\(\int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx\) [6]

Optimal result

Integrand size = 43, antiderivative size = 1780 \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] 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)/(g*x+f)^(3/2)+2/3*g*(b*(C*d*f*(3*d*g+e*f)-e*g*(-A*d*g-3*A*e 
*f+4*B*d*f))-c*(4*C*d^2*f^2-B*d*f*(d*g+3*e*f)+A*(d^2*g^2+3*e^2*f^2))-a*(C* 
(3*d^2*g^2+e^2*f^2)+e*g*(4*A*e*g-B*(3*d*g+e*f))))*(e*x+d)^(1/2)*(c*x^2+b*x 
+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(-d*g+e*f)^2/(c*f^2-g*(-a*g+b*f))/(g*x+f)^(3 
/2)+2/3*c^(1/2)*(-4*a*c+b^2)^(1/4)*(c^2*f*(2*C*d^2*f^2*(d*g+3*e*f)-B*d*f*( 
-d^2*g^2+6*d*e*f*g+3*e^2*f^2)+A*(-4*d^3*g^3+9*d^2*e*f*g^2+3*e^3*f^3))+c*(b 
*B*d*f*g*(d^2*g^2+3*d*e*f*g+12*e^2*f^2)-b*C*d*f^2*(4*d^2*g^2+9*d*e*f*g+3*e 
^2*f^2)-A*b*g*(-2*d^3*g^3+3*d^2*e*f*g^2+9*d*e^2*f^2*g+6*e^3*f^3)+a*C*f*(6* 
d^3*g^3+5*d^2*e*f*g^2+2*d*e^2*f^2*g+3*e^3*f^3)+a*g*(A*e*g*(5*d^2*g^2-4*d*e 
*f*g+15*e^2*f^2)-B*(3*d^3*g^3+2*d^2*e*f*g^2+5*d*e^2*f^2*g+6*e^3*f^3)))-e*g 
*(a^2*g*(2*e*g*(-4*A*e*g+3*B*d*g+B*e*f)+C*(-3*d^2*g^2-6*d*e*f*g+e^2*f^2))- 
b^2*(C*d*f^2*(7*d*g+e*f)-g*(B*d*f*(d*g+7*e*f)-A*(-2*d^2*g^2+7*d*e*f*g+3*e^ 
2*f^2)))+a*b*(C*f*(12*d^2*g^2+3*d*e*f*g+e^2*f^2)+g*(A*e*g*(3*d*g+13*e*f)-B 
*(3*d^2*g^2+9*d*e*f*g+4*e^2*f^2)))))*(b-(-4*a*c+b^2)^(1/2)+2*c*x)^(1/2)*(e 
*x+d)^(1/2)*((2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)*(b+(-4*a*c+b^2)^(1/2)+2*c*x) 
/c/(-4*a*c+b^2)^(1/2)/(g*x+f))^(1/2)*EllipticE(1/2*(-2*c*f+(b+(-4*a*c+b^2) 
^(1/2))*g)^(1/2)*(b-(-4*a*c+b^2)^(1/2)+2*c*x)^(1/2)/c^(1/2)/(-4*a*c+b^2)^( 
1/4)/(g*x+f)^(1/2),2*(c*(-4*a*c+b^2)^(1/2)*(-d*g+e*f)/(2*c*d-(b-(-4*a*c+b^ 
2)^(1/2))*e)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g))^(1/2))/(a*e^2-b*d*e+c*d^...
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(107718\) vs. \(2(1780)=3560\).

Time = 52.98 (sec) , antiderivative size = 107718, normalized size of antiderivative = 60.52 \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x + C*x^2)/((d + e*x)^(3/2)*(f + g*x)^(5/2)*Sqrt[a + b*x 
+ c*x^2]),x]
 

Output:

Result too large to show
 

Rubi [F]

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.

\(\displaystyle \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 2154

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx+\int \frac {\frac {B}{e}+\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 1292

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx+\int \frac {\frac {B}{e}+\frac {C x}{e}-\frac {C d}{e^2}}{\sqrt {d+e x} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 2154

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx-\frac {(2 C d-B e) \int \frac {1}{\sqrt {d+e x} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx}{e^2}+\int \frac {C \sqrt {d+e x}}{e^2 (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx-\frac {(2 C d-B e) \int \frac {1}{\sqrt {d+e x} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx}{e^2}+\frac {C \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx}{e^2}\)

\(\Big \downarrow \) 1282

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx-\frac {(2 C d-B e) \left (\frac {\int \frac {3 c f (e f-d g)-g (3 b e f-2 b d g-2 a e g)-g (3 c e f-c d g-b e g) x}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac {2 g^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} (e f-d g) \left (a g^2-b f g+c f^2\right )}\right )}{e^2}+\frac {C \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx}{e^2}\)

