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

Optimal. Leaf size=32 \[ \frac{3 (a+b x)^{4/3}}{4 b^2}-\frac{3 a \sqrt [3]{a+b x}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0074868, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{3 (a+b x)^{4/3}}{4 b^2}-\frac{3 a \sqrt [3]{a+b x}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^(2/3),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x}{(a+b x)^{2/3}} \, dx &=\int \left (-\frac{a}{b (a+b x)^{2/3}}+\frac{\sqrt [3]{a+b x}}{b}\right ) \, dx\\ &=-\frac{3 a \sqrt [3]{a+b x}}{b^2}+\frac{3 (a+b x)^{4/3}}{4 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0179814, size = 23, normalized size = 0.72 \[ \frac{3 (b x-3 a) \sqrt [3]{a+b x}}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^(2/3),x]

[Out]

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

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Maple [A]  time = 0.001, size = 21, normalized size = 0.7 \begin{align*} -{\frac{-3\,bx+9\,a}{4\,{b}^{2}}\sqrt [3]{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04654, size = 35, normalized size = 1.09 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}}}{4 \, b^{2}} - \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} a}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48452, size = 50, normalized size = 1.56 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}}{\left (b x - 3 \, a\right )}}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(b*x + a)^(1/3)*(b*x - 3*a)/b^2

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Sympy [B]  time = 1.70781, size = 162, normalized size = 5.06 \begin{align*} - \frac{9 a^{\frac{10}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac{9 a^{\frac{10}{3}}}{4 a^{2} b^{2} + 4 a b^{3} x} - \frac{6 a^{\frac{7}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac{9 a^{\frac{7}{3}} b x}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac{3 a^{\frac{4}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*a**(10/3)*(1 + b*x/a)**(1/3)/(4*a**2*b**2 + 4*a*b**3*x) + 9*a**(10/3)/(4*a**2*b**2 + 4*a*b**3*x) - 6*a**(7/
3)*b*x*(1 + b*x/a)**(1/3)/(4*a**2*b**2 + 4*a*b**3*x) + 9*a**(7/3)*b*x/(4*a**2*b**2 + 4*a*b**3*x) + 3*a**(4/3)*
b**2*x**2*(1 + b*x/a)**(1/3)/(4*a**2*b**2 + 4*a*b**3*x)

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Giac [A]  time = 1.10816, size = 31, normalized size = 0.97 \begin{align*} \frac{3 \,{\left ({\left (b x + a\right )}^{\frac{4}{3}} - 4 \,{\left (b x + a\right )}^{\frac{1}{3}} a\right )}}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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