\(\int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx\) [612]

Optimal result

Integrand size = 31, antiderivative size = 724 \[ \int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx=\frac {3 g (3 e f-d g-5 e g) \sqrt {f+g x} \sqrt {6+5 x+x^2}}{4 e \sqrt {d+e x}}+\frac {g^2 \sqrt {d+e x} \sqrt {f+g x} \sqrt {6+5 x+x^2}}{2 e}+\frac {3 \sqrt {d-3 e} \sqrt {f-3 g} g (3 e f-d g-5 e g) (2+x) \sqrt {\frac {(e f-d g) (3+x)}{(f-3 g) (d+e x)}} E\left (\arcsin \left (\frac {\sqrt {d-3 e} \sqrt {f+g x}}{\sqrt {f-3 g} \sqrt {d+e x}}\right )|\frac {(d-2 e) (f-3 g)}{(d-3 e) (f-2 g)}\right )}{4 e^2 \sqrt {\frac {(e f-d g) (2+x)}{(f-2 g) (d+e x)}} \sqrt {6+5 x+x^2}}+\frac {\sqrt {f-3 g} (e f-d g) \left (d e (7 f-4 g) g-3 d^2 g^2-e^2 \left (8 f^2-27 f g+33 g^2\right )\right ) (2+x) \sqrt {\frac {(e f-d g) (3+x)}{(f-3 g) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d-3 e} \sqrt {f+g x}}{\sqrt {f-3 g} \sqrt {d+e x}}\right ),\frac {(d-2 e) (f-3 g)}{(d-3 e) (f-2 g)}\right )}{4 \sqrt {d-3 e} e^3 (f-2 g) \sqrt {\frac {(e f-d g) (2+x)}{(f-2 g) (d+e x)}} \sqrt {6+5 x+x^2}}-\frac {\sqrt {f-3 g} (e f-d g) \left (10 d e (f-g) g-3 d^2 g^2-e^2 \left (15 f^2-50 f g+51 g^2\right )\right ) (2+x) \sqrt {\frac {(e f-d g) (3+x)}{(f-3 g) (d+e x)}} \operatorname {EllipticPi}\left (\frac {e (f-3 g)}{(d-3 e) g},\arcsin \left (\frac {\sqrt {d-3 e} \sqrt {f+g x}}{\sqrt {f-3 g} \sqrt {d+e x}}\right ),\frac {(d-2 e) (f-3 g)}{(d-3 e) (f-2 g)}\right )}{4 \sqrt {d-3 e} e^3 (f-2 g) \sqrt {\frac {(e f-d g) (2+x)}{(f-2 g) (d+e x)}} \sqrt {6+5 x+x^2}} \] Output:

3/4*g*(-d*g+3*e*f-5*e*g)*(g*x+f)^(1/2)*(x^2+5*x+6)^(1/2)/e/(e*x+d)^(1/2)+1 
/2*g^2*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(x^2+5*x+6)^(1/2)/e+3/4*(d-3*e)^(1/2)*( 
f-3*g)^(1/2)*g*(-d*g+3*e*f-5*e*g)*(2+x)*((-d*g+e*f)*(3+x)/(f-3*g)/(e*x+d)) 
^(1/2)*EllipticE((d-3*e)^(1/2)*(g*x+f)^(1/2)/(f-3*g)^(1/2)/(e*x+d)^(1/2),( 
(d-2*e)*(f-3*g)/(d-3*e)/(f-2*g))^(1/2))/e^2/((-d*g+e*f)*(2+x)/(f-2*g)/(e*x 
+d))^(1/2)/(x^2+5*x+6)^(1/2)+1/4*(f-3*g)^(1/2)*(-d*g+e*f)*(d*e*(7*f-4*g)*g 
-3*d^2*g^2-e^2*(8*f^2-27*f*g+33*g^2))*(2+x)*((-d*g+e*f)*(3+x)/(f-3*g)/(e*x 
+d))^(1/2)*EllipticF((d-3*e)^(1/2)*(g*x+f)^(1/2)/(f-3*g)^(1/2)/(e*x+d)^(1/ 
2),((d-2*e)*(f-3*g)/(d-3*e)/(f-2*g))^(1/2))/(d-3*e)^(1/2)/e^3/(f-2*g)/((-d 
*g+e*f)*(2+x)/(f-2*g)/(e*x+d))^(1/2)/(x^2+5*x+6)^(1/2)-1/4*(f-3*g)^(1/2)*( 
-d*g+e*f)*(10*d*e*(f-g)*g-3*d^2*g^2-e^2*(15*f^2-50*f*g+51*g^2))*(2+x)*((-d 
*g+e*f)*(3+x)/(f-3*g)/(e*x+d))^(1/2)*EllipticPi((d-3*e)^(1/2)*(g*x+f)^(1/2 
)/(f-3*g)^(1/2)/(e*x+d)^(1/2),e*(f-3*g)/(d-3*e)/g,((d-2*e)*(f-3*g)/(d-3*e) 
/(f-2*g))^(1/2))/(d-3*e)^(1/2)/e^3/(f-2*g)/((-d*g+e*f)*(2+x)/(f-2*g)/(e*x+ 
d))^(1/2)/(x^2+5*x+6)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(12678\) vs. \(2(724)=1448\).

