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

Optimal. Leaf size=383 \[ \frac{77 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{77 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 a^2 \sqrt{a x+b \sqrt [3]{x}}}{5 b^4 \sqrt [3]{x}}+\frac{77 a \sqrt{a x+b \sqrt [3]{x}}}{15 b^3 x}-\frac{11 \sqrt{a x+b \sqrt [3]{x}}}{3 b^2 x^{5/3}}+\frac{3}{b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}} \]

[Out]

3/(b*x^(4/3)*Sqrt[b*x^(1/3) + a*x]) + (77*a^(5/2)*(b + a*x^(2/3))*x^(1/3))/(5*b^
4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) - (11*Sqrt[b*x^(1/3) + a*x]
)/(3*b^2*x^(5/3)) + (77*a*Sqrt[b*x^(1/3) + a*x])/(15*b^3*x) - (77*a^2*Sqrt[b*x^(
1/3) + a*x])/(5*b^4*x^(1/3)) - (77*a^(9/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b +
a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticE[2*ArcTan[(a^(1/4)*x
^(1/6))/b^(1/4)], 1/2])/(5*b^(15/4)*Sqrt[b*x^(1/3) + a*x]) + (77*a^(9/4)*(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])/(10*b^(15/4)*Sqrt[b*x^(1/3
) + a*x])

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Rubi [A]  time = 0.927327, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{77 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{77 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 a^2 \sqrt{a x+b \sqrt [3]{x}}}{5 b^4 \sqrt [3]{x}}+\frac{77 a \sqrt{a x+b \sqrt [3]{x}}}{15 b^3 x}-\frac{11 \sqrt{a x+b \sqrt [3]{x}}}{3 b^2 x^{5/3}}+\frac{3}{b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

3/(b*x^(4/3)*Sqrt[b*x^(1/3) + a*x]) + (77*a^(5/2)*(b + a*x^(2/3))*x^(1/3))/(5*b^
4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) - (11*Sqrt[b*x^(1/3) + a*x]
)/(3*b^2*x^(5/3)) + (77*a*Sqrt[b*x^(1/3) + a*x])/(15*b^3*x) - (77*a^2*Sqrt[b*x^(
1/3) + a*x])/(5*b^4*x^(1/3)) - (77*a^(9/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b +
a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticE[2*ArcTan[(a^(1/4)*x
^(1/6))/b^(1/4)], 1/2])/(5*b^(15/4)*Sqrt[b*x^(1/3) + a*x]) + (77*a^(9/4)*(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])/(10*b^(15/4)*Sqrt[b*x^(1/3
) + a*x])

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Rubi in Sympy [A]  time = 83.5986, size = 354, normalized size = 0.92 \[ - \frac{77 a^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{77 a^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{10 b^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{77 a^{\frac{5}{2}} \sqrt{a x + b \sqrt [3]{x}}}{5 b^{4} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} - \frac{77 a^{2} \sqrt{a x + b \sqrt [3]{x}}}{5 b^{4} \sqrt [3]{x}} + \frac{77 a \sqrt{a x + b \sqrt [3]{x}}}{15 b^{3} x} + \frac{3}{b x^{\frac{4}{3}} \sqrt{a x + b \sqrt [3]{x}}} - \frac{11 \sqrt{a x + b \sqrt [3]{x}}}{3 b^{2} x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-77*a**(9/4)*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_e(2*atan(a**(1/4)*x**(1/6)/b**(
1/4)), 1/2)/(5*b**(15/4)*x**(1/6)*(a*x**(2/3) + b)) + 77*a**(9/4)*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)/(10*b**(15/4)*x
**(1/6)*(a*x**(2/3) + b)) + 77*a**(5/2)*sqrt(a*x + b*x**(1/3))/(5*b**4*(sqrt(a)*
x**(1/3) + sqrt(b))) - 77*a**2*sqrt(a*x + b*x**(1/3))/(5*b**4*x**(1/3)) + 77*a*s
qrt(a*x + b*x**(1/3))/(15*b**3*x) + 3/(b*x**(4/3)*sqrt(a*x + b*x**(1/3))) - 11*s
qrt(a*x + b*x**(1/3))/(3*b**2*x**(5/3))

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Mathematica [C]  time = 0.0891386, size = 108, normalized size = 0.28 \[ \frac{231 a^3 x^2 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )-231 a^3 x^2-154 a^2 b x^{4/3}+22 a b^2 x^{2/3}-10 b^3}{15 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*b^3 + 22*a*b^2*x^(2/3) - 154*a^2*b*x^(4/3) - 231*a^3*x^2 + 231*a^3*Sqrt[1 +
b/(a*x^(2/3))]*x^2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(b/(a*x^(2/3)))])/(15*b^4
*x^(4/3)*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.02, size = 339, normalized size = 0.9 \[{\frac{1}{30\,{x}^{3}{b}^{4}} \left ( 462\,{a}^{2}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -231\,{a}^{2}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -462\,\sqrt{b\sqrt [3]{x}+ax}{x}^{10/3}{a}^{3}-372\,\sqrt{b\sqrt [3]{x}+ax}{x}^{8/3}{a}^{2}b+44\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{2}a{b}^{2}+64\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{8/3}{a}^{2}b-20\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{4/3}{b}^{3} \right ) \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(462*a^2*b*((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)*x^(8/3)*(x^(1/3)*
(b+a*x^(2/3)))^(1/2)*EllipticE(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2
*2^(1/2))-231*a^2*b*((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)*x^(8/3)*(x^(1
/3)*(b+a*x^(2/3)))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)
,1/2*2^(1/2))-462*(b*x^(1/3)+a*x)^(1/2)*x^(10/3)*a^3-372*(b*x^(1/3)+a*x)^(1/2)*x
^(8/3)*a^2*b+44*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^2*a*b^2+64*(x^(1/3)*(b+a*x^(2/3)
))^(1/2)*x^(8/3)*a^2*b-20*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(4/3)*b^3)/x^3/(b+a*x^
(2/3))/b^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (a x^{3} + b x^{\frac{7}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((a*x^3 + b*x^(7/3))*sqrt(a*x + b*x^(1/3))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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