\(\int \frac {\sqrt {f+g x}}{(d+e x)^{5/2} \sqrt {6+5 x+x^2}} \, dx\) [603]

Optimal result

Integrand size = 31, antiderivative size = 434 \[ \int \frac {\sqrt {f+g x}}{(d+e x)^{5/2} \sqrt {6+5 x+x^2}} \, dx=-\frac {2 e \sqrt {f+g x} \sqrt {6+5 x+x^2}}{3 \left (d^2-5 d e+6 e^2\right ) (d+e x)^{3/2}}-\frac {2 \sqrt {f-3 g} \left (2 e^2 (5 f-3 g)+3 d^2 g-d e (4 f+5 g)\right ) (2+x) \sqrt {\frac {(e f-d g) (3+x)}{(f-3 g) (d+e x)}} E\left (\arcsin \left (\frac {\sqrt {d-3 e} \sqrt {f+g x}}{\sqrt {f-3 g} \sqrt {d+e x}}\right )|\frac {(d-2 e) (f-3 g)}{(d-3 e) (f-2 g)}\right )}{3 (d-3 e)^{3/2} (d-2 e)^2 (e f-d g) \sqrt {\frac {(e f-d g) (2+x)}{(f-2 g) (d+e x)}} \sqrt {6+5 x+x^2}}-\frac {2 (3 d-8 e) \sqrt {f-3 g} (2+x) \sqrt {\frac {(e f-d g) (3+x)}{(f-3 g) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d-3 e} \sqrt {f+g x}}{\sqrt {f-3 g} \sqrt {d+e x}}\right ),\frac {(d-2 e) (f-3 g)}{(d-3 e) (f-2 g)}\right )}{3 (d-3 e)^{3/2} (d-2 e)^2 \sqrt {\frac {(e f-d g) (2+x)}{(f-2 g) (d+e x)}} \sqrt {6+5 x+x^2}} \] Output:

-2/3*e*(g*x+f)^(1/2)*(x^2+5*x+6)^(1/2)/(d^2-5*d*e+6*e^2)/(e*x+d)^(3/2)-2/3 
*(f-3*g)^(1/2)*(2*e^2*(5*f-3*g)+3*d^2*g-d*e*(4*f+5*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))/(d-3*e)^(3/2) 
/(d-2*e)^2/(-d*g+e*f)/((-d*g+e*f)*(2+x)/(f-2*g)/(e*x+d))^(1/2)/(x^2+5*x+6) 
^(1/2)-2/3*(3*d-8*e)*(f-3*g)^(1/2)*(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)^(3/2)/(d-2*e)^2/((-d*g+e* 
f)*(2+x)/(f-2*g)/(e*x+d))^(1/2)/(x^2+5*x+6)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3571\) vs. \(2(434)=868\).

Time = 34.41 (sec) , antiderivative size = 3571, normalized size of antiderivative = 8.23 \[ \int \frac {\sqrt {f+g x}}{(d+e x)^{5/2} \sqrt {6+5 x+x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[6 + 5*x + x^2]*((-2*e)/(3*(d^2 - 5*d*e + 
6*e^2)*(d + e*x)^2) - (2*e*(-4*d*e*f + 10*e^2*f + 3*d^2*g - 5*d*e*g - 6*e^ 
2*g))/(3*(d^2 - 5*d*e + 6*e^2)^2*(-(e*f) + d*g)*(d + e*x))) + ((-2*(4*d*e* 
f - 10*e^2*f - 3*d^2*g + 5*d*e*g + 6*e^2*g)*(d + e*x)^(5/2)*(1 + d^2/(d + 
e*x)^2 - (5*d*e)/(d + e*x)^2 + (6*e^2)/(d + e*x)^2 - (2*d)/(d + e*x) + (5* 
e)/(d + e*x))*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x)))/(Sqrt[((d + e*x)^2* 
(1 + d^2/(d + e*x)^2 - (5*d*e)/(d + e*x)^2 + (6*e^2)/(d + e*x)^2 - (2*d)/( 
d + e*x) + (5*e)/(d + e*x)))/e^2]*Sqrt[f + ((d + e*x)*(g - (d*g)/(d + e*x) 
))/e]) + (2*(d - 3*e)*(d - 2*e)*(-(e*f) + d*g)*(d + e*x)^(3/2)*Sqrt[(1 + d 
^2/(d + e*x)^2 - (5*d*e)/(d + e*x)^2 + (6*e^2)/(d + e*x)^2 - (2*d)/(d + e* 
x) + (5*e)/(d + e*x))*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x))]*((e*f*(-(d 
- 3*e)^(-1) + (d + e*x)^(-1))*Sqrt[(-(d - 2*e)^(-1) + (d + e*x)^(-1))/((d 
- 3*e)^(-1) - (d - 2*e)^(-1))]*Sqrt[(-(g/(-(e*f) + d*g)) + (d + e*x)^(-1)) 
/((d - 3*e)^(-1) - g/(-(e*f) + d*g))]*EllipticF[ArcSin[Sqrt[-(((-(e*f) + d 
*g)*(-1 + d/(d + e*x) - (3*e)/(d + e*x)))/(e*(-f + 3*g)))]], ((d - 2*e)*(- 
f + 3*g))/(-(e*f) + d*g)])/(Sqrt[(-(d - 3*e)^(-1) + (d + e*x)^(-1))/(-(d - 
 3*e)^(-1) + g/(-(e*f) + d*g))]*Sqrt[-((1 + (d^2 - 5*d*e + 6*e^2)/(d + e*x 
)^2 + (-2*d + 5*e)/(d + e*x))*(-g + (-(e*f) + d*g)/(d + e*x)))]) - (3*d*g* 
(-(d - 3*e)^(-1) + (d + e*x)^(-1))*Sqrt[(-(d - 2*e)^(-1) + (d + e*x)^(-1)) 
/((d - 3*e)^(-1) - (d - 2*e)^(-1))]*Sqrt[(-(g/(-(e*f) + d*g)) + (d + e*...
 