Time = 34.86 (sec) , antiderivative size = 12678, normalized size of antiderivative = 17.51 \[ \int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(f + g*x)^(5/2)/(Sqrt[d + e*x]*Sqrt[6 + 5*x + 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.

\(\displaystyle \int \frac {(f+g x)^{5/2}}{\sqrt {x^2+5 x+6} \sqrt {d+e x}} \, dx\)

\(\Big \downarrow \) 1278

\(\displaystyle \frac {g^2 \sqrt {x^2+5 x+6} \sqrt {d+e x} \sqrt {f+g x}}{2 e}-\frac {\int -\frac {4 e f^3-5 d g^2 f+3 g^2 (3 e f-d g-5 e g) x^2-6 g^2 (e f+d g)-2 g \left (d g (f+5 g)-e \left (6 f^2-5 g f-6 g^2\right )\right ) x}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {x^2+5 x+6}}dx}{4 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {4 e f^3-5 d g^2 f+3 g^2 (3 e f-d g-5 e g) x^2-6 g^2 (e f+d g)-2 g \left (d g (f+5 g)-e \left (6 f^2-5 g f-6 g^2\right )\right ) x}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {x^2+5 x+6}}dx}{4 e}+\frac {g^2 \sqrt {x^2+5 x+6} \sqrt {d+e x} \sqrt {f+g x}}{2 e}\)

\(\Big \downarrow \) 2154

\(\displaystyle \frac {\frac {(e f-d g) \left (3 d^2 g^2-8 d e f g+5 d e g^2+4 e^2 f^2-6 e^2 g^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {x^2+5 x+6}}dx}{e^2}+\int \frac {\sqrt {d+e x} \left (\frac {5 d g^3}{e}+\frac {3 d^2 g^3}{e^2}-12 g^3-\frac {11 d f g^2}{e}-10 f g^2+12 f^2 g+\left (-\frac {3 d g^3}{e}-15 g^3+9 f g^2\right ) x\right )}{\sqrt {f+g x} \sqrt {x^2+5 x+6}}dx}{4 e}+\frac {g^2 \sqrt {x^2+5 x+6} \sqrt {d+e x} \sqrt {f+g x}}{2 e}\)

