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

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

[Out]

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

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Rubi [A]  time = 0.0037208, antiderivative size = 23, normalized size of antiderivative = 1., 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 = {43} \[ \frac{3}{7} (z-1)^{7/3}+\frac{3}{4} (z-1)^{4/3} \]

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 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+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*}

Mathematica [A]  time = 0.0047318, size = 16, normalized size = 0.7 \[ \frac{3}{28} (z-1)^{4/3} (4 z+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + z)^(1/3)*z,z]

[Out]

(3*(-1 + z)^(4/3)*(3 + 4*z))/28

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Maple [A]  time = 0.002, size = 13, normalized size = 0.6 \begin{align*}{\frac{12\,z+9}{28} \left ( -1+z \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.932029, size = 20, normalized size = 0.87 \begin{align*} \frac{3}{7} \,{\left (z - 1\right )}^{\frac{7}{3}} + \frac{3}{4} \,{\left (z - 1\right )}^{\frac{4}{3}} \end{align*}

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 = 0.415033, size = 49, normalized size = 2.13 \begin{align*} \frac{3}{28} \,{\left (4 \, z^{2} - z - 3\right )}{\left (z - 1\right )}^{\frac{1}{3}} \end{align*}

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 [A]  time = 0.85472, size = 92, normalized size = 4. \begin{align*} \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{align*}

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 = 1.14098, size = 20, normalized size = 0.87 \begin{align*} \frac{3}{7} \,{\left (z - 1\right )}^{\frac{7}{3}} + \frac{3}{4} \,{\left (z - 1\right )}^{\frac{4}{3}} \end{align*}

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)