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

Optimal. Leaf size=51 \[ \frac{9}{8} a^2 b x^{8/3}+\frac{3}{5} a^3 x^{5/3}+\frac{9}{11} a b^2 x^{11/3}+\frac{3}{14} b^3 x^{14/3} \]

[Out]

(3*a^3*x^(5/3))/5 + (9*a^2*b*x^(8/3))/8 + (9*a*b^2*x^(11/3))/11 + (3*b^3*x^(14/3))/14

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Rubi [A]  time = 0.010782, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{9}{8} a^2 b x^{8/3}+\frac{3}{5} a^3 x^{5/3}+\frac{9}{11} a b^2 x^{11/3}+\frac{3}{14} b^3 x^{14/3} \]

Antiderivative was successfully verified.

[In]

Int[x^(2/3)*(a + b*x)^3,x]

[Out]

(3*a^3*x^(5/3))/5 + (9*a^2*b*x^(8/3))/8 + (9*a*b^2*x^(11/3))/11 + (3*b^3*x^(14/3))/14

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^{2/3} (a+b x)^3 \, dx &=\int \left (a^3 x^{2/3}+3 a^2 b x^{5/3}+3 a b^2 x^{8/3}+b^3 x^{11/3}\right ) \, dx\\ &=\frac{3}{5} a^3 x^{5/3}+\frac{9}{8} a^2 b x^{8/3}+\frac{9}{11} a b^2 x^{11/3}+\frac{3}{14} b^3 x^{14/3}\\ \end{align*}

Mathematica [A]  time = 0.0103705, size = 39, normalized size = 0.76 \[ \frac{3 x^{5/3} \left (1155 a^2 b x+616 a^3+840 a b^2 x^2+220 b^3 x^3\right )}{3080} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)*(a + b*x)^3,x]

[Out]

(3*x^(5/3)*(616*a^3 + 1155*a^2*b*x + 840*a*b^2*x^2 + 220*b^3*x^3))/3080

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Maple [A]  time = 0.004, size = 36, normalized size = 0.7 \begin{align*}{\frac{660\,{b}^{3}{x}^{3}+2520\,a{b}^{2}{x}^{2}+3465\,{a}^{2}bx+1848\,{a}^{3}}{3080}{x}^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/3080*x^(5/3)*(220*b^3*x^3+840*a*b^2*x^2+1155*a^2*b*x+616*a^3)

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Maxima [A]  time = 1.18694, size = 47, normalized size = 0.92 \begin{align*} \frac{3}{14} \, b^{3} x^{\frac{14}{3}} + \frac{9}{11} \, a b^{2} x^{\frac{11}{3}} + \frac{9}{8} \, a^{2} b x^{\frac{8}{3}} + \frac{3}{5} \, a^{3} x^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/14*b^3*x^(14/3) + 9/11*a*b^2*x^(11/3) + 9/8*a^2*b*x^(8/3) + 3/5*a^3*x^(5/3)

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Fricas [A]  time = 1.45842, size = 101, normalized size = 1.98 \begin{align*} \frac{3}{3080} \,{\left (220 \, b^{3} x^{4} + 840 \, a b^{2} x^{3} + 1155 \, a^{2} b x^{2} + 616 \, a^{3} x\right )} x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/3080*(220*b^3*x^4 + 840*a*b^2*x^3 + 1155*a^2*b*x^2 + 616*a^3*x)*x^(2/3)

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Sympy [A]  time = 2.50533, size = 49, normalized size = 0.96 \begin{align*} \frac{3 a^{3} x^{\frac{5}{3}}}{5} + \frac{9 a^{2} b x^{\frac{8}{3}}}{8} + \frac{9 a b^{2} x^{\frac{11}{3}}}{11} + \frac{3 b^{3} x^{\frac{14}{3}}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a**3*x**(5/3)/5 + 9*a**2*b*x**(8/3)/8 + 9*a*b**2*x**(11/3)/11 + 3*b**3*x**(14/3)/14

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Giac [A]  time = 1.05635, size = 47, normalized size = 0.92 \begin{align*} \frac{3}{14} \, b^{3} x^{\frac{14}{3}} + \frac{9}{11} \, a b^{2} x^{\frac{11}{3}} + \frac{9}{8} \, a^{2} b x^{\frac{8}{3}} + \frac{3}{5} \, a^{3} x^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/14*b^3*x^(14/3) + 9/11*a*b^2*x^(11/3) + 9/8*a^2*b*x^(8/3) + 3/5*a^3*x^(5/3)