\(\Big \downarrow \) 1280

\(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (\frac {5 d g^3}{e}+\frac {3 d^2 g^3}{e^2}-12 g^3-\frac {11 d f g^2}{e}-10 f g^2+12 f^2 g+\left (-\frac {3 d g^3}{e}-15 g^3+9 f g^2\right ) x\right )}{\sqrt {f+g x} \sqrt {x^2+5 x+6}}dx-\frac {2 (d+e x) \left (3 d^2 g^2-8 d e f g+5 d e g^2+4 e^2 f^2-6 e^2 g^2\right ) \sqrt {\frac {\left (x^2+5 x+6\right ) (e f-d g)^2}{(f-3 g) (f-2 g) (d+e x)^2}} \int \frac {1}{\sqrt {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(2 d f-5 e f-5 d g+12 e g) (f+g x)}{(f-3 g) (f-2 g) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{e^2 \sqrt {x^2+5 x+6}}}{4 e}+\frac {g^2 \sqrt {x^2+5 x+6} \sqrt {d+e x} \sqrt {f+g x}}{2 e}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (\frac {5 d g^3}{e}+\frac {3 d^2 g^3}{e^2}-12 g^3-\frac {11 d f g^2}{e}-10 f g^2+12 f^2 g+\left (-\frac {3 d g^3}{e}-15 g^3+9 f g^2\right ) x\right )}{\sqrt {f+g x} \sqrt {x^2+5 x+6}}dx-\frac {\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} (d+e x) \left (3 d^2 g^2-8 d e f g+5 d e g^2+4 e^2 f^2-6 e^2 g^2\right ) \sqrt {\frac {\left (x^2+5 x+6\right ) (e f-d g)^2}{(f-3 g) (f-2 g) (d+e x)^2}} \left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right ) \sqrt {\frac {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(f+g x) (2 d f-5 d g-5 e f+12 e g)}{(f-3 g) (f-2 g) (d+e x)}+1}{\left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d-3 e} \sqrt [4]{d-2 e} \sqrt {f+g x}}{\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 d f-5 e f-5 d g+12 e g}{\sqrt {d-3 e} \sqrt {d-2 e} \sqrt {f-3 g} \sqrt {f-2 g}}+2\right )\right )}{e^2 \sqrt {x^2+5 x+6} \sqrt [4]{d-3 e} \sqrt [4]{d-2 e} \sqrt {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(f+g x) (2 d f-5 d g-5 e f+12 e g)}{(f-3 g) (f-2 g) (d+e x)}+1}}}{4 e}+\frac {g^2 \sqrt {x^2+5 x+6} \sqrt {d+e x} \sqrt {f+g x}}{2 e}\)

\(\Big \downarrow \) 2154

\(\displaystyle \frac {-\frac {g \left (-3 d^2 g^2+d e g (8 f-5 g)-\left (e^2 \left (3 f^2+5 f g-12 g^2\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {x^2+5 x+6}}dx}{e^2}+\int \frac {\left (-\frac {3 d g^2}{e}-15 g^2+9 f g\right ) \sqrt {d+e x} \sqrt {f+g x}}{\sqrt {x^2+5 x+6}}dx-\frac {\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} (d+e x) \left (3 d^2 g^2-8 d e f g+5 d e g^2+4 e^2 f^2-6 e^2 g^2\right ) \sqrt {\frac {\left (x^2+5 x+6\right ) (e f-d g)^2}{(f-3 g) (f-2 g) (d+e x)^2}} \left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right ) \sqrt {\frac {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(f+g x) (2 d f-5 d g-5 e f+12 e g)}{(f-3 g) (f-2 g) (d+e x)}+1}{\left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d-3 e} \sqrt [4]{d-2 e} \sqrt {f+g x}}{\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 d f-5 e f-5 d g+12 e g}{\sqrt {d-3 e} \sqrt {d-2 e} \sqrt {f-3 g} \sqrt {f-2 g}}+2\right )\right )}{e^2 \sqrt {x^2+5 x+6} \sqrt [4]{d-3 e} \sqrt [4]{d-2 e} \sqrt {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(f+g x) (2 d f-5 d g-5 e f+12 e g)}{(f-3 g) (f-2 g) (d+e x)}+1}}}{4 e}+\frac {g^2 \sqrt {x^2+5 x+6} \sqrt {d+e x} \sqrt {f+g x}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {g \left (-3 d^2 g^2+d e g (8 f-5 g)-\left (e^2 \left (3 f^2+5 f g-12 g^2\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {x^2+5 x+6}}dx}{e^2}+3 g \left (3 f-\frac {g (d+5 e)}{e}\right ) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {x^2+5 x+6}}dx-\frac {\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} (d+e x) \left (3 d^2 g^2-8 d e f g+5 d e g^2+4 e^2 f^2-6 e^2 g^2\right ) \sqrt {\frac {\left (x^2+5 x+6\right ) (e f-d g)^2}{(f-3 g) (f-2 g) (d+e x)^2}} \left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right ) \sqrt {\frac {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(f+g x) (2 d f-5 d g-5 e f+12 e g)}{(f-3 g) (f-2 g) (d+e x)}+1}{\left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d-3 e} \sqrt [4]{d-2 e} \sqrt {f+g x}}{\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 d f-5 e f-5 d g+12 e g}{\sqrt {d-3 e} \sqrt {d-2 e} \sqrt {f-3 g} \sqrt {f-2 g}}+2\right )\right )}{e^2 \sqrt {x^2+5 x+6} \sqrt [4]{d-3 e} \sqrt [4]{d-2 e} \sqrt {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(f+g x) (2 d f-5 d g-5 e f+12 e g)}{(f-3 g) (f-2 g) (d+e x)}+1}}}{4 e}+\frac {g^2 \sqrt {x^2+5 x+6} \sqrt {d+e x} \sqrt {f+g x}}{2 e}\)

