3.2296 \(\int (a+b \sqrt [3]{x})^2 x^2 \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^3}{3}+\frac{3}{5} a b x^{10/3}+\frac{3}{11} b^2 x^{11/3} \]

[Out]

(a^2*x^3)/3 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(11/3))/11

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Rubi [A]  time = 0.024967, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^2 x^3}{3}+\frac{3}{5} a b x^{10/3}+\frac{3}{11} b^2 x^{11/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^2*x^2,x]

[Out]

(a^2*x^3)/3 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(11/3))/11

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^2 \, dx &=3 \operatorname{Subst}\left (\int x^8 (a+b x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^2 x^8+2 a b x^9+b^2 x^{10}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^2 x^3}{3}+\frac{3}{5} a b x^{10/3}+\frac{3}{11} b^2 x^{11/3}\\ \end{align*}

Mathematica [A]  time = 0.0192821, size = 34, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{3}{5} a b x^{10/3}+\frac{3}{11} b^2 x^{11/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^2*x^2,x]

[Out]

(a^2*x^3)/3 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(11/3))/11

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Maple [A]  time = 0.002, size = 25, normalized size = 0.7 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{3\,ab}{5}{x}^{{\frac{10}{3}}}}+{\frac{3\,{b}^{2}}{11}{x}^{{\frac{11}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*a^2+3/5*a*b*x^(10/3)+3/11*b^2*x^(11/3)

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Maxima [B]  time = 0.988431, size = 200, normalized size = 5.88 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11}}{11 \, b^{9}} - \frac{12 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a}{5 \, b^{9}} + \frac{28 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{2}}{3 \, b^{9}} - \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{3}}{b^{9}} + \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{4}}{b^{9}} - \frac{28 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{5}}{b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{6}}{5 \, b^{9}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{7}}{b^{9}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{8}}{b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/11*(b*x^(1/3) + a)^11/b^9 - 12/5*(b*x^(1/3) + a)^10*a/b^9 + 28/3*(b*x^(1/3) + a)^9*a^2/b^9 - 21*(b*x^(1/3) +
a)^8*a^3/b^9 + 30*(b*x^(1/3) + a)^7*a^4/b^9 - 28*(b*x^(1/3) + a)^6*a^5/b^9 + 84/5*(b*x^(1/3) + a)^5*a^6/b^9 -
6*(b*x^(1/3) + a)^4*a^7/b^9 + (b*x^(1/3) + a)^3*a^8/b^9

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Fricas [A]  time = 1.44221, size = 70, normalized size = 2.06 \begin{align*} \frac{3}{11} \, b^{2} x^{\frac{11}{3}} + \frac{3}{5} \, a b x^{\frac{10}{3}} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/11*b^2*x^(11/3) + 3/5*a*b*x^(10/3) + 1/3*a^2*x^3

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Sympy [A]  time = 1.76555, size = 31, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{3 a b x^{\frac{10}{3}}}{5} + \frac{3 b^{2} x^{\frac{11}{3}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**3/3 + 3*a*b*x**(10/3)/5 + 3*b**2*x**(11/3)/11

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Giac [A]  time = 1.08502, size = 32, normalized size = 0.94 \begin{align*} \frac{3}{11} \, b^{2} x^{\frac{11}{3}} + \frac{3}{5} \, a b x^{\frac{10}{3}} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/11*b^2*x^(11/3) + 3/5*a*b*x^(10/3) + 1/3*a^2*x^3