3.2285 \(\int (a+b \sqrt [3]{x}) x^4 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^5}{5}+\frac{3}{16} b x^{16/3} \]

[Out]

(a*x^5)/5 + (3*b*x^(16/3))/16

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Rubi [A]  time = 0.0052233, 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} \[ \frac{a x^5}{5}+\frac{3}{16} b x^{16/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))*x^4,x]

[Out]

(a*x^5)/5 + (3*b*x^(16/3))/16

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^4 \, dx &=\int \left (a x^4+b x^{13/3}\right ) \, dx\\ &=\frac{a x^5}{5}+\frac{3}{16} b x^{16/3}\\ \end{align*}

Mathematica [A]  time = 0.0066342, size = 19, normalized size = 1. \[ \frac{a x^5}{5}+\frac{3}{16} b x^{16/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))*x^4,x]

[Out]

(a*x^5)/5 + (3*b*x^(16/3))/16

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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \begin{align*}{\frac{a{x}^{5}}{5}}+{\frac{3\,b}{16}{x}^{{\frac{16}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))*x^4,x)

[Out]

1/5*a*x^5+3/16*b*x^(16/3)

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Maxima [B]  time = 0.965223, size = 339, normalized size = 17.84 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16}}{16 \, b^{15}} - \frac{14 \,{\left (b x^{\frac{1}{3}} + a\right )}^{15} a}{5 \, b^{15}} + \frac{39 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{2}}{2 \, b^{15}} - \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{3}}{b^{15}} + \frac{1001 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{4}}{4 \, b^{15}} - \frac{546 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{5}}{b^{15}} + \frac{9009 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{6}}{10 \, b^{15}} - \frac{1144 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{7}}{b^{15}} + \frac{9009 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{8}}{8 \, b^{15}} - \frac{858 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{9}}{b^{15}} + \frac{1001 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{10}}{2 \, b^{15}} - \frac{1092 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{11}}{5 \, b^{15}} + \frac{273 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{12}}{4 \, b^{15}} - \frac{14 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{13}}{b^{15}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{14}}{2 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))*x^4,x, algorithm="maxima")

[Out]

3/16*(b*x^(1/3) + a)^16/b^15 - 14/5*(b*x^(1/3) + a)^15*a/b^15 + 39/2*(b*x^(1/3) + a)^14*a^2/b^15 - 84*(b*x^(1/
3) + a)^13*a^3/b^15 + 1001/4*(b*x^(1/3) + a)^12*a^4/b^15 - 546*(b*x^(1/3) + a)^11*a^5/b^15 + 9009/10*(b*x^(1/3
) + a)^10*a^6/b^15 - 1144*(b*x^(1/3) + a)^9*a^7/b^15 + 9009/8*(b*x^(1/3) + a)^8*a^8/b^15 - 858*(b*x^(1/3) + a)
^7*a^9/b^15 + 1001/2*(b*x^(1/3) + a)^6*a^10/b^15 - 1092/5*(b*x^(1/3) + a)^5*a^11/b^15 + 273/4*(b*x^(1/3) + a)^
4*a^12/b^15 - 14*(b*x^(1/3) + a)^3*a^13/b^15 + 3/2*(b*x^(1/3) + a)^2*a^14/b^15

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Fricas [A]  time = 1.45833, size = 39, normalized size = 2.05 \begin{align*} \frac{3}{16} \, b x^{\frac{16}{3}} + \frac{1}{5} \, a x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))*x^4,x, algorithm="fricas")

[Out]

3/16*b*x^(16/3) + 1/5*a*x^5

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Sympy [A]  time = 2.73064, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{5}}{5} + \frac{3 b x^{\frac{16}{3}}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))*x**4,x)

[Out]

a*x**5/5 + 3*b*x**(16/3)/16

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Giac [A]  time = 1.08523, size = 18, normalized size = 0.95 \begin{align*} \frac{3}{16} \, b x^{\frac{16}{3}} + \frac{1}{5} \, a x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))*x^4,x, algorithm="giac")

[Out]

3/16*b*x^(16/3) + 1/5*a*x^5