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

Optimal. Leaf size=136 \[ -\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]

[Out]

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

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Rubi [A]  time = 0.208611, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.2516, size = 129, normalized size = 0.95 \[ - \frac{2 \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{f + g x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{\sqrt{c} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)

[Out]

-2*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(f + g*x)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*
sqrt(-a)) + 1/2)), 2*a*g/(a*g - sqrt(c)*f*sqrt(-a)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(
-a)*(-f - g*x)/(a*g - sqrt(c)*f*sqrt(-a)))*sqrt(a + c*x**2))

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Mathematica [C]  time = 0.656388, size = 294, normalized size = 2.16 \[ \frac{2 i \sqrt{f+g x} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (\sqrt{a}+i \sqrt{c} x\right )}{\sqrt{a} g-i \sqrt{c} f}} \left (E\left (i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )-F\left (i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{\sqrt{c} g \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{g \left (\sqrt{c} x+i \sqrt{a}\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*
f + Sqrt[a]*g)]*Sqrt[f + g*x]*(EllipticE[I*ArcSinh[Sqrt[-((Sqrt[c]*(f + g*x))/(S
qrt[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])

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Maple [B]  time = 0.04, size = 396, normalized size = 2.9 \[ 2\,{\frac{\sqrt{gx+f}\sqrt{c{x}^{2}+a} \left ( cf-g\sqrt{-ac} \right ) }{g \left ( cg{x}^{3}+cf{x}^{2}+agx+fa \right ){c}^{2}}\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}} \left ( \sqrt{-ac}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) g+f{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) c-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) \sqrt{-ac}g-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) cf \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{g x + f}}{\sqrt{c x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(g*x + f)/sqrt(c*x^2 + a),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{g x + f}}{\sqrt{c x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(g*x + f)/sqrt(c*x^2 + a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{f + g x}}{\sqrt{a + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{g x + f}}{\sqrt{c x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(g*x + f)/sqrt(c*x^2 + a),x, algorithm="giac")

[Out]

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