\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx\) [40]

Optimal result

Integrand size = 44, antiderivative size = 581 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx=-\frac {2 D \sqrt {c+d x} \sqrt {a-b x^2}}{3 b d f}+\frac {2 \sqrt {a} (3 d D e-3 C d f+2 c D f) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^2 f^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (3 b d D e^2+a d D f^2-3 b d f (C e-B f)+b c f \left (3 D e-3 C f+\frac {2 c D f}{d}\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d f^3 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} f}{\sqrt {b} e+\sqrt {a} f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \left (\sqrt {b} e+\sqrt {a} f\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2/3*D*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d/f+2/3*a^(1/2)*(-3*C*d*f+2*D*c*f+ 
3*D*d*e)*(d*x+c)^(1/2)*(1-b*x^2/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^2/f^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-2/3 
*a^(1/2)*(3*b*d*D*e^2+a*d*D*f^2-3*b*d*f*(-B*f+C*e)+b*c*f*(3*D*e-3*C*f+2*c* 
D*f/d))*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*El 
lipticF(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/f^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+2*a^(1 
/2)*(D*e^3-f*(C*e^2-f*(-A*f+B*e)))*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d)) 
^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^( 
1/2),2*a^(1/2)*f/(b^(1/2)*e+a^(1/2)*f),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/ 
2)*d))^(1/2))/f^3/(b^(1/2)*e+a^(1/2)*f)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.72 (sec) , antiderivative size = 1922, normalized size of antiderivative = 3.31 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(-2*D*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b*d*f) - (2*(3*b*c^2*d^2*Sqrt[-c + 
 (Sqrt[a]*d)/Sqrt[b]]*D*e^2*f - 3*a*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*D*e 
^2*f - 3*b*c^2*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*e*f^2 + 3*a*C*d^4*Sqrt 
[-c + (Sqrt[a]*d)/Sqrt[b]]*e*f^2 - b*c^3*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]* 
D*e*f^2 + a*c*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*D*e*f^2 + 3*b*c^3*C*d*Sqr 
t[-c + (Sqrt[a]*d)/Sqrt[b]]*f^3 - 3*a*c*C*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b 
]]*f^3 - 2*b*c^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*D*f^3 + 2*a*c^2*d^2*Sqrt[- 
c + (Sqrt[a]*d)/Sqrt[b]]*D*f^3 - 6*b*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]* 
D*e^2*f*(c + d*x) + 6*b*c*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*e*f^2*(c + 
d*x) + 2*b*c^2*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*D*e*f^2*(c + d*x) - 6*b*c^ 
2*C*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*f^3*(c + d*x) + 4*b*c^3*Sqrt[-c + (Sq 
rt[a]*d)/Sqrt[b]]*D*f^3*(c + d*x) + 3*b*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] 
*D*e^2*f*(c + d*x)^2 - 3*b*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*e*f^2*(c + 
 d*x)^2 - b*c*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*D*e*f^2*(c + d*x)^2 + 3*b*c 
*C*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*f^3*(c + d*x)^2 - 2*b*c^2*Sqrt[-c + (S 
qrt[a]*d)/Sqrt[b]]*D*f^3*(c + d*x)^2 + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*f 
*(-(d*e) + c*f)*(3*d*D*e - 3*C*d*f + 2*c*D*f)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x 
))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/ 
2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqr 
t[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*d*f*(a*d*D*f*(-(d*e) +...
 

Rubi [A] (verified)

Time = 2.26 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2349, 731, 186, 413, 412, 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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 731

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

\(\Big \downarrow \) 186

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}\)

