C*d + 29*B*d^2 + 27*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...
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 883, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2180, 27, 686, 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.

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

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

\(\displaystyle \frac {\frac {d \left (\frac {2 \int -\frac {a \left (A b^2 c d \left (b c^2-33 a d^2\right )-a \left (a^2 D d^4+a b \left (-15 D c^2+9 C d c-5 B d^2\right ) d^2+b^2 c^2 \left (-18 D c^2+23 C d c-27 B d^2\right )\right )\right )+b \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (A b \left (-21 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right )-a \left (a^2 d^3 (3 C d-5 c D)+a b c d \left (-29 B d^2-27 c^2 D+27 c C d\right )+b^2 c^3 (2 c C-3 B d)\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}+\frac {x \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 d^3 D+a b d \left (-5 B d^2-9 c^2 D+6 c C d\right )+b^2 c^2 (2 c C-3 B d)\right )\right )+a \left (A b d \left (7 a d^2+b c^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (-8 B d^2-6 c^2 D+7 c C d\right )\right )\right )}{a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{3 a b \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {d \left (-\frac {\int \frac {a \left (A b^2 c d \left (b c^2-33 a d^2\right )-a \left (a^2 D d^4+a b \left (-15 D c^2+9 C d c-5 B d^2\right ) d^2+b^2 c^2 \left (-18 D c^2+23 C d c-27 B d^2\right )\right )\right )+b \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (A b \left (-21 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right )-a \left (a^2 d^3 (3 C d-5 c D)+a b c d \left (-29 B d^2-27 c^2 D+27 c C d\right )+b^2 c^3 (2 c C-3 B d)\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}+\frac {x \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 d^3 D+a b d \left (-5 B d^2-9 c^2 D+6 c C d\right )+b^2 c^2 (2 c C-3 B d)\right )\right )+a \left (A b d \left (7 a d^2+b c^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (-8 B d^2-6 c^2 D+7 c C d\right )\right )\right )}{a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{3 a b \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {d \left (-\frac {\frac {b \left (A b \left (-21 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right )-a \left (a^2 d^3 (3 C d-5 c D)+a b c d \left (-29 B d^2-27 c^2 D+27 c C d\right )+b^2 c^3 (2 c C-3 B d)\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 d^3 D+a b d \left (-5 B d^2-9 c^2 D+6 c C d\right )+b^2 c^2 (2 c C-3 B d)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (A b \left (-21 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right )-a \left (a^2 d^3 (3 C d-5 c D)+a b c d \left (-29 B d^2-27 c^2 D+27 c C d\right )+b^2 c^3 (2 c C-3 B d)\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}+\frac {x \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 d^3 D+a b d \left (-5 B d^2-9 c^2 D+6 c C d\right )+b^2 c^2 (2 c C-3 B d)\right )\right )+a \left (A b d \left (7 a d^2+b c^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (-8 B d^2-6 c^2 D+7 c C d\right )\right )\right )}{a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{3 a b \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {d \left (-\frac {\frac {b \sqrt {1-\frac {b x^2}{a}} \left (A b \left (-21 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right )-a \left (a^2 d^3 (3 C d-5 c D)+a b c d \left (-29 B d^2-27 c^2 D+27 c C d\right )+b^2 c^3 (2 c C-3 B d)\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 d^3 D+a b d \left (-5 B d^2-9 c^2 D+6 c C d\right )+b^2 c^2 (2 c C-3 B d)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (A b \left (-21 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right )-a \left (a^2 d^3 (3 C d-5 c D)+a b c d \left (-29 B d^2-27 c^2 D+27 c C d\right )+b^2 c^3 (2 c C-3 B d)\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}+\frac {x \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 d^3 D+a b d \left (-5 B d^2-9 c^2 D+6 c C d\right )+b^2 c^2 (2 c C-3 B d)\right )\right )+a \left (A b d \left (7 a d^2+b c^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (-8 B d^2-6 c^2 D+7 