\(\int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx\) [132]

Optimal result

Integrand size = 28, antiderivative size = 740 \[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {2 \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} (7 e f-3 d g) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{3 c^{3/4} e^2 \sqrt {a+c x^2}}-\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (a e^2 g^2+\sqrt {-a} \sqrt {c} e g (7 e f-3 d g)-3 c \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{3 c^{5/4} e^3 \sqrt {a+c x^2}}-\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} (e f-d g)^2 \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticPi}\left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g},\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{\sqrt [4]{c} e^3 \sqrt {a+c x^2}} \] Output:

2/3*g^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/e-2/3*(c^(1/2)*f-(-a)^(1/2)*g)*(c^ 
(1/2)*f+(-a)^(1/2)*g)^(1/2)*(-3*d*g+7*e*f)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-( 
-a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*Ell 
ipticE(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+(- 
a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2))/c^(3/4)/e^2/(c*x^2+a)^(1/2)-2 
/3*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(a*e^2*g^2+(-a)^(1/2)*c^(1/2)*e*g*(-3*d* 
g+7*e*f)-3*c*(d^2*g^2-3*d*e*f*g+3*e^2*f^2))*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f- 
(-a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*El 
lipticF(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+( 
-a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2))/c^(5/4)/e^3/(c*x^2+a)^(1/2)- 
2*(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(-d*g+e*f)^2*(1-c^(1/2)*(g*x+f)/(c^(1/2)* 
f-(-a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)* 
EllipticPi(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),e*(f+(-a)^ 
(1/2)*g/c^(1/2))/(-d*g+e*f),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2 
)*g))^(1/2))/c^(1/4)/e^3/(c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 31.20 (sec) , antiderivative size = 1440, normalized size of antiderivative = 1.95 \[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) + (2*(f + g*x)^(3/2)*(7*c*e^ 
2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/S 
qrt[c]] + (7*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (3* 
c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (7*a*e^2*f*g^2 
*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (3*a*d*e*g^3*Sqrt[-f - (I 
*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (14*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/ 
Sqrt[c]])/(f + g*x) + (6*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + 
g*x) + (Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(7*e*f - 3*d*g)*Sqrt[1 - f/ 
(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*S 
qrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g) 
/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a] 
*g)])/Sqrt[f + g*x] + (I*e*(Sqrt[c]*f + I*Sqrt[a]*g)*(I*Sqrt[a]*e*g + Sqrt 
[c]*(6*e*f - 3*d*g))*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g* 
x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticF[I 
*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*S 
qrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((3*I)*c*e^2*f^2*Sqr 
t[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x 
) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e 
*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sq 
rt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqr...
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1848\) vs. \(2(740)=1480\).

Time = 7.52 (sec) , antiderivative size = 1848, normalized size of antiderivative = 2.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {740, 2009}

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.

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

\(\Big \downarrow \) 740

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt [4]{c} \sqrt [4]{c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right ) (e f-d g)^3}{e^3 g \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {c x^2+a}}-\frac {\text {arctanh}\left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {e} \sqrt {e f-d g} \sqrt {c x^2+a}}\right ) (e f-d g)^{5/2}}{e^{5/2} \sqrt {c d^2+a e^2}}+\frac {\sqrt [4]{c f^2+a g^2} \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right ) (e f-d g)^2}{\sqrt [4]{c} e^3 \sqrt {c x^2+a}}-\frac {\sqrt [4]{c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right )^2 \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c f^2+a g^2} e+\sqrt {c} (e f-d g)\right )^2}{4 \sqrt {c} e (e f-d g) \sqrt {c f^2+a g^2}},2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right ) (e f-d g)^2}{2 \sqrt [4]{c} e^3 g \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {c x^2+a}}-\frac {2 \left (c f^2+a g^2\right )^{3/4} \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right ) (e f-d g)}{c^{3/4} e^2 \sqrt {c x^2+a}}+\frac {\left (c f^2+a g^2\right )^{3/4} \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right ) (e f-d g)}{c^{3/4} e^2 \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {f+g x} \sqrt {c x^2+a} (e f-d g)}{\sqrt {c} e^2 \sqrt {c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}-\frac {8 f \left (c f^2+a g^2\right )^{3/4} \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{3 c^{3/4} e \sqrt {c x^2+a}}-\frac {\sqrt [4]{c f^2+a g^2} \left (c f^2-4 \sqrt {c} \sqrt {c f^2+a g^2} f+a g^2\right ) \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{3 c^{5/4} e \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {f+g x} \sqrt {c x^2+a}}{3 c e}+\frac {8 f g^2 \sqrt {f+g x} \sqrt {c x^2+a}}{3 \sqrt {c} e \sqrt {c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\)

