3.1456 \(\int \sqrt [3]{3-2 x} (7+x) \, dx\)

Optimal. Leaf size=27 \[ \frac{3}{28} (3-2 x)^{7/3}-\frac{51}{16} (3-2 x)^{4/3} \]

[Out]

(-51*(3 - 2*x)^(4/3))/16 + (3*(3 - 2*x)^(7/3))/28

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Rubi [A]  time = 0.0048718, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{3}{28} (3-2 x)^{7/3}-\frac{51}{16} (3-2 x)^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 2*x)^(1/3)*(7 + x),x]

[Out]

(-51*(3 - 2*x)^(4/3))/16 + (3*(3 - 2*x)^(7/3))/28

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]{3-2 x} (7+x) \, dx &=\int \left (\frac{17}{2} \sqrt [3]{3-2 x}-\frac{1}{2} (3-2 x)^{4/3}\right ) \, dx\\ &=-\frac{51}{16} (3-2 x)^{4/3}+\frac{3}{28} (3-2 x)^{7/3}\\ \end{align*}

Mathematica [A]  time = 0.0082182, size = 18, normalized size = 0.67 \[ -\frac{3}{112} (3-2 x)^{4/3} (8 x+107) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 2*x)^(1/3)*(7 + x),x]

[Out]

(-3*(3 - 2*x)^(4/3)*(107 + 8*x))/112

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Maple [A]  time = 0.002, size = 15, normalized size = 0.6 \begin{align*} -{\frac{24\,x+321}{112} \left ( 3-2\,x \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3-2*x)^(1/3)*(7+x),x)

[Out]

-3/112*(8*x+107)*(3-2*x)^(4/3)

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Maxima [A]  time = 0.954057, size = 26, normalized size = 0.96 \begin{align*} \frac{3}{28} \,{\left (-2 \, x + 3\right )}^{\frac{7}{3}} - \frac{51}{16} \,{\left (-2 \, x + 3\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-2*x)^(1/3)*(7+x),x, algorithm="maxima")

[Out]

3/28*(-2*x + 3)^(7/3) - 51/16*(-2*x + 3)^(4/3)

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Fricas [A]  time = 1.99108, size = 63, normalized size = 2.33 \begin{align*} \frac{3}{112} \,{\left (16 \, x^{2} + 190 \, x - 321\right )}{\left (-2 \, x + 3\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-2*x)^(1/3)*(7+x),x, algorithm="fricas")

[Out]

3/112*(16*x^2 + 190*x - 321)*(-2*x + 3)^(1/3)

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Sympy [A]  time = 0.888232, size = 114, normalized size = 4.22 \begin{align*} \begin{cases} \frac{3 \left (x + 7\right )^{2} \sqrt [3]{2 x - 3} e^{\frac{i \pi }{3}}}{7} - \frac{51 \left (x + 7\right ) \sqrt [3]{2 x - 3} e^{\frac{i \pi }{3}}}{56} - \frac{2601 \sqrt [3]{2 x - 3} e^{\frac{i \pi }{3}}}{112} & \text{for}\: \frac{2 \left |{x + 7}\right |}{17} > 1 \\\frac{3 \sqrt [3]{3 - 2 x} \left (x + 7\right )^{2}}{7} - \frac{51 \sqrt [3]{3 - 2 x} \left (x + 7\right )}{56} - \frac{2601 \sqrt [3]{3 - 2 x}}{112} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-2*x)**(1/3)*(7+x),x)

[Out]

Piecewise((3*(x + 7)**2*(2*x - 3)**(1/3)*exp(I*pi/3)/7 - 51*(x + 7)*(2*x - 3)**(1/3)*exp(I*pi/3)/56 - 2601*(2*
x - 3)**(1/3)*exp(I*pi/3)/112, 2*Abs(x + 7)/17 > 1), (3*(3 - 2*x)**(1/3)*(x + 7)**2/7 - 51*(3 - 2*x)**(1/3)*(x
+ 7)/56 - 2601*(3 - 2*x)**(1/3)/112, True))

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Giac [A]  time = 1.05487, size = 35, normalized size = 1.3 \begin{align*} \frac{3}{28} \,{\left (2 \, x - 3\right )}^{2}{\left (-2 \, x + 3\right )}^{\frac{1}{3}} - \frac{51}{16} \,{\left (-2 \, x + 3\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-2*x)^(1/3)*(7+x),x, algorithm="giac")

[Out]

3/28*(2*x - 3)^2*(-2*x + 3)^(1/3) - 51/16*(-2*x + 3)^(4/3)