\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx\) [153]

Optimal result

Integrand size = 37, antiderivative size = 737 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx=\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {2 \left (5 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (2 c^2 C d+3 B c d^2-8 A d^3-7 c^3 D\right )\right ) \sqrt {a-b x^2}}{15 d^2 \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}+\frac {2 \left (15 a^2 d^4 (C d-3 c D)-b^2 c^2 \left (2 c^2 C d+3 B c d^2-23 A d^3+8 c^3 D\right )+a b d^2 \left (19 c^2 C d-29 B c d^2+9 A d^3+21 c^3 D\right )\right ) \sqrt {a-b x^2}}{15 d^2 \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \left (15 a^2 d^4 (C d-3 c D)-b^2 c^2 \left (2 c^2 C d+3 B c d^2-23 A d^3+8 c^3 D\right )+a b d^2 \left (19 c^2 C d-29 B c d^2+9 A d^3+21 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 d^3 \left (b c^2-a d^2\right )^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (15 a^2 d^4 D-5 a b d^2 \left (2 c C d-B d^2+3 c^2 D\right )+b^2 c \left (2 c^2 C d+3 B c d^2-8 A d^3+8 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {b} d^3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2/5*(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)^(5/2)+2/15*(5*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(-8*A*d^3+3*B*c*d^2+2 
*C*c^2*d-7*D*c^3))*(-b*x^2+a)^(1/2)/d^2/(-a*d^2+b*c^2)^2/(d*x+c)^(3/2)+2/1 
5*(15*a^2*d^4*(C*d-3*D*c)-b^2*c^2*(-23*A*d^3+3*B*c*d^2+2*C*c^2*d+8*D*c^3)+ 
a*b*d^2*(9*A*d^3-29*B*c*d^2+19*C*c^2*d+21*D*c^3))*(-b*x^2+a)^(1/2)/d^2/(-a 
*d^2+b*c^2)^3/(d*x+c)^(1/2)-2/15*a^(1/2)*b^(1/2)*(15*a^2*d^4*(C*d-3*D*c)-b 
^2*c^2*(-23*A*d^3+3*B*c*d^2+2*C*c^2*d+8*D*c^3)+a*b*d^2*(9*A*d^3-29*B*c*d^2 
+19*C*c^2*d+21*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))/d^3/(-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)-2/15*a^(1/2)*(15*a^2*d^4*D-5*a*b*d^2*(-B*d^2+2*C*c*d+3*D*c^2)+b 
^2*c*(-8*A*d^3+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)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 32.86 (sec) , antiderivative size = 1122, normalized size of antiderivative = 1.52 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((c + d*x)^(7/2)*Sqrt[a - b*x^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))/(5 
*d^2*(-(b*c^2) + a*d^2)*(c + d*x)^3) - (2*(2*b*c^3*C*d + 3*b*B*c^2*d^2 - 8 
*A*b*c*d^3 - 10*a*c*C*d^3 + 5*a*B*d^4 - 7*b*c^4*D + 15*a*c^2*d^2*D))/(15*d 
^2*(-(b*c^2) + a*d^2)^2*(c + d*x)^2) - (2*(-2*b^2*c^4*C*d - 3*b^2*B*c^3*d^ 
2 + 23*A*b^2*c^2*d^3 + 19*a*b*c^2*C*d^3 - 29*a*b*B*c*d^4 + 9*a*A*b*d^5 + 1 
5*a^2*C*d^5 - 8*b^2*c^5*D + 21*a*b*c^3*d^2*D - 45*a^2*c*d^4*D))/(15*d^2*(- 
(b*c^2) + a*d^2)^3*(c + d*x))) - (2*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d 
*x))^2)/d^2]*(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-15*a^2*d^4*(C*d - 3*c*D) + 
b^2*c^2*(2*c^2*C*d + 3*B*c*d^2 - 23*A*d^3 + 8*c^3*D) - a*b*d^2*(19*c^2*C*d 
 - 29*B*c*d^2 + 9*A*d^3 + 21*c^3*D))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/( 
c + d*x))^2) - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(-15*a^2*d^4*(C*d - 3*c* 
D) + b^2*c^2*(2*c^2*C*d + 3*B*c*d^2 - 23*A*d^3 + 8*c^3*D) - a*b*d^2*(19*c^ 
2*C*d - 29*B*c*d^2 + 9*A*d^3 + 21*c^3*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*d*(Sqr 
t[b]*c - Sqrt[a]*d)*(15*A*b^(5/2)*c^2*d^2 + 15*a^(5/2)*d^4*D + 15*a^2*Sqrt 
[b]*d^3*(C*d - 2*c*D) + 5*a^(3/2)*b*d^2*(-2*c*C*d + B*d^2 - 3*c^2*D) + 3*a 
*b^(3/2)*d*(3*c^2*C*d - 8*B*c*d^2 + 3*A*d^3 + 2*c^3*D) + Sqrt[a]*b^2*c*(2* 
c^2*C*d + 3*B*c*d^2 - 8*A*d^3 + 8*c^3*D))*Sqrt[1 - c/(c + d*x) - (Sqrt[...
 