\(\Big \downarrow \) 1276

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {f+g x} \sqrt {x^2+5 x+6} g^2}{2 e}+\frac {-\frac {\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} \left (4 e^2 f^2-8 d e g f+3 d^2 g^2-6 e^2 g^2+5 d e g^2\right ) (d+e x) \sqrt {\frac {(e f-d g)^2 \left (x^2+5 x+6\right )}{(f-3 g) (f-2 g) (d+e x)^2}} \left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right ) \sqrt {\frac {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(2 d f-5 e f-5 d g+12 e g) (f+g x)}{(f-3 g) (f-2 g) (d+e x)}+1}{\left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d-3 e} \sqrt [4]{d-2 e} \sqrt {f+g x}}{\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 d f-5 e f-5 d g+12 e g}{\sqrt {d-3 e} \sqrt {d-2 e} \sqrt {f-3 g} \sqrt {f-2 g}}+2\right )\right )}{\sqrt [4]{d-3 e} \sqrt [4]{d-2 e} e^2 \sqrt {x^2+5 x+6} \sqrt {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(2 d f-5 e f-5 d g+12 e g) (f+g x)}{(f-3 g) (f-2 g) (d+e x)}+1}}-\frac {2 \sqrt {f-3 g} \left (-\left (\left (3 f^2+5 g f-12 g^2\right ) e^2\right )+d (8 f-5 g) g e-3 d^2 g^2\right ) \sqrt {\frac {(e f-d g) (x+2)}{(f-2 g) (d+e x)}} \sqrt {\frac {(e f-d g) (x+3)}{(f-3 g) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e (f-3 g)}{(d-3 e) g},\arcsin \left (\frac {\sqrt {d-3 e} \sqrt {f+g x}}{\sqrt {f-3 g} \sqrt {d+e x}}\right ),\frac {(d-2 e) (f-3 g)}{(d-3 e) (f-2 g)}\right )}{\sqrt {d-3 e} e^2 \sqrt {x^2+5 x+6}}+3 g \left (3 f-\frac {(d+5 e) g}{e}\right ) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {x^2+5 x+6}}dx}{4 e}\)