Input:

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

Output:

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) + (8*f*g^2*Sqrt[f + g*x]*Sqr 
t[a + c*x^2])/(3*Sqrt[c]*e*Sqrt[c*f^2 + a*g^2]*(1 + (Sqrt[c]*(f + g*x))/Sq 
rt[c*f^2 + a*g^2])) + (2*g^2*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(S 
qrt[c]*e^2*Sqrt[c*f^2 + a*g^2]*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2 
])) - ((e*f - d*g)^(5/2)*ArcTanh[(Sqrt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt 
[e]*Sqrt[e*f - d*g]*Sqrt[a + c*x^2])])/(e^(5/2)*Sqrt[c*d^2 + a*e^2]) - (8* 
f*(c*f^2 + a*g^2)^(3/4)*Sqrt[(g^2*(a + c*x^2))/((c*f^2 + a*g^2)*(1 + (Sqrt 
[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^ 
2 + a*g^2])*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/ 
4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(3*c^(3/4)*e*Sqrt[a + c*x^2 
]) - (2*(e*f - d*g)*(c*f^2 + a*g^2)^(3/4)*Sqrt[(g^2*(a + c*x^2))/((c*f^2 + 
 a*g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*(1 + (Sqrt[c]*(f 
 + g*x))/Sqrt[c*f^2 + a*g^2])*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/( 
c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(c^(3/4)* 
e^2*Sqrt[a + c*x^2]) + ((e*f - d*g)^2*(c*f^2 + a*g^2)^(1/4)*Sqrt[(g^2*(a + 
 c*x^2))/((c*f^2 + a*g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2) 
]*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*EllipticF[2*ArcTan[(c^(1/4 
)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g 
^2])/2])/(c^(1/4)*e^3*Sqrt[a + c*x^2]) + ((e*f - d*g)*(c*f^2 + a*g^2)^(3/4 
)*Sqrt[(g^2*(a + c*x^2))/((c*f^2 + a*g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt...
 

Defintions of rubi rules used

rule 740

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^ 
2]), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f 
 + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && Inte 
gerQ[n + 1/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 7.39 (sec) , antiderivative size = 948, normalized size of antiderivative = 1.28

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 g^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{3 e c}+\frac {2 \left (\frac {g \left (d^{2} g^{2}-3 d e f g +3 e^{2} f^{2}\right )}{e^{3}}-\frac {a \,g^{3}}{3 c e}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 \left (-\frac {g^{2} \left (d g -3 e f \right )}{e^{2}}-\frac {2 f \,g^{2}}{3 e}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}-\frac {2 \left (d^{3} g^{3}-3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g -e^{3} f^{3}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{e^{4} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(948\)
risch	\(\text {Expression too large to display}\)	\(1559\)
default	\(\text {Expression too large to display}\)	\(3164\)

Input:

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

Output:

((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/3*g^2/e/c*(c*g* 
x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(g*(d^2*g^2-3*d*e*f*g+3*e^2*f^2)/e^3-1/3*a/ 
c*g^3/e)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a 
*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c) 
^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*EllipticF(((x+f/g)/(f/g 
-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2 
))+2*(-1/e^2*g^2*(d*g-3*e*f)-2/3*f*g^2/e)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f 
/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2) 
*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a 
*f)^(1/2)*((-f/g-(-a*c)^(1/2)/c)*EllipticE(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^ 
(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c* 
EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/ 
g-(-a*c)^(1/2)/c))^(1/2)))-2*(d^3*g^3-3*d^2*e*f*g^2+3*d*e^2*f^2*g-e^3*f^3) 
/e^4*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^ 
(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/ 
2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)/(-f/g+d/e)*EllipticPi(((x+f 
/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),(-f/g+(-a*c)^(1/2)/c)/(-f/g+d/e),((-f/g+(- 
a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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