3.2314 \(\int (a+b \sqrt [3]{x})^5 x^2 \, dx\)

Optimal. Leaf size=77 \[ \frac{30}{11} a^3 b^2 x^{11/3}+\frac{5}{2} a^2 b^3 x^4+\frac{3}{2} a^4 b x^{10/3}+\frac{a^5 x^3}{3}+\frac{15}{13} a b^4 x^{13/3}+\frac{3}{14} b^5 x^{14/3} \]

[Out]

(a^5*x^3)/3 + (3*a^4*b*x^(10/3))/2 + (30*a^3*b^2*x^(11/3))/11 + (5*a^2*b^3*x^4)/2 + (15*a*b^4*x^(13/3))/13 + (
3*b^5*x^(14/3))/14

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Rubi [A]  time = 0.0447806, antiderivative size = 77, 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} \[ \frac{30}{11} a^3 b^2 x^{11/3}+\frac{5}{2} a^2 b^3 x^4+\frac{3}{2} a^4 b x^{10/3}+\frac{a^5 x^3}{3}+\frac{15}{13} a b^4 x^{13/3}+\frac{3}{14} b^5 x^{14/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5*x^2,x]

[Out]

(a^5*x^3)/3 + (3*a^4*b*x^(10/3))/2 + (30*a^3*b^2*x^(11/3))/11 + (5*a^2*b^3*x^4)/2 + (15*a*b^4*x^(13/3))/13 + (
3*b^5*x^(14/3))/14

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 )^5 x^2 \, dx &=3 \operatorname{Subst}\left (\int x^8 (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^5 x^8+5 a^4 b x^9+10 a^3 b^2 x^{10}+10 a^2 b^3 x^{11}+5 a b^4 x^{12}+b^5 x^{13}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^5 x^3}{3}+\frac{3}{2} a^4 b x^{10/3}+\frac{30}{11} a^3 b^2 x^{11/3}+\frac{5}{2} a^2 b^3 x^4+\frac{15}{13} a b^4 x^{13/3}+\frac{3}{14} b^5 x^{14/3}\\ \end{align*}

Mathematica [A]  time = 0.0286882, size = 77, normalized size = 1. \[ \frac{30}{11} a^3 b^2 x^{11/3}+\frac{5}{2} a^2 b^3 x^4+\frac{3}{2} a^4 b x^{10/3}+\frac{a^5 x^3}{3}+\frac{15}{13} a b^4 x^{13/3}+\frac{3}{14} b^5 x^{14/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5*x^2,x]

[Out]

(a^5*x^3)/3 + (3*a^4*b*x^(10/3))/2 + (30*a^3*b^2*x^(11/3))/11 + (5*a^2*b^3*x^4)/2 + (15*a*b^4*x^(13/3))/13 + (
3*b^5*x^(14/3))/14

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{3}}{3}}+{\frac{3\,{a}^{4}b}{2}{x}^{{\frac{10}{3}}}}+{\frac{30\,{a}^{3}{b}^{2}}{11}{x}^{{\frac{11}{3}}}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{4}}{2}}+{\frac{15\,a{b}^{4}}{13}{x}^{{\frac{13}{3}}}}+{\frac{3\,{b}^{5}}{14}{x}^{{\frac{14}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^5*x^2,x)

[Out]

1/3*a^5*x^3+3/2*a^4*b*x^(10/3)+30/11*a^3*b^2*x^(11/3)+5/2*a^2*b^3*x^4+15/13*a*b^4*x^(13/3)+3/14*b^5*x^(14/3)

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Maxima [B]  time = 0.971497, size = 201, normalized size = 2.61 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14}}{14 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a}{13 \, b^{9}} + \frac{7 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{2}}{b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{3}}{11 \, b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{4}}{b^{9}} - \frac{56 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{5}}{3 \, b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{6}}{2 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{7}}{7 \, b^{9}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{8}}{2 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5*x^2,x, algorithm="maxima")

[Out]

3/14*(b*x^(1/3) + a)^14/b^9 - 24/13*(b*x^(1/3) + a)^13*a/b^9 + 7*(b*x^(1/3) + a)^12*a^2/b^9 - 168/11*(b*x^(1/3
) + a)^11*a^3/b^9 + 21*(b*x^(1/3) + a)^10*a^4/b^9 - 56/3*(b*x^(1/3) + a)^9*a^5/b^9 + 21/2*(b*x^(1/3) + a)^8*a^
6/b^9 - 24/7*(b*x^(1/3) + a)^7*a^7/b^9 + 1/2*(b*x^(1/3) + a)^6*a^8/b^9

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Fricas [A]  time = 1.473, size = 166, normalized size = 2.16 \begin{align*} \frac{5}{2} \, a^{2} b^{3} x^{4} + \frac{1}{3} \, a^{5} x^{3} + \frac{3}{154} \,{\left (11 \, b^{5} x^{4} + 140 \, a^{3} b^{2} x^{3}\right )} x^{\frac{2}{3}} + \frac{3}{26} \,{\left (10 \, a b^{4} x^{4} + 13 \, a^{4} b x^{3}\right )} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5*x^2,x, algorithm="fricas")

[Out]

5/2*a^2*b^3*x^4 + 1/3*a^5*x^3 + 3/154*(11*b^5*x^4 + 140*a^3*b^2*x^3)*x^(2/3) + 3/26*(10*a*b^4*x^4 + 13*a^4*b*x
^3)*x^(1/3)

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Sympy [A]  time = 1.99835, size = 75, normalized size = 0.97 \begin{align*} \frac{a^{5} x^{3}}{3} + \frac{3 a^{4} b x^{\frac{10}{3}}}{2} + \frac{30 a^{3} b^{2} x^{\frac{11}{3}}}{11} + \frac{5 a^{2} b^{3} x^{4}}{2} + \frac{15 a b^{4} x^{\frac{13}{3}}}{13} + \frac{3 b^{5} x^{\frac{14}{3}}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**5*x**2,x)

[Out]

a**5*x**3/3 + 3*a**4*b*x**(10/3)/2 + 30*a**3*b**2*x**(11/3)/11 + 5*a**2*b**3*x**4/2 + 15*a*b**4*x**(13/3)/13 +
3*b**5*x**(14/3)/14

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Giac [A]  time = 1.14717, size = 77, normalized size = 1. \begin{align*} \frac{3}{14} \, b^{5} x^{\frac{14}{3}} + \frac{15}{13} \, a b^{4} x^{\frac{13}{3}} + \frac{5}{2} \, a^{2} b^{3} x^{4} + \frac{30}{11} \, a^{3} b^{2} x^{\frac{11}{3}} + \frac{3}{2} \, a^{4} b x^{\frac{10}{3}} + \frac{1}{3} \, a^{5} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5*x^2,x, algorithm="giac")

[Out]

3/14*b^5*x^(14/3) + 15/13*a*b^4*x^(13/3) + 5/2*a^2*b^3*x^4 + 30/11*a^3*b^2*x^(11/3) + 3/2*a^4*b*x^(10/3) + 1/3
*a^5*x^3