\(\int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx\) [648]

Optimal result

Integrand size = 21, antiderivative size = 136 \[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {733, 430} \[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} \sqrt {a+c x^2} \sqrt {f+g x}} \]

[In]

Int[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]

[Out]

(-2*Sqrt[-a]*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 -
(Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(Sqrt[c]*Sqrt[f + g*x]*Sqrt[a + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 733

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*a*Rt[-c/a, 2]*(d + e*x)^m*(Sqrt[1
+ c*(x^2/a)]/(c*Sqrt[a + c*x^2]*(c*((d + e*x)/(c*d - a*e*Rt[-c/a, 2])))^m)), Subst[Int[(1 + 2*a*e*Rt[-c/a, 2]*
(x^2/(c*d - a*e*Rt[-c/a, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-c/a, 2]*x)/2]], x] /; FreeQ[{a, c, d, e},
x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {2 \sqrt {-a} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.20 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 i \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \sqrt {a+c x^2}} \]

[In]

Integrate[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]

[Out]

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

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {2 \left (c f -g \sqrt {-a c}\right ) F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}-c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}+c f}}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}\, \sqrt {c \,x^{2}+a}\, \sqrt {g x +f}}{c g \left (c g \,x^{3}+c f \,x^{2}+a g x +f a \right )}\)	\(200\)
elliptic	\(\frac {2 \sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\)	\(242\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 \, \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )}{c g} \]

[In]

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

[Out]

2*sqrt(c*g)*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]

\[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \sqrt {f + g x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} \sqrt {g x + f}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} \sqrt {g x + f}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}} \,d x \]

[In]

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

[Out]

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