3.2287 \(\int (a+b \sqrt [3]{x}) x^2 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^3}{3}+\frac{3}{10} b x^{10/3} \]

[Out]

(a*x^3)/3 + (3*b*x^(10/3))/10

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Rubi [A]  time = 0.0046951, antiderivative size = 19, 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 = {14} \[ \frac{a x^3}{3}+\frac{3}{10} b x^{10/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))*x^2,x]

[Out]

(a*x^3)/3 + (3*b*x^(10/3))/10

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx &=\int \left (a x^2+b x^{7/3}\right ) \, dx\\ &=\frac{a x^3}{3}+\frac{3}{10} b x^{10/3}\\ \end{align*}

Mathematica [A]  time = 0.0063741, size = 19, normalized size = 1. \[ \frac{a x^3}{3}+\frac{3}{10} b x^{10/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))*x^2,x]

[Out]

(a*x^3)/3 + (3*b*x^(10/3))/10

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \begin{align*}{\frac{a{x}^{3}}{3}}+{\frac{3\,b}{10}{x}^{{\frac{10}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3+3/10*b*x^(10/3)

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Maxima [B]  time = 0.959275, size = 201, normalized size = 10.58 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10}}{10 \, b^{9}} - \frac{8 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a}{3 \, b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{2}}{2 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{3}}{b^{9}} + \frac{35 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{4}}{b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{5}}{5 \, b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{6}}{b^{9}} - \frac{8 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{7}}{b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{8}}{2 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(b*x^(1/3) + a)^10/b^9 - 8/3*(b*x^(1/3) + a)^9*a/b^9 + 21/2*(b*x^(1/3) + a)^8*a^2/b^9 - 24*(b*x^(1/3) + a
)^7*a^3/b^9 + 35*(b*x^(1/3) + a)^6*a^4/b^9 - 168/5*(b*x^(1/3) + a)^5*a^5/b^9 + 21*(b*x^(1/3) + a)^4*a^6/b^9 -
8*(b*x^(1/3) + a)^3*a^7/b^9 + 3/2*(b*x^(1/3) + a)^2*a^8/b^9

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Fricas [A]  time = 1.43681, size = 39, normalized size = 2.05 \begin{align*} \frac{3}{10} \, b x^{\frac{10}{3}} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*b*x^(10/3) + 1/3*a*x^3

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Sympy [A]  time = 1.50234, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{3}}{3} + \frac{3 b x^{\frac{10}{3}}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + 3*b*x**(10/3)/10

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Giac [A]  time = 1.11108, size = 18, normalized size = 0.95 \begin{align*} \frac{3}{10} \, b x^{\frac{10}{3}} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*b*x^(10/3) + 1/3*a*x^3