\(\Big \downarrow \) 1285

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx-\frac {(2 C d-B e) \left (\frac {\int \frac {3 c f (e f-d g)-g (3 b e f-2 b d g-2 a e g)-g (3 c e f-c d g-b e g) x}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac {2 g^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} (e f-d g) \left (a g^2-b f g+c f^2\right )}\right )}{e^2}+\frac {C \left (-\frac {\int -\frac {3 c d f-2 b d g+a e g+(3 c e f-c d g-b e g) x}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a g^2-b f g+c f^2\right )}-\frac {2 g \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} \left (a g^2-b f g+c f^2\right )}\right )}{e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx-\frac {(2 C d-B e) \left (\frac {\int \frac {3 c f (e f-d g)-g (3 b e f-2 b d g-2 a e g)-g (3 c e f-c d g-b e g) x}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac {2 g^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} (e f-d g) \left (a g^2-b f g+c f^2\right )}\right )}{e^2}+\frac {C \left (\frac {\int \frac {3 c d f-2 b d g+a e g+(3 c e f-c d g-b e g) x}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a g^2-b f g+c f^2\right )}-\frac {2 g \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} \left (a g^2-b f g+c f^2\right )}\right )}{e^2}\)

\(\Big \downarrow \) 2154

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx-\frac {(2 C d-B e) \left (\frac {\int \frac {\left (b g^2+\frac {c d g^2}{e}-3 c f g\right ) \sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-\left (g (-2 a e g-b d g+3 b e f)-c \left (3 e f^2-\frac {d^2 g^2}{e}\right )\right ) \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac {2 g^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} (e f-d g) \left (a g^2-b f g+c f^2\right )}\right )}{e^2}+\frac {C \left (\frac {\int \frac {\left (3 c f-b g-\frac {c d g}{e}\right ) \sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-g \left (-a e+b d-\frac {c d^2}{e}\right ) \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a g^2-b f g+c f^2\right )}-\frac {2 g \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} \left (a g^2-b f g+c f^2\right )}\right )}{e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx-\frac {(2 C d-B e) \left (\frac {-\left (\left (g (-2 a e g-b d g+3 b e f)-c \left (3 e f^2-\frac {d^2 g^2}{e}\right )\right ) \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx\right )-g \left (-b g-\frac {c d g}{e}+3 c f\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac {2 g^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} (e f-d g) \left (a g^2-b f g+c f^2\right )}\right )}{e^2}+\frac {C \left (\frac {\left (-b g-\frac {c d g}{e}+3 c f\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-g \left (-a e+b d-\frac {c d^2}{e}\right ) \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a g^2-b f g+c f^2\right )}-\frac {2 g \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} \left (a g^2-b f g+c f^2\right )}\right )}{e^2}\)

\(\Big \downarrow \) 1281

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx-\frac {(2 C d-B e) \left (\frac {-\left (g (-2 a e g-b d g+3 b e f)-c \left (3 e f^2-\frac {d^2 g^2}{e}\right )\right ) \left (\frac {e \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e f-d g}-\frac {g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )-g \left (-b g-\frac {c d g}{e}+3 c f\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac {2 g^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} (e f-d g) \left (a g^2-b f g+c f^2\right )}\right )}{e^2}+\frac {C \left (\frac {\left (-b g-\frac {c d g}{e}+3 c f\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-g \left (-a e+b d-\frac {c d^2}{e}\right ) \left (\frac {e \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{e f-d g}-\frac {g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )}{3 \left (a g^2-b f g+c f^2\right )}-\frac {2 g \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 (f+g x)^{3/2} \left (a g^2-b f g+c f^2\right )}\right )}{e^2}\)

\(\Big \downarrow \) 1280

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx+\frac {C \left (\frac {\left (3 c f-b g-\frac {c d g}{e}\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-\left (-\frac {c d^2}{e}+b d-a e\right ) g \left (-\frac {g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}-\frac {2 e (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{(e f-d g)^2 \sqrt {c x^2+b x+a}}\right )}{3 \left (c f^2-b g f+a g^2\right )}-\frac {2 g \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}}\right )}{e^2}-\frac {(2 C d-B e) \left (\frac {2 \sqrt {d+e x} \sqrt {c x^2+b x+a} g^2}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}}+\frac {-g \left (3 c f-b g-\frac {c d g}{e}\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-\left (g (3 b e f-b d g-2 a e g)-c \left (3 e f^2-\frac {d^2 g^2}{e}\right )\right ) \left (-\frac {g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}-\frac {2 e (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{(e f-d g)^2 \sqrt {c x^2+b x+a}}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )}\right )}{e^2}\)

