3.306 \(\int x \sqrt [3]{c+x} \, dx\)

Optimal. Leaf size=24 \[ \frac{3}{7} (c+x)^{7/3}-\frac{3}{4} c (c+x)^{4/3} \]

[Out]

(-3*c*(c + x)^(4/3))/4 + (3*(c + x)^(7/3))/7

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Rubi [A]  time = 0.0044251, antiderivative size = 24, 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} \[ \frac{3}{7} (c+x)^{7/3}-\frac{3}{4} c (c+x)^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[x*(c + x)^(1/3),x]

[Out]

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

Mathematica [A]  time = 0.0064194, size = 18, normalized size = 0.75 \[ \frac{3}{28} (c+x)^{4/3} (4 x-3 c) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(c + x)^(1/3),x]

[Out]

(3*(c + x)^(4/3)*(-3*c + 4*x))/28

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Maple [A]  time = 0.003, size = 15, normalized size = 0.6 \begin{align*} -{\frac{9\,c-12\,x}{28} \left ( c+x \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c+x)^(1/3),x)

[Out]

-3/28*(c+x)^(4/3)*(3*c-4*x)

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Maxima [A]  time = 0.936892, size = 22, normalized size = 0.92 \begin{align*} \frac{3}{7} \,{\left (c + x\right )}^{\frac{7}{3}} - \frac{3}{4} \,{\left (c + x\right )}^{\frac{4}{3}} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c+x)^(1/3),x, algorithm="maxima")

[Out]

3/7*(c + x)^(7/3) - 3/4*(c + x)^(4/3)*c

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Fricas [A]  time = 1.99004, size = 58, normalized size = 2.42 \begin{align*} -\frac{3}{28} \,{\left (3 \, c^{2} - c x - 4 \, x^{2}\right )}{\left (c + x\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c+x)^(1/3),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.926045, size = 144, normalized size = 6. \begin{align*} - \frac{9 c^{\frac{13}{3}} \sqrt [3]{1 + \frac{x}{c}}}{28 c^{2} + 28 c x} + \frac{9 c^{\frac{13}{3}}}{28 c^{2} + 28 c x} - \frac{6 c^{\frac{10}{3}} x \sqrt [3]{1 + \frac{x}{c}}}{28 c^{2} + 28 c x} + \frac{9 c^{\frac{10}{3}} x}{28 c^{2} + 28 c x} + \frac{15 c^{\frac{7}{3}} x^{2} \sqrt [3]{1 + \frac{x}{c}}}{28 c^{2} + 28 c x} + \frac{12 c^{\frac{4}{3}} x^{3} \sqrt [3]{1 + \frac{x}{c}}}{28 c^{2} + 28 c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c+x)**(1/3),x)

[Out]

-9*c**(13/3)*(1 + x/c)**(1/3)/(28*c**2 + 28*c*x) + 9*c**(13/3)/(28*c**2 + 28*c*x) - 6*c**(10/3)*x*(1 + x/c)**(
1/3)/(28*c**2 + 28*c*x) + 9*c**(10/3)*x/(28*c**2 + 28*c*x) + 15*c**(7/3)*x**2*(1 + x/c)**(1/3)/(28*c**2 + 28*c
*x) + 12*c**(4/3)*x**3*(1 + x/c)**(1/3)/(28*c**2 + 28*c*x)

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Giac [A]  time = 1.05171, size = 22, normalized size = 0.92 \begin{align*} \frac{3}{7} \,{\left (c + x\right )}^{\frac{7}{3}} - \frac{3}{4} \,{\left (c + x\right )}^{\frac{4}{3}} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c+x)^(1/3),x, algorithm="giac")

[Out]

3/7*(c + x)^(7/3) - 3/4*(c + x)^(4/3)*c