c C d\right )\right )\right )}{a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{3 a b \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {a (b B c+a D c-A b d-a C d)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x}{3 a b \left (b c^2-a d^2\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {\frac {a \left (A b d \left (b c^2+7 a d^2\right )+a \left (a (C d-2 c D) d^2+b c \left (-6 D c^2+7 C d c-8 B d^2\right )\right )\right )+\left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 D d^3+a b \left (-9 D c^2+6 C d c-5 B d^2\right ) d+b^2 c^2 (2 c C-3 B d)\right )\right ) x}{a \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {d \left (-\frac {2 \sqrt {a-b x^2} \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right )}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 D d^3+a b \left (-9 D c^2+6 C d c-5 B d^2\right ) d+b^2 c^2 (2 c C-3 B d)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}}{b c^2-a d^2}\right )}{2 a \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {d \left (-\frac {-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 d^3 D+a b d \left (-5 B d^2-9 c^2 D+6 c C d\right )+b^2 c^2 (2 c C-3 B d)\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b \left (-21 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right )-a \left (a^2 d^3 (3 C d-5 c D)+a b c d \left (-29 B d^2-27 c^2 D+27 c C d\right )+b^2 c^3 (2 c C-3 B d)\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}-\frac {2 \sqrt {a-b x^2} \left (A b \left (-21 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right )-a \left (a^2 d^3 (3 C d-5 c D)+a b c d \left (-29 B d^2-27 c^2 D+27 c C d\right )+b^2 c^3 (2 c C-3 B d)\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{2 a \left (b c^2-a d^2\right )}+\frac {x \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 d^3 D+a b d \left (-5 B d^2-9 c^2 D+6 c C d\right )+b^2 c^2 (2 c C-3 B d)\right )\right )+a \left (A b d \left (7 a d^2+b c^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (-8 B d^2-6 c^2 D+7 c C d\right )\right )\right )}{a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)}{3 a b \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {a (b B c+a D c-A b d-a C d)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x}{3 a b \left (b c^2-a d^2\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {\frac {a \left (A b d \left (b c^2+7 a d^2\right )+a \left (a (C d-2 c D) d^2+b c \left (-6 D c^2+7 C d c-8 B d^2\right )\right )\right )+\left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 D d^3+a b \left (-9 D c^2+6 C d c-5 B d^2\right ) d+b^2 c^2 (2 c C-3 B d)\right )\right ) x}{a \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {d \left (-\frac {2 \sqrt {a-b x^2} \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right )}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {-\frac {2 \sqrt {a} \sqrt {b} \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 D d^3+a b \left (-9 D c^2+6 C d c-5 B d^2\right ) d+b^2 c^2 (2 c C-3 B d)\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}}{b c^2-a d^2}\right )}{2 a \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {a (b B c+a D c-A b d-a C d)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x}{3 a b \left (b c^2-a d^2\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {\frac {a \left (A b d \left (b c^2+7 a d^2\right )+a \left (a (C d-2 c D) d^2+b c \left (-6 D c^2+7 C d c-8 B d^2\right )\right )\right )+\left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 D d^3+a b \left (-9 D c^2+6 C d c-5 B d^2\right ) d+b^2 c^2 (2 c C-3 B d)\right )\right ) x}{a \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {d \left (-\frac {2 \sqrt {a-b x^2} \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right )}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 D d^3+a b \left (-9 D c^2+6 C d c-5 B d^2\right ) d+b^2 c^2 (2 c C-3 B d)\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}}{b c^2-a d^2}\right )}{2 a \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {a (b B c+a D c-A b d-a C d)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x}{3 a b \left (b c^2-a d^2\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}+\frac {\frac {a \left (A b d \left (b c^2+7 a d^2\right )+a \left (a (C d-2 c D) d^2+b c \left (-6 D c^2+7 C d c-8 B d^2\right )\right )\right )+\left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 D d^3+a b \left (-9 D c^2+6 C d c-5 B d^2\right ) d+b^2 c^2 (2 c C-3 B d)\right )\right ) x}{a \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {d \left (-\frac {2 \sqrt {a-b x^2} \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right )}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (4 A b^2 c \left (b c^2-3 a d^2\right )-a \left (a^2 D d^3+a b \left (-9 D c^2+6 C d c-5 B d^2\right ) d+b^2 c^2 (2 c C-3 B d)\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \left (A b \left (4 b^2 c^4-15 a b d^2 c^2-21 a^2 d^4\right )-a \left (b^2 (2 c C-3 B d) c^3+a b d \left (-27 D c^2+27 C d c-29 B d^2\right ) c+a^2 d^3 (3 C d-5 c D)\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}}{b c^2-a d^2}\right )}{2 a \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((c + d*x)^(3/2)*(a - b*x^2)^(5/2)),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)/(3*a*b*(b*c^2 - a*d^2)*Sqrt[c + d*x]*(a - b*x^2)^(3/2)) + ((a*(A*b*d*( 
b*c^2 + 7*a*d^2) + a*(a*d^2*(C*d - 2*c*D) + b*c*(7*c*C*d - 8*B*d^2 - 6*c^2 
*D))) + (4*A*b^2*c*(b*c^2 - 3*a*d^2) - a*(b^2*c^2*(2*c*C - 3*B*d) + a^2*d^ 
3*D + a*b*d*(6*c*C*d - 5*B*d^2 - 9*c^2*D)))*x)/(a*(b*c^2 - a*d^2)*Sqrt[c + 
 d*x]*Sqrt[a - b*x^2]) + (d*((-2*(A*b*(4*b^2*c^4 - 15*a*b*c^2*d^2 - 21*a^2 
*d^4) - a*(b^2*c^3*(2*c*C - 3*B*d) + a^2*d^3*(3*C*d - 5*c*D) + a*b*c*d*(27 
*c*C*d - 29*B*d^2 - 27*c^2*D)))*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + 
 d*x]) - ((-2*Sqrt[a]*Sqrt[b]*(A*b*(4*b^2*c^4 - 15*a*b*c^2*d^2 - 21*a^2*d^ 
4) - a*(b^2*c^3*(2*c*C - 3*B*d) + a^2*d^3*(3*C*d - 5*c*D) + a*b*c*d*(27*c* 
C*d - 29*B*d^2 - 27*c^2*D)))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[A 
rcSin[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)*(4*A*b^2*c*(b*c^2 - 3*a*d^2) - a*(b^2*c^2*( 
2*c*C - 3*B*d) + a^2*d^3*D + a*b*d*(6*c*C*d - 5*B*d^2 - 9*c^2*D)))*Sqrt[(S 
qrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[A 
rcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + 
 d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(b*c^2 - a*d^2)))/(2*a*(b 
*c^2 - a*d^2)))/(6*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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(1645\) vs. \(2(777)=1554\).

Time = 10.01 (sec) , antiderivative size = 1646, normalized size of antiderivative = 1.94

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1646\)
default	\(\text {Expression too large to display}\)	\(11864\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2)/(-b*x^2+a)^(5/2),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)*((1/3*(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)/b^2/a/(a*d^2-b*c 
^2)^2*x-1/3*(2*A*b^2*c*d-B*a*b*d^2-B*b^2*c^2+2*C*a*b*c*d-D*a^2*d^2-D*a*b*c 
^2)/b^3/(a*d^2-b*c^2)^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x^2-a/b)^2-2* 
(-b*d*x-b*c)*(1/12/b*(9*A*a^2*b*d^4+15*A*a*b^2*c^2*d^2-4*A*b^3*c^4-17*B*a^ 
2*b*c*d^3-3*B*a*b^2*c^3*d+3*C*a^3*d^4+15*C*a^2*b*c^2*d^2+2*C*a*b^2*c^4-5*D 
*a^3*c*d^3-15*D*a^2*b*c^3*d)/a^2/(a*d^2-b*c^2)^3*x-1/12*(21*A*a*b^2*c*d^3- 
A*b^3*c^3*d-5*B*a^2*b*d^4-15*B*a*b^2*c^2*d^2+9*C*a^2*b*c*d^3+11*C*a*b^2*c^ 
3*d+D*a^3*d^4-15*D*a^2*b*c^2*d^2-6*D*a*b^2*c^4)/(a*d^2-b*c^2)^3/a/b^2)/((x 
^2-a/b)*(-b*d*x-b*c))^(1/2)-2*(-b*d*x^2+a*d)*d/(a*d^2-b*c^2)^3*(A*d^3-B*c* 
d^2+C*c^2*d-D*c^3)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)+2*(-1/6/(a*d^2-b*c^2)^2/ 
b*(12*A*a*b^2*c*d^2-4*A*b^3*c^3-5*B*a^2*b*d^3-3*B*a*b^2*c^2*d+6*C*a^2*b*c* 
d^2+2*C*a*b^2*c^3+D*a^3*d^3-9*D*a^2*b*c^2*d)/a^2+1/12/b*d*(21*A*a*b^2*c*d^ 
3-A*b^3*c^3*d-5*B*a^2*b*d^4-15*B*a*b^2*c^2*d^2+9*C*a^2*b*c*d^3+11*C*a*b^2* 
c^3*d+D*a^3*d^4-15*D*a^2*b*c^2*d^2-6*D*a*b^2*c^4)/(a*d^2-b*c^2)^3/a-1/6*c* 
(9*A*a^2*b*d^4+15*A*a*b^2*c^2*d^2-4*A*b^3*c^4-17*B*a^2*b*c*d^3-3*B*a*b^2*c 
^3*d+3*C*a^3*d^4+15*C*a^2*b*c^2*d^2+2*C*a*b^2*c^4-5*D*a^3*c*d^3-15*D*a^2*b 
*c^3*d)/a^2/(a*d^2-b*c^2)^3-b*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*d*c/(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+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2873 vs. \(2 (790) = 1580\).

