3.2288 \(\int (a+b \sqrt [3]{x}) x \, dx\)

Optimal. Leaf size=19 \[ \frac {a x^2}{2}+\frac {3}{7} b x^{7/3} \]

[Out]

1/2*a*x^2+3/7*b*x^(7/3)

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \frac {a x^2}{2}+\frac {3}{7} b x^{7/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))*x,x]

[Out]

(a*x^2)/2 + (3*b*x^(7/3))/7

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 \, dx &=\int \left (a x+b x^{4/3}\right ) \, dx\\ &=\frac {a x^2}{2}+\frac {3}{7} b x^{7/3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {a x^2}{2}+\frac {3}{7} b x^{7/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))*x,x]

[Out]

(a*x^2)/2 + (3*b*x^(7/3))/7

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fricas [A]  time = 0.70, size = 13, normalized size = 0.68 \[ \frac {3}{7} \, b x^{\frac {7}{3}} + \frac {1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*b*x^(7/3) + 1/2*a*x^2

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giac [A]  time = 0.17, size = 13, normalized size = 0.68 \[ \frac {3}{7} \, b x^{\frac {7}{3}} + \frac {1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*b*x^(7/3) + 1/2*a*x^2

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maple [A]  time = 0.00, size = 14, normalized size = 0.74 \[ \frac {3 b \,x^{\frac {7}{3}}}{7}+\frac {a \,x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2+3/7*b*x^(7/3)

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maxima [B]  time = 0.82, size = 98, normalized size = 5.16 \[ \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7}}{7 \, b^{6}} - \frac {5 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a}{2 \, b^{6}} + \frac {6 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{2}}{b^{6}} - \frac {15 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{3}}{2 \, b^{6}} + \frac {5 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{4}}{b^{6}} - \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{5}}{2 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*(b*x^(1/3) + a)^7/b^6 - 5/2*(b*x^(1/3) + a)^6*a/b^6 + 6*(b*x^(1/3) + a)^5*a^2/b^6 - 15/2*(b*x^(1/3) + a)^4
*a^3/b^6 + 5*(b*x^(1/3) + a)^3*a^4/b^6 - 3/2*(b*x^(1/3) + a)^2*a^5/b^6

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mupad [B]  time = 0.03, size = 13, normalized size = 0.68 \[ \frac {a\,x^2}{2}+\frac {3\,b\,x^{7/3}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^(1/3)),x)

[Out]

(a*x^2)/2 + (3*b*x^(7/3))/7

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sympy [A]  time = 1.24, size = 15, normalized size = 0.79 \[ \frac {a x^{2}}{2} + \frac {3 b x^{\frac {7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + 3*b*x**(7/3)/7

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