3.60 \(\int \frac{x}{\sqrt{a x+b x^3}} \, dx\)

Optimal. Leaf size=229 \[ \frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b x^3}}-\frac{2 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b x^3}}+\frac{2 x \left (a+b x^2\right )}{\sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}} \]

[Out]

(2*x*(a + b*x^2))/(Sqrt[b]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[a*x + b*x^3]) - (2*a^(1/4)
*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Ellipti
cE[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(b^(3/4)*Sqrt[a*x + b*x^3]) + (a^(
1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Ell
ipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(b^(3/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.340734, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b x^3}}-\frac{2 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b x^3}}+\frac{2 x \left (a+b x^2\right )}{\sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a*x + b*x^3],x]

[Out]

(2*x*(a + b*x^2))/(Sqrt[b]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[a*x + b*x^3]) - (2*a^(1/4)
*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Ellipti
cE[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(b^(3/4)*Sqrt[a*x + b*x^3]) + (a^(
1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Ell
ipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(b^(3/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 31.567, size = 214, normalized size = 0.93 \[ - \frac{2 \sqrt [4]{a} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt{x} \left (a + b x^{2}\right )} + \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt{x} \left (a + b x^{2}\right )} + \frac{2 \sqrt{a x + b x^{3}}}{\sqrt{b} \left (\sqrt{a} + \sqrt{b} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x**3+a*x)**(1/2),x)

[Out]

-2*a**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*sq
rt(a*x + b*x**3)*elliptic_e(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(b**(3/4)*sq
rt(x)*(a + b*x**2)) + a**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt
(a) + sqrt(b)*x)*sqrt(a*x + b*x**3)*elliptic_f(2*atan(b**(1/4)*sqrt(x)/a**(1/4))
, 1/2)/(b**(3/4)*sqrt(x)*(a + b*x**2)) + 2*sqrt(a*x + b*x**3)/(sqrt(b)*(sqrt(a)
+ sqrt(b)*x))

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Mathematica [C]  time = 0.0761848, size = 108, normalized size = 0.47 \[ \frac{2 i x^2 \sqrt{\frac{b x^2}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{b} x}{\sqrt{a}}\right )^{3/2} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a*x + b*x^3],x]

[Out]

((2*I)*x^2*Sqrt[1 + (b*x^2)/a]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]]
, -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1]))/(((I*Sqrt[b]*x)/
Sqrt[a])^(3/2)*Sqrt[x*(a + b*x^2)])

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Maple [A]  time = 0.023, size = 158, normalized size = 0.7 \[{\frac{1}{b}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^3+a*x)^(1/2),x)

[Out]

1/b*(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-2*(x-1/b*(-a*b)^(
1/2))*b/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)/(b*x^3+a*x)^(1/2)*(-2/b*(-
a*b)^(1/2)*EllipticE(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/
b*(-a*b)^(1/2)*EllipticF(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2)
))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{b x^{3} + a x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*x^3 + a*x),x, algorithm="maxima")

[Out]

integrate(x/sqrt(b*x^3 + a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{b x^{3} + a x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*x^3 + a*x),x, algorithm="fricas")

[Out]

integral(x/sqrt(b*x^3 + a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x \left (a + b x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**3+a*x)**(1/2),x)

[Out]

Integral(x/sqrt(x*(a + b*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{b x^{3} + a x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(b*x^3 + a*x),x, algorithm="giac")

[Out]

integrate(x/sqrt(b*x^3 + a*x), x)