\(\int \frac {A+B x+C x^2}{\sqrt {d+e x} (f+g x)^2 \sqrt {a+b x+c x^2}} \, dx\) [61]

Optimal result

Integrand size = 41, antiderivative size = 916 \[ \int \frac {A+B x+C x^2}{\sqrt {d+e x} (f+g x)^2 \sqrt {a+b x+c x^2}} \, dx=\frac {\left (C f^2-g (B f-A g)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{(e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)}-\frac {\sqrt {b^2-4 a c} \left (C f^2-g (B f-A g)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} g (e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 C g (b f-a g)-c \left (C f^2+g (B f-A g)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c g^2 \left (c f^2-b f g+a g^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c \left (C f^3 (e f-2 d g)+f g \left (B e f^2-A g (3 e f-2 d g)\right )\right )-g \left (b \left (C f^2 (2 e f-3 d g)+g^2 (B d f-2 A e f+A d g)\right )-a g (C f (3 e f-4 d g)-g (B e f-2 B d g+A e g))\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{g^2 \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

(C*f^2-g*(-A*g+B*f))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/(-d*g+e*f)/(a*g^2-b 
*f*g+c*f^2)/(g*x+f)-1/2*(-4*a*c+b^2)^(1/2)*(C*f^2-g*(-A*g+B*f))*(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))*2^(1/2)/g/(-d*g+e*f)/(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*x^2+b*x+a)^(1/2)-2^(1/2)*(-4*a*c 
+b^2)^(1/2)*(2*C*g*(-a*g+b*f)-c*(C*f^2+g*(-A*g+B*f)))*(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)*Ellip 
ticF(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/g^2/(a*g^2-b*f*g+c*f^2)/ 
(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)-2*2^(1/2)*(-4*a*c+b^2)^(1/2)*(c*(C*f^3*( 
-2*d*g+e*f)+f*g*(B*e*f^2-A*g*(-2*d*g+3*e*f)))-g*(b*(C*f^2*(-3*d*g+2*e*f)+g 
^2*(A*d*g-2*A*e*f+B*d*f))-a*g*(C*f*(-4*d*g+3*e*f)-g*(A*e*g-2*B*d*g+B*e*f)) 
))*(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/(2*c*f-(b+(-4*a 
*c+b^2)^(1/2))*g)/(-d*g+e*f)/(a*g^2-b*f*g+c*f^2)/(e*x+d)^(1/2)/(c*x^2+b*x+ 
a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 39.07 (sec) , antiderivative size = 32746, normalized size of antiderivative = 35.75 \[ \int \frac {A+B x+C x^2}{\sqrt {d+e x} (f+g x)^2 \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 5.63 (sec) , antiderivative size = 1534, normalized size of antiderivative = 1.67, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.390, Rules used = {2154, 1282, 2154, 27, 1172, 321, 1269, 1172, 321, 327, 1279, 187, 25, 413, 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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 1282

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 1279

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

\(\Big \downarrow \) 187

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 187

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] &&  !SimplerQ[e + f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*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 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 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 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 1279

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b 
 - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2])   Int[1/((d + e*x 
)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g}, x]
 

rule 1282

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x 
]*(Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))   Int[((d + e*x)^ 
(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[2*d*(c*e*f - c*d*g + b* 
e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*(c*f + b* 
g)*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, 
f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
 

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 [A] (verified)

Time = 47.53 (sec) , antiderivative size = 1508, normalized size of antiderivative = 1.65

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1508\)
default	\(\text {Expression too large to display}\)	\(50098\)

Input:

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

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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