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

Optimal. Leaf size=34 \[ \frac{3 (a+b x)^{8/3}}{8 b^2}-\frac{3 a (a+b x)^{5/3}}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0078079, 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} \[ \frac{3 (a+b x)^{8/3}}{8 b^2}-\frac{3 a (a+b x)^{5/3}}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(2/3),x]

[Out]

(-3*a*(a + b*x)^(5/3))/(5*b^2) + (3*(a + b*x)^(8/3))/(8*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 x (a+b x)^{2/3} \, dx &=\int \left (-\frac{a (a+b x)^{2/3}}{b}+\frac{(a+b x)^{5/3}}{b}\right ) \, dx\\ &=-\frac{3 a (a+b x)^{5/3}}{5 b^2}+\frac{3 (a+b x)^{8/3}}{8 b^2}\\ \end{align*}

Mathematica [A]  time = 0.023607, size = 24, normalized size = 0.71 \[ \frac{3 (a+b x)^{5/3} (5 b x-3 a)}{40 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(5/3)*(-3*a + 5*b*x))/(40*b^2)

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Maple [A]  time = 0.001, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-15\,bx+9\,a}{40\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/40*(b*x+a)^(5/3)*(-5*b*x+3*a)/b^2

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Maxima [A]  time = 1.06682, size = 35, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{8}{3}}}{8 \, b^{2}} - \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}} a}{5 \, 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]

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

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Fricas [A]  time = 1.83083, size = 76, normalized size = 2.24 \begin{align*} \frac{3 \,{\left (5 \, b^{2} x^{2} + 2 \, a b x - 3 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{40 \, 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]

3/40*(5*b^2*x^2 + 2*a*b*x - 3*a^2)*(b*x + a)^(2/3)/b^2

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Sympy [B]  time = 1.5898, size = 202, normalized size = 5.94 \begin{align*} - \frac{9 a^{\frac{14}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac{9 a^{\frac{14}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} - \frac{3 a^{\frac{11}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac{9 a^{\frac{11}{3}} b x}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac{21 a^{\frac{8}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{2} b^{2} + 40 a b^{3} x} + \frac{15 a^{\frac{5}{3}} b^{3} x^{3} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{2} b^{2} + 40 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**(14/3)*(1 + b*x/a)**(2/3)/(40*a**2*b**2 + 40*a*b**3*x) + 9*a**(14/3)/(40*a**2*b**2 + 40*a*b**3*x) - 3*a*
*(11/3)*b*x*(1 + b*x/a)**(2/3)/(40*a**2*b**2 + 40*a*b**3*x) + 9*a**(11/3)*b*x/(40*a**2*b**2 + 40*a*b**3*x) + 2
1*a**(8/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(40*a**2*b**2 + 40*a*b**3*x) + 15*a**(5/3)*b**3*x**3*(1 + b*x/a)**(2/3
)/(40*a**2*b**2 + 40*a*b**3*x)

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Giac [A]  time = 1.19636, size = 34, normalized size = 1. \begin{align*} \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{8}{3}} - 8 \,{\left (b x + a\right )}^{\frac{5}{3}} a\right )}}{40 \, 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]

3/40*(5*(b*x + a)^(8/3) - 8*(b*x + a)^(5/3)*a)/b^2