Rubi [A] (verified)

Time = 1.56 (sec) , antiderivative size = 767, 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 = {2182, 27, 2182, 27, 688, 27, 600, 509, 508, 327, 512, 511, 321}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

\(\displaystyle \frac {\frac {\frac {2 \int \frac {d \left (A b^2 c d \left (15 b c^2+17 a d^2\right )-a \left (15 a^2 D d^4-5 a b \left (-3 D c^2+5 C d c-B d^2\right ) d^2-b^2 c^2 \left (-2 D c^2+7 C d c-27 B d^2\right )\right )\right )+b \left (15 a^2 (C d-3 c D) d^4+a b \left (21 D c^3+19 C d c^2-29 B d^2 c+9 A d^3\right ) d^2-b^2 c^2 \left (8 D c^3+2 C d c^2+3 B d^2 c-23 A d^3\right )\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 d^2 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-8 A d^3+3 B c d^2-7 c^3 D+2 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {d \left (A b^2 c d \left (15 b c^2+17 a d^2\right )-a \left (15 a^2 D d^4-5 a b \left (-3 D c^2+5 C d c-B d^2\right ) d^2-b^2 c^2 \left (-2 D c^2+7 C d c-27 B d^2\right )\right )\right )+b \left (15 a^2 (C d-3 c D) d^4+a b \left (21 D c^3+19 C d c^2-29 B d^2 c+9 A d^3\right ) d^2-b^2 c^2 \left (8 D c^3+2 C d c^2+3 B d^2 c-23 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 d^2 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-8 A d^3+3 B c d^2-7 c^3 D+2 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {\frac {\frac {\left (b c^2-a d^2\right ) \left (15 a^2 d^4 D-5 a b d^2 \left (-B d^2+3 c^2 D+2 c C d\right )+b^2 c \left (-8 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 d^2 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-8 A d^3+3 B c d^2-7 c^3 D+2 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {\frac {\frac {\left (b c^2-a d^2\right ) \left (15 a^2 d^4 D-5 a b d^2 \left (-B d^2+3 c^2 D+2 c C d\right )+b^2 c \left (-8 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {b \sqrt {1-\frac {b x^2}{a}} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 d^2 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-8 A d^3+3 B c d^2-7 c^3 D+2 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {\frac {\frac {\left (b c^2-a d^2\right ) \left (15 a^2 d^4 D-5 a b d^2 \left (-B d^2+3 c^2 D+2 c C d\right )+b^2 c \left (-8 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 d^2 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-8 A d^3+3 B c d^2-7 c^3 D+2 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {\frac {\left (b c^2-a d^2\right ) \left (15 a^2 d^4 D-5 a b d^2 \left (-B d^2+3 c^2 D+2 c C d\right )+b^2 c \left (-8 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 d^2 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-8 A d^3+3 B c d^2-7 c^3 D+2 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (15 a^2 d^4 D-5 a b d^2 \left (-B d^2+3 c^2 D+2 c C d\right )+b^2 c \left (-8 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 d^2 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-8 A d^3+3 B c d^2-7 c^3 D+2 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 \sqrt {a-b x^2} \left (-D c^3+C d c^2-B d^2 c+A d^3\right )}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-7 D c^3+2 C d c^2+3 B d^2 c-8 A d^3\right )\right )}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \sqrt {a-b x^2} \left (15 a^2 (C d-3 c D) d^4+a b \left (21 D c^3+19 C d c^2-29 B d^2 c+9 A d^3\right ) d^2-b^2 c^2 \left (8 D c^3+2 C d c^2+3 B d^2 c-23 A d^3\right )\right )}{\left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {-\frac {2 \sqrt {a} \sqrt {b} \left (15 a^2 (C d-3 c D) d^4+a b \left (21 D c^3+19 C d c^2-29 B d^2 c+9 A d^3\right ) d^2-b^2 c^2 \left (8 D c^3+2 C d c^2+3 B d^2 c-23 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (15 a^2 D d^4-5 a b \left (3 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (8 D c^3+2 C d c^2+3 B d^2 c-8 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}}{b c^2-a d^2}}{3 d^2 \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\frac {-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (15 a^2 d^4 D-5 a b d^2 \left (-B d^2+3 c^2 D+2 c C d\right )+b^2 c \left (-8 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (15 a^2 d^4 (C d-3 c D)+a b d^2 \left (9 A d^3-29 B c d^2+21 c^3 D+19 c^2 C d\right )-b^2 c^2 \left (-23 A d^3+3 B c d^2+8 c^3 D+2 c^2 C d\right )\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 d^2 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-8 A d^3+3 B c d^2-7 c^3 D+2 c^2 C d\right )\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{5 \left (b c^2-a d^2\right )}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((c + d*x)^(7/2)*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])/(5*d^2*(b*c^2 - a* 
d^2)*(c + d*x)^(5/2)) + ((2*(5*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) - b*c*(2* 
c^2*C*d + 3*B*c*d^2 - 8*A*d^3 - 7*c^3*D))*Sqrt[a - b*x^2])/(3*d^2*(b*c^2 - 
 a*d^2)*(c + d*x)^(3/2)) + ((2*(15*a^2*d^4*(C*d - 3*c*D) - b^2*c^2*(2*c^2* 
C*d + 3*B*c*d^2 - 23*A*d^3 + 8*c^3*D) + a*b*d^2*(19*c^2*C*d - 29*B*c*d^2 + 
 9*A*d^3 + 21*c^3*D))*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) + ( 
(-2*Sqrt[a]*Sqrt[b]*(15*a^2*d^4*(C*d - 3*c*D) - b^2*c^2*(2*c^2*C*d + 3*B*c 
*d^2 - 23*A*d^3 + 8*c^3*D) + a*b*d^2*(19*c^2*C*d - 29*B*c*d^2 + 9*A*d^3 + 
21*c^3*D))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sq 
rt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[(Sqr 
t[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*(b* 
c^2 - a*d^2)*(15*a^2*d^4*D - 5*a*b*d^2*(2*c*C*d - B*d^2 + 3*c^2*D) + b^2*c 
*(2*c^2*C*d + 3*B*c*d^2 - 8*A*d^3 + 8*c^3*D))*Sqrt[(Sqrt[b]*(c + d*x))/(Sq 
rt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[ 
b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt 
[c + d*x]*Sqrt[a - b*x^2]))/(b*c^2 - a*d^2))/(3*d^2*(b*c^2 - a*d^2)))/(5*( 
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 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 = 9.60 (sec) , antiderivative size = 1206, normalized size of antiderivative = 1.64

