3.15.57 \(\int \sqrt [3]{1-x} (1+x)^2 \, dx\) [1457]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} -\frac {3}{10} (1-x)^{10/3}+\frac {12}{7} (1-x)^{7/3}-3 (1-x)^{4/3} \end {gather*}

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 45

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 = 0.02, size = 23, normalized size = 0.61 \begin {gather*} -\frac {3}{70} (1-x)^{4/3} \left (37+26 x+7 x^2\right ) \end {gather*}

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

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 29, normalized size = 0.76

method	result	size
gosper	\(-\frac {3 \left (7 x^{2}+26 x +37\right ) \left (1-x \right )^{\frac {4}{3}}}{70}\)	\(20\)
trager	\(\left (\frac {3}{10} x^{3}+\frac {57}{70} x^{2}+\frac {33}{70} x -\frac {111}{70}\right ) \left (1-x \right )^{\frac {1}{3}}\)	\(24\)
risch	\(-\frac {3 \left (7 x^{3}+19 x^{2}+11 x -37\right ) \left (-1+x \right )}{70 \left (1-x \right )^{\frac {2}{3}}}\)	\(28\)
derivativedivides	\(-3 \left (1-x \right )^{\frac {4}{3}}+\frac {12 \left (1-x \right )^{\frac {7}{3}}}{7}-\frac {3 \left (1-x \right )^{\frac {10}{3}}}{10}\)	\(29\)
default	\(-3 \left (1-x \right )^{\frac {4}{3}}+\frac {12 \left (1-x \right )^{\frac {7}{3}}}{7}-\frac {3 \left (1-x \right )^{\frac {10}{3}}}{10}\)	\(29\)
meijerg	\(x \hypergeom \left (\left [-\frac {1}{3}, 1\right ], \left [2\right ], x\right )+x^{2} \hypergeom \left (\left [-\frac {1}{3}, 2\right ], \left [3\right ], x\right )+\frac {x^{3} \hypergeom \left (\left [-\frac {1}{3}, 3\right ], \left [4\right ], x\right )}{3}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 28, normalized size = 0.74 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.71, size = 24, normalized size = 0.63 \begin {gather*} \frac {3}{70} \, {\left (7 \, x^{3} + 19 \, x^{2} + 11 \, x - 37\right )} {\left (-x + 1\right )}^{\frac {1}{3}} \end {gather*}

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)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.69, size = 144, normalized size = 3.79 \begin {gather*} \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}\: \left |{x + 1}\right | > 2 \\\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 {gather*}

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), (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, True
))

________________________________________________________________________________________

Giac [A]
time = 1.84, size = 38, normalized size = 1.00 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 21, normalized size = 0.55 \begin {gather*} -\frac {3\,{\left (1-x\right )}^{4/3}\,\left (40\,x+7\,{\left (x-1\right )}^2+30\right )}{70} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/3)*(x + 1)^2,x)

[Out]

-(3*(1 - x)^(4/3)*(40*x + 7*(x - 1)^2 + 30))/70

________________________________________________________________________________________