\(\Big \downarrow \) 1292

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx+\frac {C \left (\frac {\left (3 c f-b g-\frac {c d g}{e}\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-\left (-\frac {c d^2}{e}+b d-a e\right ) g \left (-\frac {g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}-\frac {2 e (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{(e f-d g)^2 \sqrt {c x^2+b x+a}}\right )}{3 \left (c f^2-b g f+a g^2\right )}-\frac {2 g \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}}\right )}{e^2}-\frac {(2 C d-B e) \left (\frac {2 \sqrt {d+e x} \sqrt {c x^2+b x+a} g^2}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}}+\frac {-g \left (3 c f-b g-\frac {c d g}{e}\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-\left (g (3 b e f-b d g-2 a e g)-c \left (3 e f^2-\frac {d^2 g^2}{e}\right )\right ) \left (-\frac {g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}-\frac {2 e (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \int \frac {1}{\sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{(e f-d g)^2 \sqrt {c x^2+b x+a}}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )}\right )}{e^2}\)

\(\Big \downarrow \) 1416

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {c x^2+b x+a}}dx+\frac {C \left (\frac {\left (3 c f-b g-\frac {c d g}{e}\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-\left (-\frac {c d^2}{e}+b d-a e\right ) g \left (-\frac {e \sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt [4]{c d^2-b e d+a e^2} (e f-d g)^2 \sqrt {c x^2+b x+a} \sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}-\frac {g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )}{3 \left (c f^2-b g f+a g^2\right )}-\frac {2 g \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}}\right )}{e^2}-\frac {(2 C d-B e) \left (\frac {2 \sqrt {d+e x} \sqrt {c x^2+b x+a} g^2}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}}+\frac {-g \left (3 c f-b g-\frac {c d g}{e}\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx-\left (g (3 b e f-b d g-2 a e g)-c \left (3 e f^2-\frac {d^2 g^2}{e}\right )\right ) \left (-\frac {e \sqrt [4]{c f^2-g (b f-a g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (c x^2+b x+a\right )}{\left (c f^2-b g f+a g^2\right ) (d+e x)^2}} \left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right ) \sqrt {\frac {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}{\left (\frac {\sqrt {c d^2-b e d+a e^2} (f+g x)}{\sqrt {c f^2-g (b f-a g)} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt [4]{c d^2-b e d+a e^2} (e f-d g)^2 \sqrt {c x^2+b x+a} \sqrt {\frac {\left (c d^2-b e d+a e^2\right ) (f+g x)^2}{\left (c f^2-g (b f-a g)\right ) (d+e x)^2}-\frac {(2 c d f+2 a e g-b (e f+d g)) (f+g x)}{\left (c f^2-b g f+a g^2\right ) (d+e x)}+1}}-\frac {g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )}\right )}{e^2}\)

Input:

Int[(A + B*x + C*x^2)/((d + e*x)^(3/2)*(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^ 
2]),x]
 

Output:

$Aborted
 

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 1280

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

rule 1281

Int[1/(((d_.) + (e_.)*(x_))^(3/2)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_ 
.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-g/(e*f - d*g)   Int[1/(Sqrt[d 
+ e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] + Simp[e/(e*f - d*g) 
Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]), 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 1285

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_.) + (b_.) 
*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*Sqrt[f + g*x]* 
(Sqrt[a + b*x + c*x^2]/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*( 
m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqr 
t[a + b*x + c*x^2]))*Simp[2*c*d*f*(m + 1) - e*(a*g + b*f*(2*m + 3)) - 2*(b* 
e*g*(2 + m) - c*(d*g*(m + 1) - e*f*(m + 2)))*x - c*e*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 1292

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* 
(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

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] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4997\) vs. \(2(1665)=3330\).

Time = 33.69 (sec) , antiderivative size = 4998, normalized size of antiderivative = 2.81

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(4998\)
default	\(\text {Expression too large to display}\)	\(1786054\)

Input:

int((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(5/2)/(c*x^2+b*x+a)^(1/2),x,method 
=_RETURNVERBOSE)
 

Output:

((g*x+f)*(c*x^2+b*x+a)*(e*x+d))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(1/2)/(c*x^2+b 
*x+a)^(1/2)*(2*(c*e*g*x^3+b*e*g*x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f)/(a*d* 
e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)*e*(A*e^2-B*d*e+C*d^2) 
/(d*g-e*f)^2/((x+d/e)*(c*e*g*x^3+b*e*g*x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f 
))^(1/2)-2/3/(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*( 
A*g^2-B*f*g+C*f^2)/(d*g-e*f)*(c*e*g*x^4+b*e*g*x^3+c*d*g*x^3+c*e*f*x^3+a*e* 
g*x^2+b*d*g*x^2+b*e*f*x^2+c*d*f*x^2+a*d*g*x+a*e*f*x+b*d*f*x+a*d*f)^(1/2)/( 
x+f/g)^2+2/3*(c*e*g*x^3+b*e*g*x^2+c*d*g*x^2+a*e*g*x+b*d*g*x+a*d*g)/(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)^2*(5*A*a*e*g^4+2*A*b*d* 
g^4-7*A*b*e*f*g^3-4*A*c*d*f*g^3+9*A*c*e*f^2*g^2-3*B*a*d*g^4-2*B*a*e*f*g^3+ 
B*b*d*f*g^3+4*B*b*e*f^2*g^2+B*c*d*f^2*g^2-6*B*c*e*f^3*g+6*C*a*d*f*g^3-C*a* 
e*f^2*g^2-4*C*b*d*f^2*g^2-C*b*e*f^3*g+2*C*c*d*f^3*g+3*C*c*e*f^4)/(d*g-e*f) 
/((x+f/g)*(c*e*g*x^3+b*e*g*x^2+c*d*g*x^2+a*e*g*x+b*d*g*x+a*d*g))^(1/2)+2*( 
(a*e^2*g-b*d*e*g+b*e^2*f+c*d^2*g-c*d*e*f)*(A*e^2-B*d*e+C*d^2)/(a*d*e^2*g-a 
*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)/(d*g-e*f)^2-(a*e*g+b*e*f)/(a 
*d*e^2*g-a*e^3*f-b*d^2*e*g+b*d*e^2*f+c*d^3*g-c*d^2*e*f)*e*(A*e^2-B*d*e+C*d 
^2)/(d*g-e*f)^2-1/3*(A*b*e*g^3+A*c*d*g^3-3*A*c*e*f*g^2-B*b*e*f*g^2-B*c*d*f 
*g^2+3*B*c*e*f^2*g+C*b*e*f^2*g+C*c*d*f^2*g-3*C*c*e*f^3)/g/(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*g-e*f)+1/3/g*(a*e*g^2+b*d*g^2 
-b*e*f*g-c*d*f*g+c*e*f^2)*(5*A*a*e*g^4+2*A*b*d*g^4-7*A*b*e*f*g^3-4*A*c*...
 

Fricas [F]

\[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(5/2)/(c*x^2+b*x+a)^(1/2),x, 
 algorithm="fricas")
 

Output:

integral((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*sqrt(g*x + 
f)/(c*e^2*g^3*x^7 + (3*c*e^2*f*g^2 + (2*c*d*e + b*e^2)*g^3)*x^6 + a*d^2*f^ 
3 + (3*c*e^2*f^2*g + 3*(2*c*d*e + b*e^2)*f*g^2 + (c*d^2 + 2*b*d*e + a*e^2) 
*g^3)*x^5 + (c*e^2*f^3 + 3*(2*c*d*e + b*e^2)*f^2*g + 3*(c*d^2 + 2*b*d*e + 
a*e^2)*f*g^2 + (b*d^2 + 2*a*d*e)*g^3)*x^4 + (a*d^2*g^3 + (2*c*d*e + b*e^2) 
*f^3 + 3*(c*d^2 + 2*b*d*e + a*e^2)*f^2*g + 3*(b*d^2 + 2*a*d*e)*f*g^2)*x^3 
+ (3*a*d^2*f*g^2 + (c*d^2 + 2*b*d*e + a*e^2)*f^3 + 3*(b*d^2 + 2*a*d*e)*f^2 
*g)*x^2 + (3*a*d^2*f^2*g + (b*d^2 + 2*a*d*e)*f^3)*x), x)
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x + C x^{2}}{\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{\frac {5}{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((C*x**2+B*x+A)/(e*x+d)**(3/2)/(g*x+f)**(5/2)/(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)**(5/2)*sqrt(a + b* 
x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(5/2)/(c*x^2+b*x+a)^(1/2),x, 
 algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)*(g*x + 
f)^(5/2)), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(5/2)/(c*x^2+b*x+a)^(1/2),x, 
 algorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)*(g*x + 
f)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {C\,x^2+B\,x+A}{{\left (f+g\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int((A + B*x + C*x^2)/((f + g*x)^(5/2)*(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)^(5/2)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^( 
1/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {C \,x^{2}+B x +A}{\left (e x +d \right )^{\frac {3}{2}} \left (g x +f \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}d x \] Input:

int((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(5/2)/(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)^(5/2)/(c*x^2+b*x+a)^(1/2),x)