\(\Big \downarrow \) 1292

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {f+g x} \sqrt {x^2+5 x+6} g^2}{2 e}+\frac {-\frac {\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} \left (4 e^2 f^2-8 d e g f+3 d^2 g^2-6 e^2 g^2+5 d e g^2\right ) (d+e x) \sqrt {\frac {(e f-d g)^2 \left (x^2+5 x+6\right )}{(f-3 g) (f-2 g) (d+e x)^2}} \left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right ) \sqrt {\frac {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(2 d f-5 e f-5 d g+12 e g) (f+g x)}{(f-3 g) (f-2 g) (d+e x)}+1}{\left (\frac {\sqrt {d-3 e} \sqrt {d-2 e} (f+g x)}{\sqrt {f-3 g} \sqrt {f-2 g} (d+e x)}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d-3 e} \sqrt [4]{d-2 e} \sqrt {f+g x}}{\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 d f-5 e f-5 d g+12 e g}{\sqrt {d-3 e} \sqrt {d-2 e} \sqrt {f-3 g} \sqrt {f-2 g}}+2\right )\right )}{\sqrt [4]{d-3 e} \sqrt [4]{d-2 e} e^2 \sqrt {x^2+5 x+6} \sqrt {\frac {(d-3 e) (d-2 e) (f+g x)^2}{(f-3 g) (f-2 g) (d+e x)^2}-\frac {(2 d f-5 e f-5 d g+12 e g) (f+g x)}{(f-3 g) (f-2 g) (d+e x)}+1}}-\frac {2 \sqrt {f-3 g} \left (-\left (\left (3 f^2+5 g f-12 g^2\right ) e^2\right )+d (8 f-5 g) g e-3 d^2 g^2\right ) \sqrt {\frac {(e f-d g) (x+2)}{(f-2 g) (d+e x)}} \sqrt {\frac {(e f-d g) (x+3)}{(f-3 g) (d+e x)}} (d+e x) \operatorname {EllipticPi}\left (\frac {e (f-3 g)}{(d-3 e) g},\arcsin \left (\frac {\sqrt {d-3 e} \sqrt {f+g x}}{\sqrt {f-3 g} \sqrt {d+e x}}\right ),\frac {(d-2 e) (f-3 g)}{(d-3 e) (f-2 g)}\right )}{\sqrt {d-3 e} e^2 \sqrt {x^2+5 x+6}}+3 g \left (3 f-\frac {(d+5 e) g}{e}\right ) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {x^2+5 x+6}}dx}{4 e}\)

Input:

Int[(f + g*x)^(5/2)/(Sqrt[d + e*x]*Sqrt[6 + 5*x + 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 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 1278

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

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 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 [A] (verified)

Time = 26.38 (sec) , antiderivative size = 1094, normalized size of antiderivative = 1.51

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1094\)
risch	\(\text {Expression too large to display}\)	\(1607\)
default	\(\text {Expression too large to display}\)	\(11317\)

Input:

int((g*x+f)^(5/2)/(e*x+d)^(1/2)/(x^2+5*x+6)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((x^2+5*x+6)*(e*x+d)*(g*x+f))^(1/2)/(x^2+5*x+6)^(1/2)/(e*x+d)^(1/2)/(g*x+f 
)^(1/2)*(1/2*g^2/e*(e*g*x^4+d*g*x^3+e*f*x^3+5*e*g*x^3+d*f*x^2+5*d*g*x^2+5* 
e*f*x^2+6*e*g*x^2+5*d*f*x+6*d*g*x+6*e*f*x+6*d*f)^(1/2)+2*(f^3-1/2*g^2/e*(5 
/2*d*f+3*d*g+3*e*f))*(f/g-3)*((-f/g+2)*(3+x)/(-f/g+3)/(2+x))^(1/2)*(2+x)^2 
*((x+d/e)/(-d/e+3)/(2+x))^(1/2)*((x+f/g)/(-f/g+3)/(2+x))^(1/2)/(-f/g+2)/(e 
*g*(3+x)*(2+x)*(x+d/e)*(x+f/g))^(1/2)*EllipticF(((-f/g+2)*(3+x)/(-f/g+3)/( 
2+x))^(1/2),((-2+d/e)*(f/g-3)/(d/e-3)/(f/g-2))^(1/2))+2*(3*f^2*g-1/2*g^2/e 
*(d*f+5*d*g+5*e*f+6*e*g))*(f/g-3)*((-f/g+2)*(3+x)/(-f/g+3)/(2+x))^(1/2)*(2 
+x)^2*((x+d/e)/(-d/e+3)/(2+x))^(1/2)*((x+f/g)/(-f/g+3)/(2+x))^(1/2)/(-f/g+ 
2)/(e*g*(3+x)*(2+x)*(x+d/e)*(x+f/g))^(1/2)*(-2*EllipticF(((-f/g+2)*(3+x)/( 
-f/g+3)/(2+x))^(1/2),((-2+d/e)*(f/g-3)/(d/e-3)/(f/g-2))^(1/2))-EllipticPi( 
((-f/g+2)*(3+x)/(-f/g+3)/(2+x))^(1/2),(-f/g+3)/(-f/g+2),((-2+d/e)*(f/g-3)/ 
(d/e-3)/(f/g-2))^(1/2)))+(3*f*g^2-1/2*g^2/e*(3/2*d*g+3/2*e*f+15/2*e*g))*(( 
3+x)*(x+d/e)*(x+f/g)+(f/g-3)*((-f/g+2)*(3+x)/(-f/g+3)/(2+x))^(1/2)*(2+x)^2 
*((x+d/e)/(-d/e+3)/(2+x))^(1/2)*((x+f/g)/(-f/g+3)/(2+x))^(1/2)*((10-f/g)/( 
-f/g+2)*EllipticF(((-f/g+2)*(3+x)/(-f/g+3)/(2+x))^(1/2),((-2+d/e)*(f/g-3)/ 
(d/e-3)/(f/g-2))^(1/2))+(d/e-3)*EllipticE(((-f/g+2)*(3+x)/(-f/g+3)/(2+x))^ 
(1/2),((-2+d/e)*(f/g-3)/(d/e-3)/(f/g-2))^(1/2))+(d*g+e*f+5*e*g)/e/g/(-f/g+ 
2)*EllipticPi(((-f/g+2)*(3+x)/(-f/g+3)/(2+x))^(1/2),(f/g-3)/(f/g-2),((-2+d 
/e)*(f/g-3)/(d/e-3)/(f/g-2))^(1/2))))/(e*g*(3+x)*(2+x)*(x+d/e)*(x+f/g))...
 

