3.1457 \(\int \sqrt [3]{1-x} (1+x)^2 \, dx\)

Optimal. Leaf size=38 \[ -\frac{3}{10} (1-x)^{10/3}+\frac{12}{7} (1-x)^{7/3}-3 (1-x)^{4/3} \]

[Out]

-3*(1 - x)^(4/3) + (12*(1 - x)^(7/3))/7 - (3*(1 - x)^(10/3))/10

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Rubi [A]  time = 0.0068215, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{3}{10} (1-x)^{10/3}+\frac{12}{7} (1-x)^{7/3}-3 (1-x)^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(1/3)*(1 + x)^2,x]

[Out]

-3*(1 - x)^(4/3) + (12*(1 - x)^(7/3))/7 - (3*(1 - x)^(10/3))/10

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 \sqrt [3]{1-x} (1+x)^2 \, dx &=\int \left (4 \sqrt [3]{1-x}-4 (1-x)^{4/3}+(1-x)^{7/3}\right ) \, dx\\ &=-3 (1-x)^{4/3}+\frac{12}{7} (1-x)^{7/3}-\frac{3}{10} (1-x)^{10/3}\\ \end{align*}

Mathematica [A]  time = 0.0094329, size = 23, normalized size = 0.61 \[ -\frac{3}{70} (1-x)^{4/3} \left (7 x^2+26 x+37\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(1/3)*(1 + x)^2,x]

[Out]

(-3*(1 - x)^(4/3)*(37 + 26*x + 7*x^2))/70

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Maple [A]  time = 0.003, size = 20, normalized size = 0.5 \begin{align*} -{\frac{21\,{x}^{2}+78\,x+111}{70} \left ( 1-x \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/70*(7*x^2+26*x+37)*(1-x)^(4/3)

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Maxima [A]  time = 0.953849, size = 38, normalized size = 1. \begin{align*} -\frac{3}{10} \,{\left (-x + 1\right )}^{\frac{10}{3}} + \frac{12}{7} \,{\left (-x + 1\right )}^{\frac{7}{3}} - 3 \,{\left (-x + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/10*(-x + 1)^(10/3) + 12/7*(-x + 1)^(7/3) - 3*(-x + 1)^(4/3)

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Fricas [A]  time = 2.04995, size = 68, normalized size = 1.79 \begin{align*} \frac{3}{70} \,{\left (7 \, x^{3} + 19 \, x^{2} + 11 \, x - 37\right )}{\left (-x + 1\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/70*(7*x^3 + 19*x^2 + 11*x - 37)*(-x + 1)^(1/3)

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Sympy [A]  time = 1.3091, size = 146, normalized size = 3.84 \begin{align*} \begin{cases} - \frac{3 \sqrt [3]{x - 1} \left (x + 1\right )^{3} e^{- \frac{2 i \pi }{3}}}{10} + \frac{3 \sqrt [3]{x - 1} \left (x + 1\right )^{2} e^{- \frac{2 i \pi }{3}}}{35} + \frac{9 \sqrt [3]{x - 1} \left (x + 1\right ) e^{- \frac{2 i \pi }{3}}}{35} + \frac{54 \sqrt [3]{x - 1} e^{- \frac{2 i \pi }{3}}}{35} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{3 \sqrt [3]{1 - x} \left (x + 1\right )^{3}}{10} - \frac{3 \sqrt [3]{1 - x} \left (x + 1\right )^{2}}{35} - \frac{9 \sqrt [3]{1 - x} \left (x + 1\right )}{35} - \frac{54 \sqrt [3]{1 - x}}{35} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(1/3)*(1+x)**2,x)

[Out]

Piecewise((-3*(x - 1)**(1/3)*(x + 1)**3*exp(-2*I*pi/3)/10 + 3*(x - 1)**(1/3)*(x + 1)**2*exp(-2*I*pi/3)/35 + 9*
(x - 1)**(1/3)*(x + 1)*exp(-2*I*pi/3)/35 + 54*(x - 1)**(1/3)*exp(-2*I*pi/3)/35, Abs(x + 1)/2 > 1), (3*(1 - x)*
*(1/3)*(x + 1)**3/10 - 3*(1 - x)**(1/3)*(x + 1)**2/35 - 9*(1 - x)**(1/3)*(x + 1)/35 - 54*(1 - x)**(1/3)/35, Tr
ue))

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Giac [A]  time = 1.05786, size = 51, normalized size = 1.34 \begin{align*} \frac{3}{10} \,{\left (x - 1\right )}^{3}{\left (-x + 1\right )}^{\frac{1}{3}} + \frac{12}{7} \,{\left (x - 1\right )}^{2}{\left (-x + 1\right )}^{\frac{1}{3}} - 3 \,{\left (-x + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(x - 1)^3*(-x + 1)^(1/3) + 12/7*(x - 1)^2*(-x + 1)^(1/3) - 3*(-x + 1)^(4/3)