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.

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

\(\Big \downarrow \) 1285

\(\displaystyle -\frac {\int -\frac {3 d f-10 e f+6 e g+(3 d g-e (f+5 g)) x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {x^2+5 x+6}}dx}{3 (d-3 e) (d-2 e)}-\frac {2 e \sqrt {x^2+5 x+6} \sqrt {f+g x}}{3 \left (d^2-5 d e+6 e^2\right ) (d+e x)^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {3 d f-10 e f+6 e g+(3 d g-e (f+5 g)) x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {x^2+5 x+6}}dx}{3 (d-3 e) (d-2 e)}-\frac {2 e \sqrt {x^2+5 x+6} \sqrt {f+g x}}{3 \left (d^2-5 d e+6 e^2\right ) (d+e x)^{3/2}}\)

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1280

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

\(\Big \downarrow \) 1281

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

\(\Big \downarrow \) 1280

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

\(\Big \downarrow \) 1292

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

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} \left (f+\left (5-\frac {3 d}{e}\right ) g\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 f-d g) \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}}+\left (-\frac {3 g d^2}{e}+4 f d+5 g d-10 e f+6 e g\right ) \left (\frac {\sqrt [4]{f-3 g} \sqrt [4]{f-2 g} g (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 f-d g)^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 {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {x^2+5 x+6}}dx}{e f-d g}\right )}{3 (d-3 e) (d-2 e)}-\frac {2 e \sqrt {f+g x} \sqrt {x^2+5 x+6}}{3 \left (d^2-5 e d+6 e^2\right ) (d+e x)^{3/2}}\)

Input:

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

Leaf count of result is larger than twice the leaf count of optimal. \(1733\) vs. \(2(388)=776\).

Time = 16.09 (sec) , antiderivative size = 1734, normalized size of antiderivative = 4.00

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1734\)
default	\(\text {Expression too large to display}\)	\(12644\)

Input:

int((g*x+f)^(1/2)/(e*x+d)^(5/2)/(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)*(-2/3/e/(d^2-5*d*e+6*e^2)*(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)/(x+d 
/e)^2-2/3*(e*g*x^3+e*f*x^2+5*e*g*x^2+5*e*f*x+6*e*g*x+6*e*f)/(d^3*g-d^2*e*f 
-5*d^2*e*g+5*d*e^2*f+6*d*e^2*g-6*e^3*f)*(3*d^2*g-4*d*e*f-5*d*e*g+10*e^2*f- 
6*e^2*g)/(d^2-5*d*e+6*e^2)/((x+d/e)*(e*g*x^3+e*f*x^2+5*e*g*x^2+5*e*f*x+6*e 
*g*x+6*e*f))^(1/2)+2*(1/3/e*(3*d*g-e*f-5*e*g)/(d^2-5*d*e+6*e^2)-1/3/e*(d^2 
*g-d*e*f-5*d*e*g+5*e^2*f+6*e^2*g)*(3*d^2*g-4*d*e*f-5*d*e*g+10*e^2*f-6*e^2* 
g)/(d^3*g-d^2*e*f-5*d^2*e*g+5*d*e^2*f+6*d*e^2*g-6*e^3*f)/(d^2-5*d*e+6*e^2) 
+1/3*(5*e*f+6*e*g)/(d^3*g-d^2*e*f-5*d^2*e*g+5*d*e^2*f+6*d*e^2*g-6*e^3*f)*( 
3*d^2*g-4*d*e*f-5*d*e*g+10*e^2*f-6*e^2*g)/(d^2-5*d*e+6*e^2))*(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*(1/3*(d*g-e*f-5*e*g)*(3*d^2*g-4*d*e*f-5*d*e*g+10*e^2 
*f-6*e^2*g)/(d^3*g-d^2*e*f-5*d^2*e*g+5*d*e^2*f+6*d*e^2*g-6*e^3*f)/(d^2-5*d 
*e+6*e^2)+1/3*(2*e*f+10*e*g)/(d^3*g-d^2*e*f-5*d^2*e*g+5*d*e^2*f+6*d*e^2*g- 
6*e^3*f)*(3*d^2*g-4*d*e*f-5*d*e*g+10*e^2*f-6*e^2*g)/(d^2-5*d*e+6*e^2))*(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...
 

Fricas [F]

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

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

Output:

integral(sqrt(e*x + d)*sqrt(g*x + f)*sqrt(x^2 + 5*x + 6)/(e^3*x^5 + (3*d*e 
^2 + 5*e^3)*x^4 + 3*(d^2*e + 5*d*e^2 + 2*e^3)*x^3 + 6*d^3 + (d^3 + 15*d^2* 
e + 18*d*e^2)*x^2 + (5*d^3 + 18*d^2*e)*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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