3.2297 \(\int (a+b \sqrt [3]{x})^2 x \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]

[Out]

(a^2*x^2)/2 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(8/3))/8

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Rubi [A]  time = 0.0202066, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^2*x,x]

[Out]

(a^2*x^2)/2 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(8/3))/8

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 \left (a+b \sqrt [3]{x}\right )^2 x \, dx &=3 \operatorname{Subst}\left (\int x^5 (a+b x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^2 x^5+2 a b x^6+b^2 x^7\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3}\\ \end{align*}

Mathematica [A]  time = 0.0150255, size = 34, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^2*x,x]

[Out]

(a^2*x^2)/2 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(8/3))/8

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Maple [A]  time = 0.001, size = 25, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{6\,ab}{7}{x}^{{\frac{7}{3}}}}+{\frac{3\,{b}^{2}}{8}{x}^{{\frac{8}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^2*x,x)

[Out]

1/2*a^2*x^2+6/7*a*b*x^(7/3)+3/8*b^2*x^(8/3)

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Maxima [B]  time = 0.970035, size = 132, normalized size = 3.88 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8}}{8 \, b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a}{7 \, b^{6}} + \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{2}}{b^{6}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{3}}{b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{4}}{4 \, b^{6}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{5}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^2*x,x, algorithm="maxima")

[Out]

3/8*(b*x^(1/3) + a)^8/b^6 - 15/7*(b*x^(1/3) + a)^7*a/b^6 + 5*(b*x^(1/3) + a)^6*a^2/b^6 - 6*(b*x^(1/3) + a)^5*a
^3/b^6 + 15/4*(b*x^(1/3) + a)^4*a^4/b^6 - (b*x^(1/3) + a)^3*a^5/b^6

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Fricas [A]  time = 1.45875, size = 66, normalized size = 1.94 \begin{align*} \frac{3}{8} \, b^{2} x^{\frac{8}{3}} + \frac{6}{7} \, a b x^{\frac{7}{3}} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^2*x,x, algorithm="fricas")

[Out]

3/8*b^2*x^(8/3) + 6/7*a*b*x^(7/3) + 1/2*a^2*x^2

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Sympy [A]  time = 1.61413, size = 31, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{6 a b x^{\frac{7}{3}}}{7} + \frac{3 b^{2} x^{\frac{8}{3}}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**2*x,x)

[Out]

a**2*x**2/2 + 6*a*b*x**(7/3)/7 + 3*b**2*x**(8/3)/8

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Giac [A]  time = 1.13096, size = 32, normalized size = 0.94 \begin{align*} \frac{3}{8} \, b^{2} x^{\frac{8}{3}} + \frac{6}{7} \, a b x^{\frac{7}{3}} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^2*x,x, algorithm="giac")

[Out]

3/8*b^2*x^(8/3) + 6/7*a*b*x^(7/3) + 1/2*a^2*x^2