3.180 \(\int \frac{\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=78 \[ -6 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )+\frac{6 b \sqrt{a x+b x^{2/3}}}{\sqrt [3]{x}}+\frac{2 \left (a x+b x^{2/3}\right )^{3/2}}{x} \]

[Out]

(6*b*Sqrt[b*x^(2/3) + a*x])/x^(1/3) + (2*(b*x^(2/3) + a*x)^(3/2))/x - 6*b^(3/2)*
ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]]

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Rubi [A]  time = 0.228961, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -6 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )+\frac{6 b \sqrt{a x+b x^{2/3}}}{\sqrt [3]{x}}+\frac{2 \left (a x+b x^{2/3}\right )^{3/2}}{x} \]

Antiderivative was successfully verified.

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

[Out]

(6*b*Sqrt[b*x^(2/3) + a*x])/x^(1/3) + (2*(b*x^(2/3) + a*x)^(3/2))/x - 6*b^(3/2)*
ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]]

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Rubi in Sympy [A]  time = 19.4475, size = 70, normalized size = 0.9 \[ - 6 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )} + \frac{6 b \sqrt{a x + b x^{\frac{2}{3}}}}{\sqrt [3]{x}} + \frac{2 \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*b**(3/2)*atanh(sqrt(b)*x**(1/3)/sqrt(a*x + b*x**(2/3))) + 6*b*sqrt(a*x + b*x*
*(2/3))/x**(1/3) + 2*(a*x + b*x**(2/3))**(3/2)/x

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Mathematica [A]  time = 0.118225, size = 63, normalized size = 0.81 \[ \left (2 a+\frac{8 b}{\sqrt [3]{x}}\right ) \sqrt{a x+b x^{2/3}}-6 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*a + (8*b)/x^(1/3))*Sqrt[b*x^(2/3) + a*x] - 6*b^(3/2)*ArcTanh[Sqrt[b*x^(2/3) +
a*x]/(Sqrt[b]*x^(1/3))]

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Maple [A]  time = 0.006, size = 69, normalized size = 0.9 \[ -2\,{\frac{ \left ( b{x}^{2/3}+ax \right ) ^{3/2}}{x \left ( b+a\sqrt [3]{x} \right ) ^{3/2}} \left ( 3\,{b}^{3/2}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ) - \left ( b+a\sqrt [3]{x} \right ) ^{3/2}-3\,\sqrt{b+a\sqrt [3]{x}}b \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b*x^(2/3)+a*x)^(3/2)*(3*b^(3/2)*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))-(b+a*x^
(1/3))^(3/2)-3*(b+a*x^(1/3))^(1/2)*b)/x/(b+a*x^(1/3))^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227871, size = 134, normalized size = 1.72 \[ \frac{6 \, b^{2} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right ){\rm sign}\left (x^{\frac{1}{3}}\right )}{\sqrt{-b}} + 2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 6 \, \sqrt{a x^{\frac{1}{3}} + b} b{\rm sign}\left (x^{\frac{1}{3}}\right ) - \frac{2 \,{\left (3 \, b^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b} b^{\frac{3}{2}}\right )}{\rm sign}\left (x^{\frac{1}{3}}\right )}{\sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*b^2*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))*sign(x^(1/3))/sqrt(-b) + 2*(a*x^(1/3)
+ b)^(3/2)*sign(x^(1/3)) + 6*sqrt(a*x^(1/3) + b)*b*sign(x^(1/3)) - 2*(3*b^2*arc
tan(sqrt(b)/sqrt(-b)) + 4*sqrt(-b)*b^(3/2))*sign(x^(1/3))/sqrt(-b)