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

Optimal. Leaf size=68 \[ -\frac{16 b \sqrt{a x+b x^{2/3}}}{a^3 \sqrt [3]{x}}+\frac{8 \sqrt{a x+b x^{2/3}}}{a^2}-\frac{6 x}{a \sqrt{a x+b x^{2/3}}} \]

[Out]

(-6*x)/(a*Sqrt[b*x^(2/3) + a*x]) + (8*Sqrt[b*x^(2/3) + a*x])/a^2 - (16*b*Sqrt[b*
x^(2/3) + a*x])/(a^3*x^(1/3))

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Rubi [A]  time = 0.142879, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{16 b \sqrt{a x+b x^{2/3}}}{a^3 \sqrt [3]{x}}+\frac{8 \sqrt{a x+b x^{2/3}}}{a^2}-\frac{6 x}{a \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*x)/(a*Sqrt[b*x^(2/3) + a*x]) + (8*Sqrt[b*x^(2/3) + a*x])/a^2 - (16*b*Sqrt[b*
x^(2/3) + a*x])/(a^3*x^(1/3))

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Rubi in Sympy [A]  time = 12.8231, size = 61, normalized size = 0.9 \[ - \frac{6 x}{a \sqrt{a x + b x^{\frac{2}{3}}}} + \frac{8 \sqrt{a x + b x^{\frac{2}{3}}}}{a^{2}} - \frac{16 b \sqrt{a x + b x^{\frac{2}{3}}}}{a^{3} \sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*x/(a*sqrt(a*x + b*x**(2/3))) + 8*sqrt(a*x + b*x**(2/3))/a**2 - 16*b*sqrt(a*x
+ b*x**(2/3))/(a**3*x**(1/3))

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Mathematica [A]  time = 0.0546141, size = 45, normalized size = 0.66 \[ \frac{2 \left (\frac{3 a b}{a \sqrt [3]{x}+b}+a-\frac{8 b}{\sqrt [3]{x}}\right ) \sqrt{a x+b x^{2/3}}}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 45, normalized size = 0.7 \[ 2\,{\frac{x \left ( b+a\sqrt [3]{x} \right ) \left ({a}^{2}{x}^{2/3}-4\,ab\sqrt [3]{x}-8\,{b}^{2} \right ) }{ \left ( b{x}^{2/3}+ax \right ) ^{3/2}{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^(2/3)+a*x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43205, size = 63, normalized size = 0.93 \[ \frac{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}}}{a^{3}} - \frac{12 \, \sqrt{a x^{\frac{1}{3}} + b} b}{a^{3}} - \frac{6 \, b^{2}}{\sqrt{a x^{\frac{1}{3}} + b} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*x^(1/3) + b)^(3/2)/a^3 - 12*sqrt(a*x^(1/3) + b)*b/a^3 - 6*b^2/(sqrt(a*x^(1/
3) + b)*a^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224354, size = 103, normalized size = 1.51 \[ \frac{16 \, b^{\frac{3}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{3}} - \frac{6 \, b^{2}}{\sqrt{a x^{\frac{1}{3}} + b} a^{3}{\rm sign}\left (x^{\frac{1}{3}}\right )} + \frac{2 \,{\left ({\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{6} - 6 \, \sqrt{a x^{\frac{1}{3}} + b} a^{6} b\right )}}{a^{9}{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*b^(3/2)*sign(x^(1/3))/a^3 - 6*b^2/(sqrt(a*x^(1/3) + b)*a^3*sign(x^(1/3))) + 2
*((a*x^(1/3) + b)^(3/2)*a^6 - 6*sqrt(a*x^(1/3) + b)*a^6*b)/(a^9*sign(x^(1/3)))