Fricas [F]

\[ \int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {e x + d} \sqrt {x^{2} + 5 \, x + 6}} \,d x } \] Input:

integrate((g*x+f)^(5/2)/(e*x+d)^(1/2)/(x^2+5*x+6)^(1/2),x, algorithm="fric 
as")
 

Output:

integral((g^2*x^2 + 2*f*g*x + f^2)*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(x^2 + 
5*x + 6)/(e*x^3 + (d + 5*e)*x^2 + (5*d + 6*e)*x + 6*d), x)
 

Sympy [F]

\[ \int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx=\int \frac {\left (f + g x\right )^{\frac {5}{2}}}{\sqrt {\left (x + 2\right ) \left (x + 3\right )} \sqrt {d + e x}}\, dx \] Input:

integrate((g*x+f)**(5/2)/(e*x+d)**(1/2)/(x**2+5*x+6)**(1/2),x)
 

Output:

Integral((f + g*x)**(5/2)/(sqrt((x + 2)*(x + 3))*sqrt(d + e*x)), x)
 

Maxima [F]

\[ \int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {e x + d} \sqrt {x^{2} + 5 \, x + 6}} \,d x } \] Input:

integrate((g*x+f)^(5/2)/(e*x+d)^(1/2)/(x^2+5*x+6)^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate((g*x + f)^(5/2)/(sqrt(e*x + d)*sqrt(x^2 + 5*x + 6)), x)
 

Giac [F]

\[ \int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {e x + d} \sqrt {x^{2} + 5 \, x + 6}} \,d x } \] Input:

integrate((g*x+f)^(5/2)/(e*x+d)^(1/2)/(x^2+5*x+6)^(1/2),x, algorithm="giac 
")
 

Output:

integrate((g*x + f)^(5/2)/(sqrt(e*x + d)*sqrt(x^2 + 5*x + 6)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{5/2}}{\sqrt {d+e\,x}\,\sqrt {x^2+5\,x+6}} \,d x \] Input:

int((f + g*x)^(5/2)/((d + e*x)^(1/2)*(5*x + x^2 + 6)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int((f + g*x)^(5/2)/((d + e*x)^(1/2)*(5*x + x^2 + 6)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(f+g x)^{5/2}}{\sqrt {d+e x} \sqrt {6+5 x+x^2}} \, dx=\int \frac {\left (g x +f \right )^{\frac {5}{2}}}{\sqrt {e x +d}\, \sqrt {x^{2}+5 x +6}}d x \] Input:

int((g*x+f)^(5/2)/(e*x+d)^(1/2)/(x^2+5*x+6)^(1/2),x)
 

Output:

int((g*x+f)^(5/2)/(e*x+d)^(1/2)/(x^2+5*x+6)^(1/2),x)