\(\int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx\) [131]

Optimal result
Mathematica [C] (warning: unable to verify)
Rubi [A] (warning: unable to verify)
Maple [A] (verified)
Fricas [F(-1)]
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 28, antiderivative size = 726 \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\sqrt {c} e \left (\sqrt {c f^2+a g^2}+\sqrt {c} (f+g x)\right )}-\frac {2 \left (c f^2+a g^2\right )^{3/4} \sqrt {\frac {g^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) \left (1+\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}\right )^2}} \left (1+\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}\right )\right )}{c^{3/4} e \sqrt {a+c x^2}}+\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (c e f^2+a e g^2+\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{c^{3/4} e^2 \sqrt {c f^2+a g^2} \sqrt {a+c x^2}}-\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} (e f-d g) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticPi}\left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g},\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{\sqrt [4]{c} e^2 \sqrt {a+c x^2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.25 (sec) , antiderivative size = 927, normalized size of antiderivative = 1.28 \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}} \left (\frac {2 i \sqrt {a} f g \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{\sqrt {c} e}-\frac {i \sqrt {a} d g^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{\sqrt {c} e^2}+\frac {g \sqrt {\frac {g \left (\sqrt {a}+i \sqrt {c} x\right )}{-i \sqrt {c} f+\sqrt {a} g}} \left (i \sqrt {a}+\sqrt {c} x\right ) \left (\left (\sqrt {c} f+i \sqrt {a} g\right ) E\left (\arcsin \left (\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-i \sqrt {a} g \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c e \sqrt {\frac {g \left (\sqrt {a}-i \sqrt {c} x\right )}{i \sqrt {c} f+\sqrt {a} g}}}-\frac {\sqrt {a} f^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{i \sqrt {c} d+\sqrt {a} e}+\frac {2 \sqrt {a} d f g \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{i \sqrt {c} d e+\sqrt {a} e^2}-\frac {\sqrt {a} d^2 g^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{e^2 \left (i \sqrt {c} d+\sqrt {a} e\right )}\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \] Input:

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

Output:

(2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g)]*(((2*I)*Sqrt[a]*f*g 
*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt 
[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(Sqrt[c]*e) - (I*Sqrt[a]*d 
*g^2*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/ 
Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(Sqrt[c]*e^2) + (g*Sqr 
t[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*f + Sqrt[a]*g)]*(I*Sqrt[a] + S 
qrt[c]*x)*((Sqrt[c]*f + I*Sqrt[a]*g)*EllipticE[ArcSin[Sqrt[(Sqrt[c]*(f + g 
*x))/(Sqrt[c]*f - I*Sqrt[a]*g)]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I 
*Sqrt[a]*g)] - I*Sqrt[a]*g*EllipticF[ArcSin[Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt 
[c]*f - I*Sqrt[a]*g)]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g 
)]))/(c*e*Sqrt[(g*(Sqrt[a] - I*Sqrt[c]*x))/(I*Sqrt[c]*f + Sqrt[a]*g)]) - ( 
Sqrt[a]*f^2*Sqrt[1 + (c*x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sq 
rt[a]*e), ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/( 
I*Sqrt[c]*f + Sqrt[a]*g)])/(I*Sqrt[c]*d + Sqrt[a]*e) + (2*Sqrt[a]*d*f*g*Sq 
rt[1 + (c*x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sqrt[a]*e), ArcS 
in[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + 
Sqrt[a]*g)])/(I*Sqrt[c]*d*e + Sqrt[a]*e^2) - (Sqrt[a]*d^2*g^2*Sqrt[1 + (c* 
x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sqrt[a]*e), ArcSin[Sqrt[1 
- (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g) 
])/(e^2*(I*Sqrt[c]*d + Sqrt[a]*e))))/(Sqrt[f + g*x]*Sqrt[a + c*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 5.82 (sec) , antiderivative size = 1299, normalized size of antiderivative = 1.79, 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.

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

\(\Big \downarrow \) 740

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} g^2}{\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 )}-\frac {(e f-d g)^{3/2} \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^{3/2} \sqrt {c d^2+a e^2}}-\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 )}{c^{3/4} e \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 )}{c^{3/4} e \sqrt {c x^2+a}}+\frac {(e f-d g) \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 )}{\sqrt [4]{c} e^2 \sqrt {c x^2+a}}+\frac {\sqrt [4]{c} (e f-d g)^2 \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^2 \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {c x^2+a} g}-\frac {(e f-d g) \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 )}{2 \sqrt [4]{c} e^2 \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {c x^2+a} g}\)

Input:

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

Output:

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(Sqrt[c]*e*Sqrt[c*f^2 + a*g^2]*(1 + 
(Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])) - ((e*f - d*g)^(3/2)*ArcTanh[(Sq 
rt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[e]*Sqrt[e*f - d*g]*Sqrt[a + c*x^2]) 
])/(e^(3/2)*Sqrt[c*d^2 + a*e^2]) - (2*(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*Sqrt[a + c*x^2]) + ((e*f - d*g)*(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)/Sq 
rt[c*f^2 + a*g^2])/2])/(c^(1/4)*e^2*Sqrt[a + c*x^2]) + ((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])*Elliptic 
F[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*Sqrt[a + c*x^2]) + (c^(1/4)*(e*f - d 
*g)^2*(c*f^2 + a*g^2)^(1/4)*(Sqrt[c]*(e*f - d*g) - e*Sqrt[c*f^2 + a*g^2])* 
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)...
 

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 = 3.78 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.17

method result size
elliptic \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 g \left (d g -2 e f \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 )}{e^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 g^{2} \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 )}{e \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\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^{3} \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}}\) \(852\)
default \(\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{\sqrt {-a c}\, g +c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{\sqrt {-a c}\, g -c f}}\, \left (\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) d \,g^{2}-\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) e f g -\sqrt {-a c}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \frac {\left (\sqrt {-a c}\, g -c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) d \,g^{2}+\sqrt {-a c}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \frac {\left (\sqrt {-a c}\, g -c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) e f g +\operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) a e \,g^{2}-\operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c d f g +2 \operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c e \,f^{2}-\operatorname {EllipticE}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) a e \,g^{2}-\operatorname {EllipticE}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c e \,f^{2}+\operatorname {EllipticPi}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \frac {\left (\sqrt {-a c}\, g -c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c d f g -\operatorname {EllipticPi}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \frac {\left (\sqrt {-a c}\, g -c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c e \,f^{2}\right )}{c \,e^{2} \left (c g \,x^{3}+c f \,x^{2}+a g x +a f \right )}\) \(959\)

Input:

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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