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

Optimal result

Integrand size = 9, antiderivative size = 23 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{4} (-1+z)^{4/3}+\frac {3}{7} (-1+z)^{7/3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{7} (z-1)^{7/3}+\frac {3}{4} (z-1)^{4/3} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{28} (7+4 (-1+z)) (-1+z)^{4/3} \]

[In]

Integrate[(-1 + z)^(1/3)*z,z]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57

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	\(\frac {\operatorname {signum}\left (-1+z \right )^{\frac {1}{3}} z^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},2;3;z \right )}{2 \left (-\operatorname {signum}\left (-1+z \right )\right )^{\frac {1}{3}}}\)	\(27\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{28} \, {\left (4 \, z^{2} - z - 3\right )} {\left (z - 1\right )}^{\frac {1}{3}} \]

[In]

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

[Out]

3/28*(4*z^2 - z - 3)*(z - 1)^(1/3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int \sqrt [3]{-1+z} z \, dz=\begin {cases} \frac {3 z^{2} \sqrt [3]{z - 1}}{7} - \frac {3 z \sqrt [3]{z - 1}}{28} - \frac {9 \sqrt [3]{z - 1}}{28} & \text {for}\: \left |{z}\right | > 1 \\\frac {3 z^{2} \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{7} - \frac {3 z \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{28} - \frac {9 \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{28} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{7} \, {\left (z - 1\right )}^{\frac {7}{3}} + \frac {3}{4} \, {\left (z - 1\right )}^{\frac {4}{3}} \]

[In]

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

[Out]

3/7*(z - 1)^(7/3) + 3/4*(z - 1)^(4/3)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{7} \, {\left (z - 1\right )}^{\frac {7}{3}} + \frac {3}{4} \, {\left (z - 1\right )}^{\frac {4}{3}} \]

[In]

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

[Out]

3/7*(z - 1)^(7/3) + 3/4*(z - 1)^(4/3)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3\,\left (4\,z+3\right )\,{\left (z-1\right )}^{4/3}}{28} \]

[In]

int(z*(z - 1)^(1/3),z)

[Out]

(3*(4*z + 3)*(z - 1)^(4/3))/28