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

Optimal. Leaf size=322 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 g} \]

[Out]

(2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*g) + (4*Sqrt[-a]*Sqrt[c]*f*Sqrt[f + g*x]*Sq
rt[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)])/(3*g^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + S
qrt[-a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(c*f^2 + a*g^2)*Sqrt[(Sqrt[c]*(f + g*
x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqr
t[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*Sqrt[c]*g^2
*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.635278, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 g} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*g) + (4*Sqrt[-a]*Sqrt[c]*f*Sqrt[f + g*x]*Sq
rt[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)])/(3*g^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + S
qrt[-a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(c*f^2 + a*g^2)*Sqrt[(Sqrt[c]*(f + g*
x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqr
t[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*Sqrt[c]*g^2
*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 112.412, size = 303, normalized size = 0.94 \[ \frac{4 \sqrt{c} f \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{f + g x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{3 g^{2} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{2 \sqrt{a + c x^{2}} \sqrt{f + g x}}{3 g} - \frac{4 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a g^{2} + c f^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{3 \sqrt{c} g^{2} \sqrt{a + c x^{2}} \sqrt{f + g x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(c)*f*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)))/(3*g**2*sqrt(sqrt(
c)*sqrt(-a)*(-f - g*x)/(a*g - sqrt(c)*f*sqrt(-a)))*sqrt(a + c*x**2)) + 2*sqrt(a
+ c*x**2)*sqrt(f + g*x)/(3*g) - 4*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-f - g*x)/(a*g
- sqrt(c)*f*sqrt(-a)))*sqrt(1 + c*x**2/a)*(a*g**2 + c*f**2)*elliptic_f(asin(sqr
t(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*g/(a*g - sqrt(c)*f*sqrt(-a)))/(3*sqrt(c)*
g**2*sqrt(a + c*x**2)*sqrt(f + g*x))

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Mathematica [C]  time = 3.7268, size = 456, normalized size = 1.42 \[ \frac{2 \sqrt{f+g x} \left (g^2 \left (a+c x^2\right )-\frac{2 \left (f g^2 \left (a+c x^2\right ) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}-\sqrt{a} g (f+g x)^{3/2} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+\sqrt{c} f (f+g x)^{3/2} \left (\sqrt{a} g-i \sqrt{c} f\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{(f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right )}{3 g^3 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[f + g*x]*(g^2*(a + c*x^2) - (2*(f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(
a + c*x^2) + Sqrt[c]*f*((-I)*Sqrt[c]*f + Sqrt[a]*g)*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)^(3/
2)*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[a]*g*(Sqrt[c]*f + I*Sqrt[a]*
g)*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)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sq
rt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(S
qrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(3*g^3*Sqrt[a + c*x^2])

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Maple [B]  time = 0.024, size = 688, normalized size = 2.1 \[ -{\frac{2}{3\,c \left ( cg{x}^{3}+cf{x}^{2}+agx+fa \right ){g}^{3}}\sqrt{gx+f}\sqrt{c{x}^{2}+a} \left ( 2\,\sqrt{-ac}\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) a{g}^{3}+2\,\sqrt{-ac}\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) c{f}^{2}g-2\,ac\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}}{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) f{g}^{2}-2\,\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}}{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ){c}^{2}{f}^{3}-{x}^{3}{c}^{2}{g}^{3}-{x}^{2}{c}^{2}f{g}^{2}-xac{g}^{3}-acf{g}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*x^2+a)^(1/2)*(g*x+f)^(1/2)*(2*(-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)*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))*a*g^3+2*(-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)*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*
f^2*g-2*a*c*(-(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)*Ellipti
cE((-(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))*f*g^2-2*(-(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)*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^2*f^3-x^3*c^2*g^3-x^2*c^2*f*g^2-x*a*c*g^3-a*c*f*g^2)
/c/(c*g*x^3+c*f*x^2+a*g*x+a*f)/g^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError