3.163 \(\int \frac{1}{x \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ -\frac{5 a^{3/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{5 \sqrt{a x+b \sqrt [3]{x}}}{b^2 x^{2/3}}+\frac{3}{b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}} \]

[Out]

3/(b*x^(1/3)*Sqrt[b*x^(1/3) + a*x]) - (5*Sqrt[b*x^(1/3) + a*x])/(b^2*x^(2/3)) -
(5*a^(3/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x
^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(2*b^(9/
4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.393298, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{5 a^{3/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{5 \sqrt{a x+b \sqrt [3]{x}}}{b^2 x^{2/3}}+\frac{3}{b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

3/(b*x^(1/3)*Sqrt[b*x^(1/3) + a*x]) - (5*Sqrt[b*x^(1/3) + a*x])/(b^2*x^(2/3)) -
(5*a^(3/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x
^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(2*b^(9/
4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 31.6078, size = 151, normalized size = 0.96 \[ - \frac{5 a^{\frac{3}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{2 b^{\frac{9}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{3}{b \sqrt [3]{x} \sqrt{a x + b \sqrt [3]{x}}} - \frac{5 \sqrt{a x + b \sqrt [3]{x}}}{b^{2} x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**(3/4)*sqrt((a*x**(2/3) + b)/(sqrt(a)*x**(1/3) + sqrt(b))**2)*(sqrt(a)*x**(
1/3) + sqrt(b))*sqrt(a*x + b*x**(1/3))*elliptic_f(2*atan(a**(1/4)*x**(1/6)/b**(1
/4)), 1/2)/(2*b**(9/4)*x**(1/6)*(a*x**(2/3) + b)) + 3/(b*x**(1/3)*sqrt(a*x + b*x
**(1/3))) - 5*sqrt(a*x + b*x**(1/3))/(b**2*x**(2/3))

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Mathematica [C]  time = 0.0784883, size = 81, normalized size = 0.51 \[ \frac{5 a x^{2/3} \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )-5 a x^{2/3}-2 b}{b^2 \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b - 5*a*x^(2/3) + 5*a*Sqrt[1 + b/(a*x^(2/3))]*x^(2/3)*Hypergeometric2F1[1/4,
1/2, 5/4, -(b/(a*x^(2/3)))])/(b^2*x^(1/3)*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.017, size = 180, normalized size = 1.1 \[ -{\frac{1}{2\,{b}^{2}x} \left ( 5\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{2/3}\sqrt{-ab}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +4\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }\sqrt [3]{x}b+4\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }xa+6\,x\sqrt{b\sqrt [3]{x}+ax}a \right ) \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(5*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(2/3)*(-a*b)^(1/2)*((a*x^(1/3)+(-a*b)^(1
/2))/(-a*b)^(1/2))^(1/2)*(-2*(a*x^(1/3)-(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x^(1
/3)/(-a*b)^(1/2)*a)^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2
),1/2*2^(1/2))+4*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(1/3)*b+4*(x^(1/3)*(b+a*x^(2/3)
))^(1/2)*x*a+6*x*(b*x^(1/3)+a*x)^(1/2)*a)/b^2/x/(b+a*x^(2/3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*x^(1/3))^(3/2)*x),x, algorithm="maxima")

[Out]

integrate(1/((a*x + b*x^(1/3))^(3/2)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (a x^{2} + b x^{\frac{4}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*x^(1/3))^(3/2)*x),x, algorithm="fricas")

[Out]

integral(1/((a*x^2 + b*x^(4/3))*sqrt(a*x + b*x^(1/3))), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(a*x + b*x**(1/3))**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*x^(1/3))^(3/2)*x),x, algorithm="giac")

[Out]

integrate(1/((a*x + b*x^(1/3))^(3/2)*x), x)