\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx\) [151]

Optimal result

Integrand size = 37, antiderivative size = 449 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}-\frac {2 D \sqrt {c+d x} \sqrt {a-b x^2}}{3 b d^2}+\frac {2 \sqrt {a} \left (a d^2 (3 C d-5 c D)-b \left (6 c^2 C d-3 B c d^2+3 A d^3-8 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^3 \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (a d^2 D-b \left (6 c C d-3 B d^2-8 c^2 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2*(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 
)^(1/2)-2/3*D*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^2+2/3*a^(1/2)*(a*d^2*(3*C 
*d-5*D*c)-b*(3*A*d^3-3*B*c*d^2+6*C*c^2*d-8*D*c^3))*(d*x+c)^(1/2)*((-b*x^2+ 
a)/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^( 
1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/d^3/(-a*d^2+b*c^2)/((d*x+c)/( 
c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-2/3*a^(1/2)*(a*d^2*D-b*(-3*B* 
d^2+6*C*c*d-8*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))/b^(3/2)/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.73 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a d^2 (3 C d-5 c D)+b \left (-6 c^2 C d+3 B c d^2-3 A d^3+8 c^3 D\right )\right ) \left (a-b x^2\right )+d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right ) \left (a d^2 D (c+d x)-b \left (3 B c d^2-3 A d^3+4 c^3 D+c^2 (-3 C d+d D x)\right )\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2 (3 C d-5 c D)+b \left (-6 c^2 C d+3 B c d^2-3 A d^3+8 c^3 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 A b^{3/2} d^2+a^{3/2} d^2 D-3 a \sqrt {b} d (C d-2 c D)+\sqrt {a} b \left (-6 c C d+3 B d^2+8 c^2 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{3 b d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:

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

Output:

(2*(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a*d^2*(3*C*d - 5*c*D) + b*(-6*c^2* 
C*d + 3*B*c*d^2 - 3*A*d^3 + 8*c^3*D))*(a - b*x^2) + d^2*Sqrt[-c + (Sqrt[a] 
*d)/Sqrt[b]]*(a - b*x^2)*(a*d^2*D*(c + d*x) - b*(3*B*c*d^2 - 3*A*d^3 + 4*c 
^3*D + c^2*(-3*C*d + d*D*x))) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(a*d^2*( 
3*C*d - 5*c*D) + b*(-6*c^2*C*d + 3*B*c*d^2 - 3*A*d^3 + 8*c^3*D))*Sqrt[(d*( 
Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d 
*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/S 
qrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*d*(Sqr 
t[b]*c - Sqrt[a]*d)*(3*A*b^(3/2)*d^2 + a^(3/2)*d^2*D - 3*a*Sqrt[b]*d*(C*d 
- 2*c*D) + Sqrt[a]*b*(-6*c*C*d + 3*B*d^2 + 8*c^2*D))*Sqrt[(d*(Sqrt[a]/Sqrt 
[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d 
*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x] 
], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(3*b*d^4*Sqrt[-c + ( 
Sqrt[a]*d)/Sqrt[b]]*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2182, 27, 2185, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {\frac {\left (b c^2-a d^2\right ) \left (a d^2 D-b \left (-3 B d^2-8 c^2 D+6 c C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \left (a d^2 (3 C d-5 c D)-b \left (3 A d^3-3 B c d^2-8 c^3 D+6 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}}{3 b d^2}+\frac {2}{3} D \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {\frac {\left (b c^2-a d^2\right ) \left (a d^2 D-b \left (-3 B d^2-8 c^2 D+6 c C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \sqrt {1-\frac {b x^2}{a}} \left (a d^2 (3 C d-5 c D)-b \left (3 A d^3-3 B c d^2-8 c^3 D+6 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}}{3 b d^2}+\frac {2}{3} D \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (3 C d-5 c D)-b \left (3 A d^3-3 B c d^2-8 c^3 D+6 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {\left (b c^2-a d^2\right ) \left (a d^2 D-b \left (-3 B d^2-8 c^2 D+6 c C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b d^2}+\frac {2}{3} D \sqrt {a-b x^2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \sqrt {a-b x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((c + d*x)^(3/2)*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])/(d^2*(b*c^2 - a*d^ 
2)*Sqrt[c + d*x]) + ((2*(a/b - c^2/d^2)*D*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 
 + ((2*Sqrt[a]*Sqrt[b]*(a*d^2*(3*C*d - 5*c*D) - b*(6*c^2*C*d - 3*B*c*d^2 + 
 3*A*d^3 - 8*c^3*D))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sq 
rt[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*S 
qrt[a]*(b*c^2 - a*d^2)*(a*d^2*D - b*(6*c*C*d - 3*B*d^2 - 8*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]))/(3*b*d^2))/(b*c^2 - a*d^2 
)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(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. \(787\) vs. \(2(385)=770\).