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1206\)
default	\(\text {Expression too large to display}\)	\(17283\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2)/(-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)*(-2/5/d^5/(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)^3+2/15*(8*A*b*c*d^3-5*B*a*d^4-3*B*b*c^2*d^2+10*C*a*c*d^3-2*C*b*c^ 
3*d-15*D*a*c^2*d^2+7*D*b*c^4)/d^4/(a*d^2-b*c^2)^2*(-b*d*x^3-b*c*x^2+a*d*x+ 
a*c)^(1/2)/(x+c/d)^2-2/15*(-b*d*x^2+a*d)/d^3/(a*d^2-b*c^2)^3*(9*A*a*b*d^5+ 
23*A*b^2*c^2*d^3-29*B*a*b*c*d^4-3*B*b^2*c^3*d^2+15*C*a^2*d^5+19*C*a*b*c^2* 
d^3-2*C*b^2*c^4*d-45*D*a^2*c*d^4+21*D*a*b*c^3*d^2-8*D*b^2*c^5)/((x+c/d)*(- 
b*d*x^2+a*d))^(1/2)+2*(D/d^3-1/15*b*(8*A*b*c*d^3-5*B*a*d^4-3*B*b*c^2*d^2+1 
0*C*a*c*d^3-2*C*b*c^3*d-15*D*a*c^2*d^2+7*D*b*c^4)/d^3/(a*d^2-b*c^2)^2-1/15 
*b*c/d^3*(9*A*a*b*d^5+23*A*b^2*c^2*d^3-29*B*a*b*c*d^4-3*B*b^2*c^3*d^2+15*C 
*a^2*d^5+19*C*a*b*c^2*d^3-2*C*b^2*c^4*d-45*D*a^2*c*d^4+21*D*a*b*c^3*d^2-8* 
D*b^2*c^5)/(a*d^2-b*c^2)^3)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^ 
(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*( 
a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/ 
2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2)) 
/(-c/d-1/b*(a*b)^(1/2)))^(1/2))-2/15*b/d^2*(9*A*a*b*d^5+23*A*b^2*c^2*d^3-2 
9*B*a*b*c*d^4-3*B*b^2*c^3*d^2+15*C*a^2*d^5+19*C*a*b*c^2*d^3-2*C*b^2*c^4*d- 
45*D*a^2*c*d^4+21*D*a*b*c^3*d^2-8*D*b^2*c^5)/(a*d^2-b*c^2)^3*(c/d-1/b*(a*b 
)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d- 
1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1718 vs. \(2 (670) = 1340\).

Time = 0.20 (sec) , antiderivative size = 1718, normalized size of antiderivative = 2.33 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2)/(-b*x^2+a)^(1/2),x, algorithm= 
"fricas")
 

Output:

-2/45*((8*D*b^3*c^9 + 2*C*b^3*c^8*d - 52*B*a*b^2*c^5*d^4 - 3*(9*D*a*b^2 - 
B*b^3)*c^7*d^2 + 2*(C*a*b^2 + 11*A*b^3)*c^6*d^3 + 6*(10*C*a^2*b + 7*A*a*b^ 
2)*c^4*d^5 - 15*(3*D*a^3 + B*a^2*b)*c^3*d^6 + (8*D*b^3*c^6*d^3 + 2*C*b^3*c 
^5*d^4 - 52*B*a*b^2*c^2*d^7 - 3*(9*D*a*b^2 - B*b^3)*c^4*d^5 + 2*(C*a*b^2 + 
 11*A*b^3)*c^3*d^6 + 6*(10*C*a^2*b + 7*A*a*b^2)*c*d^8 - 15*(3*D*a^3 + B*a^ 
2*b)*d^9)*x^3 + 3*(8*D*b^3*c^7*d^2 + 2*C*b^3*c^6*d^3 - 52*B*a*b^2*c^3*d^6 
- 3*(9*D*a*b^2 - B*b^3)*c^5*d^4 + 2*(C*a*b^2 + 11*A*b^3)*c^4*d^5 + 6*(10*C 
*a^2*b + 7*A*a*b^2)*c^2*d^7 - 15*(3*D*a^3 + B*a^2*b)*c*d^8)*x^2 + 3*(8*D*b 
^3*c^8*d + 2*C*b^3*c^7*d^2 - 52*B*a*b^2*c^4*d^5 - 3*(9*D*a*b^2 - B*b^3)*c^ 
6*d^3 + 2*(C*a*b^2 + 11*A*b^3)*c^5*d^4 + 6*(10*C*a^2*b + 7*A*a*b^2)*c^3*d^ 
6 - 15*(3*D*a^3 + B*a^2*b)*c^2*d^7)*x)*sqrt(-b*d)*weierstrassPInverse(4/3* 
(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + 
 c)/d) + 3*(8*D*b^3*c^8*d + 2*C*b^3*c^7*d^2 - 3*(7*D*a*b^2 - B*b^3)*c^6*d^ 
3 - (19*C*a*b^2 + 23*A*b^3)*c^5*d^4 + (45*D*a^2*b + 29*B*a*b^2)*c^4*d^5 - 
3*(5*C*a^2*b + 3*A*a*b^2)*c^3*d^6 + (8*D*b^3*c^5*d^4 + 2*C*b^3*c^4*d^5 - 3 
*(7*D*a*b^2 - B*b^3)*c^3*d^6 - (19*C*a*b^2 + 23*A*b^3)*c^2*d^7 + (45*D*a^2 
*b + 29*B*a*b^2)*c*d^8 - 3*(5*C*a^2*b + 3*A*a*b^2)*d^9)*x^3 + 3*(8*D*b^3*c 
^6*d^3 + 2*C*b^3*c^5*d^4 - 3*(7*D*a*b^2 - B*b^3)*c^4*d^5 - (19*C*a*b^2 + 2 
3*A*b^3)*c^3*d^6 + (45*D*a^2*b + 29*B*a*b^2)*c^2*d^7 - 3*(5*C*a^2*b + 3*A* 
a*b^2)*c*d^8)*x^2 + 3*(8*D*b^3*c^7*d^2 + 2*C*b^3*c^6*d^3 - 3*(7*D*a*b^2...
 

