\(\int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx\) [127]

Optimal result

Integrand size = 21, antiderivative size = 423 \[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{c^{3/4} g \sqrt {a+c x^2}}+\frac {2 \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{c^{3/4} g \sqrt {a+c x^2}} \] Output:

-2*(c^(1/2)*f-(-a)^(1/2)*g)*(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)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/ 
2)*g))^(1/2)*EllipticE(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)/g/(c* 
x^2+a)^(1/2)+2*(c^(1/2)*f-(-a)^(1/2)*g)*(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)*(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)/g/(c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.57 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {2 i \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {\frac {g \left (\sqrt {a}+i \sqrt {c} x\right )}{-i \sqrt {c} f+\sqrt {a} g}} \sqrt {f+g x} \left (E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{\sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{g \left (i \sqrt {a}+\sqrt {c} x\right )}} \sqrt {a+c x^2}} \] Input:

Integrate[Sqrt[f + g*x]/Sqrt[a + c*x^2],x]
 

Output:

((2*I)*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sq 
rt[c]*f + Sqrt[a]*g)]*Sqrt[f + g*x]*(EllipticE[I*ArcSinh[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)] - EllipticF[I*ArcSinh[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 
)]))/(Sqrt[c]*g*Sqrt[(Sqrt[c]*(f + g*x))/(g*(I*Sqrt[a] + Sqrt[c]*x))]*Sqrt 
[a + c*x^2])
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.38, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {507, 1459, 1416, 1509}

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.

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

\(\Big \downarrow \) 507

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

\(\Big \downarrow \) 1459

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

Int[Sqrt[f + g*x]/Sqrt[a + c*x^2],x]
 

Output:

(2*(-((Sqrt[c*f^2 + a*g^2]*(-((Sqrt[f + g*x]*Sqrt[a + (c*f^2)/g^2 - (2*c*f 
*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*( 
f + g*x))/Sqrt[c*f^2 + a*g^2]))) + ((c*f^2 + a*g^2)^(1/4)*(1 + (Sqrt[c]*(f 
 + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^ 
2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[ 
c*f^2 + a*g^2])^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^(1/4)*Sqrt[a + ( 
c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])))/Sqrt[c]) + (( 
c*f^2 + a*g^2)^(3/4)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a 
 + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2 
)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*EllipticF[2*ArcTa 
n[(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])/(2*c^(3/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 
+ (c*(f + g*x)^2)/g^2])))/g
 

Defintions of rubi rules used

rule 507

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[2/ 
d   Subst[Int[x^2/Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)] 
, x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
 

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 1459

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 2]}, Simp[1/q   Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[1/q 
 Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \left (c f -\sqrt {-a c}\, g \right ) \sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{\sqrt {-a c}\, g +c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{\sqrt {-a c}\, g -c f}}\, \left (\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) g -\sqrt {-a c}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) g +f \operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c -\operatorname {EllipticE}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c f \right )}{g \left (c g \,x^{3}+c f \,x^{2}+a g x +a f \right ) c^{2}}\)	\(396\)
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 f \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 g \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(556\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {c g} f {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) - 3 \, \sqrt {c g} g {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right )\right )}}{3 \, c g} \] Input:

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

Output:

2/3*(2*sqrt(c*g)*f*weierstrassPInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/2 
7*(c*f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g) - 3*sqrt(c*g)*g*weierstr 
assZeta(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), 
weierstrassPInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^ 
2)/(c*g^3), 1/3*(3*g*x + f)/g)))/(c*g)
 

Sympy [F]

\[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f + g x}}{\sqrt {a + c x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c \,x^{2}+a}d x \] Input:

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

Output:

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