3.394 \(\int \frac{x}{\sqrt [3]{a+b x}} \, dx\)

Optimal. Leaf size=34 \[ \frac{3 (a+b x)^{5/3}}{5 b^2}-\frac{3 a (a+b x)^{2/3}}{2 b^2} \]

[Out]

(-3*a*(a + b*x)^(2/3))/(2*b^2) + (3*(a + b*x)^(5/3))/(5*b^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0078093, antiderivative size = 34, 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} \[ \frac{3 (a+b x)^{5/3}}{5 b^2}-\frac{3 a (a+b x)^{2/3}}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^(1/3),x]

[Out]

(-3*a*(a + b*x)^(2/3))/(2*b^2) + (3*(a + b*x)^(5/3))/(5*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}{\sqrt [3]{a+b x}} \, dx &=\int \left (-\frac{a}{b \sqrt [3]{a+b x}}+\frac{(a+b x)^{2/3}}{b}\right ) \, dx\\ &=-\frac{3 a (a+b x)^{2/3}}{2 b^2}+\frac{3 (a+b x)^{5/3}}{5 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0273163, size = 24, normalized size = 0.71 \[ \frac{3 (a+b x)^{2/3} (2 b x-3 a)}{10 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(2/3)*(-3*a + 2*b*x))/(10*b^2)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-6\,bx+9\,a}{10\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x+a)^(1/3),x)

[Out]

-3/10*(b*x+a)^(2/3)*(-2*b*x+3*a)/b^2

________________________________________________________________________________________

Maxima [A]  time = 1.08437, size = 35, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}}}{5 \, b^{2}} - \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^(1/3),x, algorithm="maxima")

[Out]

3/5*(b*x + a)^(5/3)/b^2 - 3/2*(b*x + a)^(2/3)*a/b^2

________________________________________________________________________________________

Fricas [A]  time = 1.64726, size = 54, normalized size = 1.59 \begin{align*} \frac{3 \,{\left (2 \, b x - 3 \, a\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{10 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^(1/3),x, algorithm="fricas")

[Out]

3/10*(2*b*x - 3*a)*(b*x + a)^(2/3)/b^2

________________________________________________________________________________________

Sympy [B]  time = 1.44533, size = 162, normalized size = 4.76 \begin{align*} - \frac{9 a^{\frac{11}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac{9 a^{\frac{11}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} - \frac{3 a^{\frac{8}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac{9 a^{\frac{8}{3}} b x}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac{6 a^{\frac{5}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)**(1/3),x)

[Out]

-9*a**(11/3)*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x) + 9*a**(11/3)/(10*a**2*b**2 + 10*a*b**3*x) - 3*a*
*(8/3)*b*x*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x) + 9*a**(8/3)*b*x/(10*a**2*b**2 + 10*a*b**3*x) + 6*a
**(5/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x)

________________________________________________________________________________________

Giac [A]  time = 1.15929, size = 34, normalized size = 1. \begin{align*} \frac{3 \,{\left (2 \,{\left (b x + a\right )}^{\frac{5}{3}} - 5 \,{\left (b x + a\right )}^{\frac{2}{3}} a\right )}}{10 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^(1/3),x, algorithm="giac")

[Out]

3/10*(2*(b*x + a)^(5/3) - 5*(b*x + a)^(2/3)*a)/b^2