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

Optimal. Leaf size=60 \[ \frac{6 \sqrt [3]{x}}{b \sqrt{a x+b x^{2/3}}}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.103153, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{6 \sqrt [3]{x}}{b \sqrt{a x+b x^{2/3}}}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^(2/3) + a*x)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.95854, size = 53, normalized size = 0.88 \[ \frac{6 \sqrt [3]{x}}{b \sqrt{a x + b x^{\frac{2}{3}}}} - \frac{6 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.107749, size = 71, normalized size = 1.18 \[ \frac{6 \sqrt{a x+b x^{2/3}}}{b \sqrt [3]{x} \left (a \sqrt [3]{x}+b\right )}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 55, normalized size = 0.9 \[ 6\,{\frac{x \left ( b+a\sqrt [3]{x} \right ) }{ \left ( b{x}^{2/3}+ax \right ) ^{3/2}{b}^{5/2}} \left ({b}^{3/2}-{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ) b\sqrt{b+a\sqrt [3]{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^(2/3)+a*x)^(3/2),x)

[Out]

6*x*(b+a*x^(1/3))*(b^(3/2)-arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*b*(b+a*x^(1/3))^
(1/2))/(b*x^(2/3)+a*x)^(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, 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, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226018, size = 117, normalized size = 1.95 \[ -\frac{6 \,{\left (\sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b}\right )}{\rm sign}\left (x^{\frac{1}{3}}\right )}{\sqrt{-b} b^{\frac{3}{2}}} + \frac{6 \, \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b{\rm sign}\left (x^{\frac{1}{3}}\right )} + \frac{6}{\sqrt{a x^{\frac{1}{3}} + b} b{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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