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

Optimal. Leaf size=113 \[ \frac{3 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{8 b^{3/2}}-\frac{3 a^2 \sqrt{a x+b x^{2/3}}}{8 b x^{2/3}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{4 x}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{x^2} \]

[Out]

(-3*a*Sqrt[b*x^(2/3) + a*x])/(4*x) - (3*a^2*Sqrt[b*x^(2/3) + a*x])/(8*b*x^(2/3))
- (b*x^(2/3) + a*x)^(3/2)/x^2 + (3*a^3*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3)
+ a*x]])/(8*b^(3/2))

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Rubi [A]  time = 0.31782, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{8 b^{3/2}}-\frac{3 a^2 \sqrt{a x+b x^{2/3}}}{8 b x^{2/3}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{4 x}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a*Sqrt[b*x^(2/3) + a*x])/(4*x) - (3*a^2*Sqrt[b*x^(2/3) + a*x])/(8*b*x^(2/3))
- (b*x^(2/3) + a*x)^(3/2)/x^2 + (3*a^3*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3)
+ a*x]])/(8*b^(3/2))

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Rubi in Sympy [A]  time = 26.3269, size = 100, normalized size = 0.88 \[ \frac{3 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{8 b^{\frac{3}{2}}} - \frac{3 a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{8 b x^{\frac{2}{3}}} - \frac{3 a \sqrt{a x + b x^{\frac{2}{3}}}}{4 x} - \frac{\left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**3*atanh(sqrt(b)*x**(1/3)/sqrt(a*x + b*x**(2/3)))/(8*b**(3/2)) - 3*a**2*sqrt
(a*x + b*x**(2/3))/(8*b*x**(2/3)) - 3*a*sqrt(a*x + b*x**(2/3))/(4*x) - (a*x + b*
x**(2/3))**(3/2)/x**2

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Mathematica [A]  time = 0.134653, size = 92, normalized size = 0.81 \[ \frac{3 a^3 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{8 b^{3/2}}-\frac{\left (3 a^2 x^{2/3}+14 a b \sqrt [3]{x}+8 b^2\right ) \sqrt{a x+b x^{2/3}}}{8 b x^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

-((8*b^2 + 14*a*b*x^(1/3) + 3*a^2*x^(2/3))*Sqrt[b*x^(2/3) + a*x])/(8*b*x^(4/3))
+ (3*a^3*ArcTanh[Sqrt[b*x^(2/3) + a*x]/(Sqrt[b]*x^(1/3))])/(8*b^(3/2))

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Maple [A]  time = 0.005, size = 93, normalized size = 0.8 \[{\frac{1}{8\,{x}^{2}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{7/2}\sqrt{b+a\sqrt [3]{x}}-8\,{b}^{5/2} \left ( b+a\sqrt [3]{x} \right ) ^{3/2}-3\,{b}^{3/2} \left ( b+a\sqrt [3]{x} \right ) ^{5/2}+3\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ) x{a}^{3}b \right ){b}^{-{\frac{5}{2}}} \left ( b+a\sqrt [3]{x} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x^(2/3)+a*x)^(3/2)*(3*b^(7/2)*(b+a*x^(1/3))^(1/2)-8*b^(5/2)*(b+a*x^(1/3))
^(3/2)-3*b^(3/2)*(b+a*x^(1/3))^(5/2)+3*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*x*a^
3*b)/x^2/(b+a*x^(1/3))^(3/2)/b^(5/2)

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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^3,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^3,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.255469, size = 146, normalized size = 1.29 \[ -\frac{\frac{3 \, a^{4} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right ){\rm sign}\left (x^{\frac{1}{3}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{4}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 8 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{4} b{\rm sign}\left (x^{\frac{1}{3}}\right ) - 3 \, \sqrt{a x^{\frac{1}{3}} + b} a^{4} b^{2}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{3} b x}}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(3*a^4*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))*sign(x^(1/3))/(sqrt(-b)*b) + (3
*(a*x^(1/3) + b)^(5/2)*a^4*sign(x^(1/3)) + 8*(a*x^(1/3) + b)^(3/2)*a^4*b*sign(x^
(1/3)) - 3*sqrt(a*x^(1/3) + b)*a^4*b^2*sign(x^(1/3)))/(a^3*b*x))/a