\(\int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx\) [10]

Optimal result

Integrand size = 41, antiderivative size = 748 \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (C d^2-e (B d-A e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e \left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} C \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e g \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (C f^2-B f g+A g^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{g \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (e f-d g) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.40 (sec) , antiderivative size = 980, normalized size of antiderivative = 1.31 \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (C d^2+e (-B d+A e)\right ) g (-e f+d g) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-g \left (4 a C d e^3 f-2 a B e^4 f-C d^2 e \sqrt {\left (b^2-4 a c\right ) e^2} f+B d e^2 \sqrt {\left (b^2-4 a c\right ) e^2} f-A e^3 \sqrt {\left (b^2-4 a c\right ) e^2} f-2 a C d^2 e^2 g+2 a A e^4 g+C d^3 \sqrt {\left (b^2-4 a c\right ) e^2} g-B d^2 e \sqrt {\left (b^2-4 a c\right ) e^2} g+A d e^2 \sqrt {\left (b^2-4 a c\right ) e^2} g+2 c d e \left (C d^2 f-B d^2 g+A e (-e f+2 d g)\right )+b e \left (C d^2 (-3 e f+d g)+e (A e (e f-3 d g)+B d (e f+d g))\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+2 e^2 \left (c d^2+e (-b d+a e)\right ) \left (C f^2+g (-B f+A g)\right ) \operatorname {EllipticPi}\left (\frac {\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (e f-d g)}{2 \left (c d^2+e (-b d+a e)\right ) g},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 e^2 \left (c d^2+e (-b d+a e)\right ) \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} g (e f-d g)^2 \sqrt {a+x (b+c x)}} \] Input:

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

Output:

((I/2)*(d + e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + S 
qrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)) 
)/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sq 
rt[(b^2 - 4*a*c)*e^2])*(C*d^2 + e*(-(B*d) + A*e))*g*(-(e*f) + d*g)*Ellipti 
cE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b 
^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e 
^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - g*(4*a*C*d*e^3*f - 2*a*B* 
e^4*f - C*d^2*e*Sqrt[(b^2 - 4*a*c)*e^2]*f + B*d*e^2*Sqrt[(b^2 - 4*a*c)*e^2 
]*f - A*e^3*Sqrt[(b^2 - 4*a*c)*e^2]*f - 2*a*C*d^2*e^2*g + 2*a*A*e^4*g + C* 
d^3*Sqrt[(b^2 - 4*a*c)*e^2]*g - B*d^2*e*Sqrt[(b^2 - 4*a*c)*e^2]*g + A*d*e^ 
2*Sqrt[(b^2 - 4*a*c)*e^2]*g + 2*c*d*e*(C*d^2*f - B*d^2*g + A*e*(-(e*f) + 2 
*d*g)) + b*e*(C*d^2*(-3*e*f + d*g) + e*(A*e*(e*f - 3*d*g) + B*d*(e*f + d*g 
))))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b 
*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^ 
2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + 2*e^2*(c*d^2 
+ e*(-(b*d) + a*e))*(C*f^2 + g*(-(B*f) + A*g))*EllipticPi[((-2*c*d + b*e + 
 Sqrt[(b^2 - 4*a*c)*e^2])*(e*f - d*g))/(2*(c*d^2 + e*(-(b*d) + a*e))*g), I 
*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 
 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2]) 
/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(e^2*(c*d^2 + e*(-(b*d) + ...
 

Rubi [A] (warning: unable to verify)

Time = 4.27 (sec) , antiderivative size = 1065, normalized size of antiderivative = 1.42, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {2154, 1237, 27, 1269, 1172, 321, 327, 1288, 2009}

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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (-B e g+C d g+C e f) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} C g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{g^2 \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {a+b x+c x^2} (-B e g+C d g+C e f)}{g^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (-B e g+C d g+C e f) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} C g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{g^2 \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {a+b x+c x^2} (-B e g+C d g+C e f)}{g^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (-B e g+C d g+C e f) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} C g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{g^2 \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {a+b x+c x^2} (-B e g+C d g+C e f)}{g^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1288

