\(\int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx\) [39]

Optimal result

Integrand size = 39, antiderivative size = 481 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx=-\frac {2 \sqrt {a} C \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} d f \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} (C d e+c C f-B d f) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} d f^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (C e^2-B e f+A f^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} f}{\sqrt {b} e+\sqrt {a} f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f^2 \left (\sqrt {b} e+\sqrt {a} f\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2*a^(1/2)*C*(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/f/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+2*a 
^(1/2)*(-B*d*f+C*c*f+C*d*e)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)* 
(1-b*x^2/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/f^2/(d*x+c)^(1/2)/(-b 
*x^2+a)^(1/2)-2*a^(1/2)*(A*f^2-B*e*f+C*e^2)*(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^2/(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 = 24.52 (sec) , antiderivative size = 1214, normalized size of antiderivative = 2.52 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(-2*(b*c^2*C*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*e*f - a*C*d^3*Sqrt[-c + (Sqr 
t[a]*d)/Sqrt[b]]*e*f - b*c^3*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*f^2 + a*c*C* 
d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*f^2 - 2*b*c*C*d*Sqrt[-c + (Sqrt[a]*d)/S 
qrt[b]]*e*f*(c + d*x) + 2*b*c^2*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*f^2*(c + 
d*x) + b*C*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*e*f*(c + d*x)^2 - b*c*C*Sqrt[- 
c + (Sqrt[a]*d)/Sqrt[b]]*f^2*(c + d*x)^2 - I*Sqrt[b]*C*(Sqrt[b]*c - Sqrt[a 
]*d)*f*(d*e - c*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]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b 
]*c - Sqrt[a]*d)] + I*Sqrt[b]*d*f*(Sqrt[a]*C*(-(d*e) + c*f) + Sqrt[b]*(c*C 
*e - B*c*f + A*d*f))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sq 
rt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sq 
rt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt 
[b]*c - Sqrt[a]*d)] - I*b*C*d^2*e^2*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)*Ellipti 
cPi[(Sqrt[b]*(-(d*e) + c*f))/((Sqrt[b]*c - Sqrt[a]*d)*f), 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)] + I*b*B*d^2*e*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)*EllipticPi[ 
(Sqrt[b]*(-(d*e) + c*f))/((Sqrt[b]*c - Sqrt[a]*d)*f), I*ArcSinh[Sqrt[-c...
 

Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2349, 600, 509, 508, 327, 512, 511, 321, 731, 186, 413, 412}

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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 731

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

\(\Big \downarrow \) 186

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(-2*Sqrt[a]*C*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - 
(Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]* 
d*f*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + ( 
2*Sqrt[a]*(C*d*e + c*C*f - B*d*f)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sq 
rt[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*f^2*Sqrt[c + d*x 
]*Sqrt[a - b*x^2]) - (2*(A + (e*(C*e - B*f))/f^2)*Sqrt[1 - (b*x^2)/a]*Sqrt 
[1 - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/(Sqrt[b]*c + Sqrt[a]*d)]*Ellipt 
icPi[(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)])/(((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 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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(818\) vs. \(2(392)=784\).

Time = 4.87 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.70

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (\frac {2 \left (B f -C e \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 )}{f^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 C \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 )}{f \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (A \,f^{2}-B e f +C \,e^{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}}}\, \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^{3} \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}}\)	\(819\)
default	\(\text {Expression too large to display}\)	\(1606\)

Input:

int((C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)/(-b*x^2+a)^(1/2),x,method=_RETURNV 
ERBOSE)
 

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(2*(B*f-C*e)/f^2 
*(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*C/f*(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^2-B*e*f+C*e^2)/f^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)/(-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. \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

integrate((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)/(sqrt(a - b*x**2)*sqrt(c + d*x)*(e + f*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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