Time = 0.30 (sec) , antiderivative size = 2873, normalized size of antiderivative = 3.38 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/18*((2*(C*a^3*b^3 - 2*A*a^2*b^4)*c^6 + 3*(9*D*a^4*b^2 - B*a^3*b^3)*c^5*d 
 - 6*(7*C*a^4*b^2 - 3*A*a^3*b^3)*c^4*d^2 + 4*(10*D*a^5*b + 13*B*a^4*b^2)*c 
^3*d^3 - 6*(4*C*a^5*b + 13*A*a^4*b^2)*c^2*d^4 - 3*(D*a^6 - 5*B*a^5*b)*c*d^ 
5 + (2*(C*a*b^5 - 2*A*b^6)*c^5*d + 3*(9*D*a^2*b^4 - B*a*b^5)*c^4*d^2 - 6*( 
7*C*a^2*b^4 - 3*A*a*b^5)*c^3*d^3 + 4*(10*D*a^3*b^3 + 13*B*a^2*b^4)*c^2*d^4 
 - 6*(4*C*a^3*b^3 + 13*A*a^2*b^4)*c*d^5 - 3*(D*a^4*b^2 - 5*B*a^3*b^3)*d^6) 
*x^5 + (2*(C*a*b^5 - 2*A*b^6)*c^6 + 3*(9*D*a^2*b^4 - B*a*b^5)*c^5*d - 6*(7 
*C*a^2*b^4 - 3*A*a*b^5)*c^4*d^2 + 4*(10*D*a^3*b^3 + 13*B*a^2*b^4)*c^3*d^3 
- 6*(4*C*a^3*b^3 + 13*A*a^2*b^4)*c^2*d^4 - 3*(D*a^4*b^2 - 5*B*a^3*b^3)*c*d 
^5)*x^4 - 2*(2*(C*a^2*b^4 - 2*A*a*b^5)*c^5*d + 3*(9*D*a^3*b^3 - B*a^2*b^4) 
*c^4*d^2 - 6*(7*C*a^3*b^3 - 3*A*a^2*b^4)*c^3*d^3 + 4*(10*D*a^4*b^2 + 13*B* 
a^3*b^3)*c^2*d^4 - 6*(4*C*a^4*b^2 + 13*A*a^3*b^3)*c*d^5 - 3*(D*a^5*b - 5*B 
*a^4*b^2)*d^6)*x^3 - 2*(2*(C*a^2*b^4 - 2*A*a*b^5)*c^6 + 3*(9*D*a^3*b^3 - B 
*a^2*b^4)*c^5*d - 6*(7*C*a^3*b^3 - 3*A*a^2*b^4)*c^4*d^2 + 4*(10*D*a^4*b^2 
+ 13*B*a^3*b^3)*c^3*d^3 - 6*(4*C*a^4*b^2 + 13*A*a^3*b^3)*c^2*d^4 - 3*(D*a^ 
5*b - 5*B*a^4*b^2)*c*d^5)*x^2 + (2*(C*a^3*b^3 - 2*A*a^2*b^4)*c^5*d + 3*(9* 
D*a^4*b^2 - B*a^3*b^3)*c^4*d^2 - 6*(7*C*a^4*b^2 - 3*A*a^3*b^3)*c^3*d^3 + 4 
*(10*D*a^5*b + 13*B*a^4*b^2)*c^2*d^4 - 6*(4*C*a^5*b + 13*A*a^4*b^2)*c*d^5 
- 3*(D*a^6 - 5*B*a^5*b)*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)/...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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