3.1.7 \(\int \sqrt [3]{-1+z} z \, dz\) [7]

Optimal. Leaf size=23 \[ \frac {3}{4} (-1+z)^{4/3}+\frac {3}{7} (-1+z)^{7/3} \]

[Out]

3/4*(-1+z)^(4/3)+3/7*(-1+z)^(7/3)

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} \frac {3}{7} (z-1)^{7/3}+\frac {3}{4} (z-1)^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + z)^(1/3)*z,z]

[Out]

(3*(-1 + z)^(4/3))/4 + (3*(-1 + z)^(7/3))/7

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+z} z \, dz &=\int \left (\sqrt [3]{-1+z}+(-1+z)^{4/3}\right ) \, dz\\ &=\frac {3}{4} (-1+z)^{4/3}+\frac {3}{7} (-1+z)^{7/3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.78 \begin {gather*} \frac {3}{28} (7+4 (-1+z)) (-1+z)^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + z)^(1/3)*z,z]

[Out]

(3*(7 + 4*(-1 + z))*(-1 + z)^(4/3))/28

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Maple [A]
time = 0.08, size = 16, normalized size = 0.70

method	result	size
gosper	\(\frac {3 \left (-1+z \right )^{\frac {4}{3}} \left (4 z +3\right )}{28}\)	\(13\)
derivativedivides	\(\frac {3 \left (-1+z \right )^{\frac {4}{3}}}{4}+\frac {3 \left (-1+z \right )^{\frac {7}{3}}}{7}\)	\(16\)
default	\(\frac {3 \left (-1+z \right )^{\frac {4}{3}}}{4}+\frac {3 \left (-1+z \right )^{\frac {7}{3}}}{7}\)	\(16\)
trager	\(\left (\frac {3}{7} z^{2}-\frac {3}{28} z -\frac {9}{28}\right ) \left (-1+z \right )^{\frac {1}{3}}\)	\(17\)
risch	\(\frac {3 \left (-1+z \right )^{\frac {1}{3}} \left (4 z^{2}-z -3\right )}{28}\)	\(18\)
meijerg	\(\frac {\mathrm {signum}\left (-1+z \right )^{\frac {1}{3}} z^{2} \hypergeom \left (\left [-\frac {1}{3}, 2\right ], \left [3\right ], z\right )}{2 \left (-\mathrm {signum}\left (-1+z \right )\right )^{\frac {1}{3}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(-1+z)^(4/3)+3/7*(-1+z)^(7/3)

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Maxima [A]
time = 2.65, size = 15, normalized size = 0.65 \begin {gather*} \frac {3}{7} \, {\left (z - 1\right )}^{\frac {7}{3}} + \frac {3}{4} \, {\left (z - 1\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*(z - 1)^(7/3) + 3/4*(z - 1)^(4/3)

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Fricas [A]
time = 0.87, size = 17, normalized size = 0.74 \begin {gather*} \frac {3}{28} \, {\left (4 \, z^{2} - z - 3\right )} {\left (z - 1\right )}^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/28*(4*z^2 - z - 3)*(z - 1)^(1/3)

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Sympy [C] Result contains complex when optimal does not.
time = 0.47, size = 92, normalized size = 4.00 \begin {gather*} \begin {cases} \frac {3 z^{2} \sqrt [3]{z - 1}}{7} - \frac {3 z \sqrt [3]{z - 1}}{28} - \frac {9 \sqrt [3]{z - 1}}{28} & \text {for}\: \left |{z}\right | > 1 \\\frac {3 z^{2} \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{7} - \frac {3 z \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{28} - \frac {9 \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{28} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+z)**(1/3)*z,z)

[Out]

Piecewise((3*z**2*(z - 1)**(1/3)/7 - 3*z*(z - 1)**(1/3)/28 - 9*(z - 1)**(1/3)/28, Abs(z) > 1), (3*z**2*(1 - z)
**(1/3)*exp(I*pi/3)/7 - 3*z*(1 - z)**(1/3)*exp(I*pi/3)/28 - 9*(1 - z)**(1/3)*exp(I*pi/3)/28, True))

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Giac [A]
time = 0.81, size = 15, normalized size = 0.65 \begin {gather*} \frac {3}{7} \, {\left (z - 1\right )}^{\frac {7}{3}} + \frac {3}{4} \, {\left (z - 1\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*(z - 1)^(7/3) + 3/4*(z - 1)^(4/3)

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Mupad [B]
time = 0.03, size = 12, normalized size = 0.52 \begin {gather*} \frac {3\,\left (4\,z+3\right )\,{\left (z-1\right )}^{4/3}}{28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(z*(z - 1)^(1/3),z)

[Out]

(3*(4*z + 3)*(z - 1)^(4/3))/28

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