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \left (\frac {e}{(e f-d g) (d+e x)^{3/2} \sqrt {c x^2+b x+a}}-\frac {g}{(e f-d g) \sqrt {d+e x} (f+g x) \sqrt {c x^2+b x+a}}\right )dx-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (-B e g+C d g+C e f) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} C g \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{g^2 \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {a+b x+c x^2} (-B e g+C d g+C e f)}{g^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {c x^2+b x+a} (C e f+C d g-B e g)}{\left (c d^2-b e d+a e^2\right ) g^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} (C e f+C d g-B e g) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} C \left (c d^2-b e d+a e^2\right ) g \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{\left (c d^2-b e d+a e^2\right ) g^2}+\left (A+\frac {f (C f-B g)}{g^2}\right ) \left (-\frac {2 \sqrt {c x^2+b x+a} e^2}{\left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) e}{\left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} g \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \operatorname {EllipticPi}\left (-\frac {\left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) g}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}\right )}{\sqrt {c} (e f-d g)^2 \sqrt {c x^2+b x+a}}\right )\)

Input:

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

Output:

(2*(C*e*f + C*d*g - B*e*g)*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2) 
*g^2*Sqrt[d + e*x]) - ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(C*e*f + C*d*g - B*e*g)* 
Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSi 
n[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sq 
rt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(e*Sqrt[(c*(d + e 
*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt 
[2]*Sqrt[b^2 - 4*a*c]*C*(c*d^2 - b*d*e + a*e^2)*g*Sqrt[(c*(d + e*x))/(2*c* 
d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c) 
)]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c] 
]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]) 
/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]))/((c*d^2 - b*d*e + a*e^2)*g^2) 
+ (A + (f*(C*f - B*g))/g^2)*((-2*e^2*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d* 
e + a*e^2)*(e*f - d*g)*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[ 
d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqr 
t[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^ 
2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/((c*d^2 - b*d*e + a*e^ 
2)*(e*f - d*g)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqr 
t[a + b*x + c*x^2]) - (Sqrt[2]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*g*S 
qrt[1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*c 
*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[-1/2*((2*c*...
 

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 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1237

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

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1288

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a 
 + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, b, c, 
d, e, f, g}, x] && IntegerQ[n + 1/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2154

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + 
 e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn 
omialRemainder[Px, d + e*x, x]   Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x 
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, 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. \(1352\) vs. \(2(672)=1344\).

Time = 7.76 (sec) , antiderivative size = 1353, normalized size of antiderivative = 1.81

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1353\)
default	\(\text {Expression too large to display}\)	\(17980\)

Input:

int((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(1/2),x,method=_RETU 
RNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2*(c*e*x^ 
2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)/e*(A*e^2-B*d*e+C*d^2)/(d*g-e*f)/((x+d/e)* 
(c*e*x^2+b*e*x+a*e))^(1/2)+2*(C/e/g+1/e*(b*e-c*d)*(A*e^2-B*d*e+C*d^2)/(d*g 
-e*f)/(a*e^2-b*d*e+c*d^2)-b/(a*e^2-b*d*e+c*d^2)*(A*e^2-B*d*e+C*d^2)/(d*g-e 
*f))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^ 
(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a* 
c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a 
*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*E 
llipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+ 
(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))-2*(A*e 
^2-B*d*e+C*d^2)*c/(d*g-e*f)/(a*e^2-b*d*e+c*d^2)*(d/e-1/2*(b+(-4*a*c+b^2)^( 
1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+ 
(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*( 
b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x 
^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1 
/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+ 
1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)) 
+1/2/c*(-b+(-4*a*c+b^2)^(1/2))*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2) 
^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4 
*a*c+b^2)^(1/2))))^(1/2)))+2*(A*g^2-B*f*g+C*f^2)/(d*g-e*f)/g^2*(d/e-1/2...
 

Fricas [F(-1)]

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

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(1/2),x, algor 
ithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(1/2),x, algor 
ithm="maxima")
 

Output:

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

Giac [F]

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

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(1/2),x, algor 
ithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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