Time = 5.31 (sec) , antiderivative size = 788, normalized size of antiderivative = 1.76

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d \,x^{2}+a d \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{d^{3} \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}-\frac {2 D \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{2} b}+\frac {2 \left (\frac {B \,d^{2}-C c d +D c^{2}}{d^{3}}-\frac {b c \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{d^{3} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {D a}{3 d b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {C d -D c}{d^{2}}-\frac {b \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{d^{2} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {2 D c}{3 d^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(788\)
default	\(\text {Expression too large to display}\)	\(3564\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, D b^{2} c^{5} - 6 \, C b^{2} c^{4} d - {\left (11 \, D a b - 3 \, B b^{2}\right )} c^{3} d^{2} + 6 \, {\left (2 \, C a b + A b^{2}\right )} c^{2} d^{3} - 3 \, {\left (D a^{2} + 3 \, B a b\right )} c d^{4} + {\left (8 \, D b^{2} c^{4} d - 6 \, C b^{2} c^{3} d^{2} - {\left (11 \, D a b - 3 \, B b^{2}\right )} c^{2} d^{3} + 6 \, {\left (2 \, C a b + A b^{2}\right )} c d^{4} - 3 \, {\left (D a^{2} + 3 \, B a b\right )} d^{5}\right )} x\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, D b^{2} c^{4} d - 6 \, C b^{2} c^{3} d^{2} - {\left (5 \, D a b - 3 \, B b^{2}\right )} c^{2} d^{3} + 3 \, {\left (C a b - A b^{2}\right )} c d^{4} + {\left (8 \, D b^{2} c^{3} d^{2} - 6 \, C b^{2} c^{2} d^{3} - {\left (5 \, D a b - 3 \, B b^{2}\right )} c d^{4} + 3 \, {\left (C a b - A b^{2}\right )} d^{5}\right )} x\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (4 \, D b^{2} c^{3} d^{2} - 3 \, C b^{2} c^{2} d^{3} - 3 \, A b^{2} d^{5} - {\left (D a b - 3 \, B b^{2}\right )} c d^{4} + {\left (D b^{2} c^{2} d^{3} - D a b d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{9 \, {\left (b^{3} c^{3} d^{4} - a b^{2} c d^{6} + {\left (b^{3} c^{2} d^{5} - a b^{2} d^{7}\right )} x\right )}} \] Input:

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

Output:

-2/9*((8*D*b^2*c^5 - 6*C*b^2*c^4*d - (11*D*a*b - 3*B*b^2)*c^3*d^2 + 6*(2*C 
*a*b + A*b^2)*c^2*d^3 - 3*(D*a^2 + 3*B*a*b)*c*d^4 + (8*D*b^2*c^4*d - 6*C*b 
^2*c^3*d^2 - (11*D*a*b - 3*B*b^2)*c^2*d^3 + 6*(2*C*a*b + A*b^2)*c*d^4 - 3* 
(D*a^2 + 3*B*a*b)*d^5)*x)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a* 
d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 3*(8 
*D*b^2*c^4*d - 6*C*b^2*c^3*d^2 - (5*D*a*b - 3*B*b^2)*c^2*d^3 + 3*(C*a*b - 
A*b^2)*c*d^4 + (8*D*b^2*c^3*d^2 - 6*C*b^2*c^2*d^3 - (5*D*a*b - 3*B*b^2)*c* 
d^4 + 3*(C*a*b - A*b^2)*d^5)*x)*sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3* 
a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3 
*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x 
+ c)/d)) + 3*(4*D*b^2*c^3*d^2 - 3*C*b^2*c^2*d^3 - 3*A*b^2*d^5 - (D*a*b - 3 
*B*b^2)*c*d^4 + (D*b^2*c^2*d^3 - D*a*b*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + 
 c))/(b^3*c^3*d^4 - a*b^2*c*d^6 + (b^3*c^2*d^5 - a*b^2*d^7)*x)
 

Sympy [F]

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

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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