3.2286 \(\int (a+b \sqrt [3]{x}) x^3 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^4}{4}+\frac{3}{13} b x^{13/3} \]

[Out]

(a*x^4)/4 + (3*b*x^(13/3))/13

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Rubi [A]  time = 0.0047911, 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} \[ \frac{a x^4}{4}+\frac{3}{13} b x^{13/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))*x^3,x]

[Out]

(a*x^4)/4 + (3*b*x^(13/3))/13

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^3 \, dx &=\int \left (a x^3+b x^{10/3}\right ) \, dx\\ &=\frac{a x^4}{4}+\frac{3}{13} b x^{13/3}\\ \end{align*}

Mathematica [A]  time = 0.005672, size = 19, normalized size = 1. \[ \frac{a x^4}{4}+\frac{3}{13} b x^{13/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))*x^3,x]

[Out]

(a*x^4)/4 + (3*b*x^(13/3))/13

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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \begin{align*}{\frac{a{x}^{4}}{4}}+{\frac{3\,b}{13}{x}^{{\frac{13}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4+3/13*b*x^(13/3)

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Maxima [B]  time = 0.962903, size = 270, normalized size = 14.21 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13}}{13 \, b^{12}} - \frac{11 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a}{4 \, b^{12}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{2}}{b^{12}} - \frac{99 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{3}}{2 \, b^{12}} + \frac{110 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{4}}{b^{12}} - \frac{693 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{5}}{4 \, b^{12}} + \frac{198 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{6}}{b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{7}}{b^{12}} + \frac{99 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{8}}{b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{9}}{4 \, b^{12}} + \frac{11 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{10}}{b^{12}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{11}}{2 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/13*(b*x^(1/3) + a)^13/b^12 - 11/4*(b*x^(1/3) + a)^12*a/b^12 + 15*(b*x^(1/3) + a)^11*a^2/b^12 - 99/2*(b*x^(1/
3) + a)^10*a^3/b^12 + 110*(b*x^(1/3) + a)^9*a^4/b^12 - 693/4*(b*x^(1/3) + a)^8*a^5/b^12 + 198*(b*x^(1/3) + a)^
7*a^6/b^12 - 165*(b*x^(1/3) + a)^6*a^7/b^12 + 99*(b*x^(1/3) + a)^5*a^8/b^12 - 165/4*(b*x^(1/3) + a)^4*a^9/b^12
+ 11*(b*x^(1/3) + a)^3*a^10/b^12 - 3/2*(b*x^(1/3) + a)^2*a^11/b^12

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Fricas [A]  time = 1.48174, size = 39, normalized size = 2.05 \begin{align*} \frac{3}{13} \, b x^{\frac{13}{3}} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/13*b*x^(13/3) + 1/4*a*x^4

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Sympy [A]  time = 1.85691, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{4}}{4} + \frac{3 b x^{\frac{13}{3}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**4/4 + 3*b*x**(13/3)/13

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Giac [A]  time = 1.12167, size = 18, normalized size = 0.95 \begin{align*} \frac{3}{13} \, b x^{\frac{13}{3}} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/13*b*x^(13/3) + 1/4*a*x^4