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {2 \int -\frac {d \left (d \left (a D f^2+3 b \left (D e^2-f (C e-B f)\right )\right )-b f (3 d D e-3 C d f+2 c D f) x\right )}{2 f^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {d \left (a D f^2+3 b \left (D e^2-f (C e-B f)\right )\right )-b f (3 d D e-3 C d f+2 c D f) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d f^3}+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {\left (d^2 \left (a D f^2+3 b \left (D e^2-f (C e-B f)\right )\right )+b c f (2 c D f-3 C d f+3 d D e)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b f (2 c D f-3 C d f+3 d D e) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}}{3 b d f^3}+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {\left (d^2 \left (a D f^2+3 b \left (D e^2-f (C e-B f)\right )\right )+b c f (2 c D f-3 C d f+3 d D e)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b f \sqrt {1-\frac {b x^2}{a}} (2 c D f-3 C d f+3 d D e) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}}{3 b d f^3}+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {\left (d^2 \left (a D f^2+3 b \left (D e^2-f (C e-B f)\right )\right )+b c f (2 c D f-3 C d f+3 d D e)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {b} f \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 c D f-3 C d f+3 d D e) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b d f^3}+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\left (d^2 \left (a D f^2+3 b \left (D e^2-f (C e-B f)\right )\right )+b c f (2 c D f-3 C d f+3 d D e)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {b} f \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 c D f-3 C d f+3 d D e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b d f^3}+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (d^2 \left (a D f^2+3 b \left (D e^2-f (C e-B f)\right )\right )+b c f (2 c D f-3 C d f+3 d D e)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} f \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 c D f-3 C d f+3 d D e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b d f^3}+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {2 \sqrt {a} \sqrt {b} f \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 c D f-3 C d f+3 d D e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (d^2 \left (a D f^2+3 b \left (D e^2-f (C e-B f)\right )\right )+b c f (2 c D f-3 C d f+3 d D e)\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}}{3 b d f^3}+\frac {2 \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^3 \sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b d f}\)

\(\Big \downarrow \) 321

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

Input:

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

Output:

(-2*D*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b*d*f) + ((2*Sqrt[a]*Sqrt[b]*f*(3* 
d*D*e - 3*C*d*f + 2*c*D*f)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[Arc 
Sin[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*f*(3*d*D*e - 3*C*d*f + 2*c*D*f) + d^2*(a*D*f^2 + 3*b*(D* 
e^2 - f*(C*e - B*f))))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*S 
qrt[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]))/(3*b*d*f^3) + (2*(D*e^3 - f*(C*e^2 - f*(B*e - A*f)))*Sqrt[1 - (b*x^2) 
/a]*Sqrt[1 - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/(Sqrt[b]*c + Sqrt[a]*d) 
]*EllipticPi[(2*f)/((Sqrt[b]*e)/Sqrt[a] + f), ArcSin[Sqrt[1 - (Sqrt[b]*x)/ 
Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(f^3*((Sqrt[b]* 
e)/Sqrt[a] + f)*Sqrt[a - b*x^2]*Sqrt[c + (Sqrt[a]*d)/Sqrt[b] - (Sqrt[a]*d* 
(1 - (Sqrt[b]*x)/Sqrt[a]))/Sqrt[b]])
 

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 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

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 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 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 731

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

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]))
 

rule 2349

Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. 
)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d 
*x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c 
+ d*x, x]   Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] &&  !IntegerQ[n 
] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [A] (verified)

Time = 4.97 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.55

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 D \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 f b d}+\frac {2 \left (\frac {B \,f^{2}-C e f +D e^{2}}{f^{3}}+\frac {D a}{3 f 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 f -D e}{f^{2}}-\frac {2 D c}{3 f d}\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}}+\frac {2 \left (A \,f^{3}-B e \,f^{2}+C \,e^{2} f -D e^{3}\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 {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {e}{f}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{f^{4} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}\, \left (-\frac {c}{d}+\frac {e}{f}\right )}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\)	\(903\)
default	\(\text {Expression too large to display}\)	\(5813\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)/(-b*x^2+a)^(1/2),x,method=_R 
ETURNVERBOSE)
 

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(-2/3*D/f/b/d*(- 
b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*((B*f^2-C*e*f+D*e^2)/f^3+1/3*D/f/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/f^2*(C*f-D*e)-2/3*D/f/d*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)))+2*(A*f^3-B*e*f^2+C*e^2*f-D*e^3)/f^4*(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+e/f)*EllipticPi(((x+c/ 
d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),(-c/d+1/b*(a*b)^(1/2))/(-c/d+e/f),((-c/d+1 
/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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