\(\int \frac {x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx\) [69]

Optimal result

Integrand size = 36, antiderivative size = 1363 \[ \int \frac {x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3222\) vs. \(2(1363)=2726\).

Time = 35.51 (sec) , antiderivative size = 3222, normalized size of antiderivative = 2.36 \[ \int \frac {x^2}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

(-2*d^2*Sqrt[f + g*x]*(a + b*x + c*x^2))/((c*d^2 - b*d*e + a*e^2)*(e*f - d 
*g)*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)]) + (Sqrt[a + b*x + c*x^2]*((-2*d^2 
*(d + e*x)^(5/2)*(c + (c*d^2)/(d + e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/ 
(d + e*x)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x))*(g + (e*f)/(d + e*x) - 
(d*g)/(d + e*x)))/(Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b* 
d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2]*Sqrt[f + ((d + e*x)*(g - 
 (d*g)/(d + e*x)))/e]) - (2*(c*d^2 - b*d*e + a*e^2)*(-(e*f) + d*g)*(d + e* 
x)^(3/2)*Sqrt[(c + (c*d^2)/(d + e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/(d 
+ e*x)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x))*(g + (e*f)/(d + e*x) - (d* 
g)/(d + e*x))]*((-2*d*(-1/2*(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(c*d 
^2 - b*d*e + a*e^2) + (d + e*x)^(-1))*Sqrt[(-1/2*(2*c*d - b*e + Sqrt[b^2*e 
^2 - 4*a*c*e^2])/(c*d^2 - b*d*e + a*e^2) + (d + e*x)^(-1))/((2*c*d - b*e - 
 Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*(c*d^2 - b*d*e + a*e^2)) - (2*c*d - b*e + S 
qrt[b^2*e^2 - 4*a*c*e^2])/(2*(c*d^2 - b*d*e + a*e^2)))]*Sqrt[(-(g/(-(e*f) 
+ d*g)) + (d + e*x)^(-1))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*(c 
*d^2 - b*d*e + a*e^2)) - g/(-(e*f) + d*g))]*EllipticF[ArcSin[Sqrt[(-2*c*d 
+ b*e + Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*d^2)/(d + e*x) - (2*b*d*e)/(d + e*x 
) + (2*a*e^2)/(d + e*x))/Sqrt[(b^2 - 4*a*c)*e^2]]/Sqrt[2]], (2*Sqrt[(b^2 - 
 4*a*c)*e^2]*(e*f - d*g))/(-2*c*d*e*f + b*e^2*f + e*Sqrt[(b^2 - 4*a*c)*e^2 
]*f + b*d*e*g - 2*a*e^2*g - d*Sqrt[(b^2 - 4*a*c)*e^2]*g)])/(Sqrt[(-1/2*...
 

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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 1281

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

\(\Big \downarrow \) 1280

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

\(\Big \downarrow \) 1292

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1276

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

\(\Big \downarrow \) 1280

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

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\left (\frac {g \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 {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}\right ) d^2}{e^2}+\frac {2 \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 ) d}{e^2 \sqrt [4]{c d^2-b e d+a e^2} (e f-d g) \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 {\sqrt {2} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {\frac {(e f-d g) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt {b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g},\arcsin \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right ),\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{e^2 \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} g \sqrt {x c+\frac {2 a c}{b+\sqrt {b^2-4 a c}}} \sqrt {c x^2+b x+a}}\)

Input:

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

Output:

$Aborted
 

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 1276

Int[Sqrt[(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 
[2]*Sqrt[2*c*f - g*(b + q)]*Sqrt[b - q + 2*c*x]*(d + e*x)*Sqrt[(e*f - d*g)* 
((b + q + 2*c*x)/((2*c*f - g*(b + q))*(d + e*x)))]*(Sqrt[(e*f - d*g)*((2*a 
+ (b + q)*x)/((b*f + q*f - 2*a*g)*(d + e*x)))]/(g*Sqrt[2*c*d - e*(b + q)]*S 
qrt[2*a*(c/(b + q)) + c*x]*Sqrt[a + b*x + c*x^2]))*EllipticPi[e*((2*c*f - g 
*(b + q))/(g*(2*c*d - e*(b + q)))), ArcSin[Sqrt[2*c*d - e*(b + q)]*(Sqrt[f 
+ g*x]/(Sqrt[2*c*f - g*(b + q)]*Sqrt[d + e*x]))], (b*d + q*d - 2*a*e)*((2*c 
*f - g*(b + q))/((b*f + q*f - 2*a*g)*(2*c*d - e*(b + q))))], x]] /; FreeQ[{ 
a, b, c, d, e, f, g}, 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 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. \(3125\) vs. \(2(1222)=2444\).

Time = 434.05 (sec) , antiderivative size = 3126, normalized size of antiderivative = 2.29

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3126\)
default	\(\text {Expression too large to display}\)	\(104836\)

Input:

int(x^2/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

((g*x+f)*(c*x^2+b*x+a)*(e*x+d))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)/(e 
*x+d)^(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*d^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*(-d/e^2+1/e^2*(a* 
e^2*g-b*d*e*g+b*e^2*f+c*d^2*g-c*d*e*f)*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)-(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*d^2)*(-f/g+d/e)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^( 
1/2)))*(x+f/g)/(f/g-d/e)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(x-1/2/c 
*(-b+(-4*a*c+b^2)^(1/2)))^2*((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+f/g)*(x+1/2*(b 
+(-4*a*c+b^2)^(1/2))/c)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+f/g)/(x-1/2/c*(-b+( 
-4*a*c+b^2)^(1/2))))^(1/2)*((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+f/g)*(x+d/e)/(f 
/g-d/e)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)/(-d/e-1/2/c*(-b+(-4*a*c+b 
^2)^(1/2)))/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+f/g)/(c*e*g*(x+f/g)*(x-1/2/c*(- 
b+(-4*a*c+b^2)^(1/2)))*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*(x+d/e))^(1/2)*Ell 
ipticF(((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+f/g)/(f/g-d/e)/(x-1/2/c*(- 
b+(-4*a*c+b^2)^(1/2))))^(1/2),((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a 
*c+b^2)^(1/2))/c)*(-f/g+d/e)/(1/2*(b+(-4*a*c+b^2)^(1/2))/c-f/g)/(d/e+1/2/c 
*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*(1/e+1/e*(b*e*g-c*d*g+c*e*f)*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)-(2*b*e*g+2*c*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*d^2)*(-f/g+d/e...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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