Sympy [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3}}{\sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(7/2)/(-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)**(7/2)), 
x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2)/(-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)^(7/2)), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2)/(-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)^(7/2)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((a - b*x^2)^(1/2)*(c + d*x)^(7/2)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a - b*x^2)^(1/2)*(c + d*x)^(7/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{7/2} \sqrt {a-b x^2}} \, dx=\text {too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2)/(-b*x^2+a)^(1/2),x)
 

Output:

( - 6*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d - 2*sqrt(c + d*x)*sqrt(a - b*x**2 
)*b**2 + 4*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c*x + 9*int((sqrt(c + d*x)*sqr 
t(a - b*x**2)*x**2)/(a*c**4 + 4*a*c**3*d*x + 6*a*c**2*d**2*x**2 + 4*a*c*d* 
*3*x**3 + a*d**4*x**4 - b*c**4*x**2 - 4*b*c**3*d*x**3 - 6*b*c**2*d**2*x**4 
 - 4*b*c*d**3*x**5 - b*d**4*x**6),x)*a*b*c**3*d**2 + 27*int((sqrt(c + d*x) 
*sqrt(a - b*x**2)*x**2)/(a*c**4 + 4*a*c**3*d*x + 6*a*c**2*d**2*x**2 + 4*a* 
c*d**3*x**3 + a*d**4*x**4 - b*c**4*x**2 - 4*b*c**3*d*x**3 - 6*b*c**2*d**2* 
x**4 - 4*b*c*d**3*x**5 - b*d**4*x**6),x)*a*b*c**2*d**3*x + 27*int((sqrt(c 
+ d*x)*sqrt(a - b*x**2)*x**2)/(a*c**4 + 4*a*c**3*d*x + 6*a*c**2*d**2*x**2 
+ 4*a*c*d**3*x**3 + a*d**4*x**4 - b*c**4*x**2 - 4*b*c**3*d*x**3 - 6*b*c**2 
*d**2*x**4 - 4*b*c*d**3*x**5 - b*d**4*x**6),x)*a*b*c*d**4*x**2 + 9*int((sq 
rt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**4 + 4*a*c**3*d*x + 6*a*c**2*d**2* 
x**2 + 4*a*c*d**3*x**3 + a*d**4*x**4 - b*c**4*x**2 - 4*b*c**3*d*x**3 - 6*b 
*c**2*d**2*x**4 - 4*b*c*d**3*x**5 - b*d**4*x**6),x)*a*b*d**5*x**3 + 3*int( 
(sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**4 + 4*a*c**3*d*x + 6*a*c**2*d* 
*2*x**2 + 4*a*c*d**3*x**3 + a*d**4*x**4 - b*c**4*x**2 - 4*b*c**3*d*x**3 - 
6*b*c**2*d**2*x**4 - 4*b*c*d**3*x**5 - b*d**4*x**6),x)*b**3*c**3*d + 9*int 
((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**4 + 4*a*c**3*d*x + 6*a*c**2*d 
**2*x**2 + 4*a*c*d**3*x**3 + a*d**4*x**4 - b*c**4*x**2 - 4*b*c**3*d*x**3 - 
 6*b*c**2*d**2*x**4 - 4*b*c*d**3*x**5 - b*d**4*x**6),x)*b**3*c**2*d**2*...