3.4.6 \(\int x \sqrt [3]{c+x} \, dx\) [306]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, 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} (c+x)^{7/3}-\frac {3}{4} c (c+x)^{4/3} \end {gather*}

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 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 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*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.75 \begin {gather*} \frac {3}{28} (c+x)^{4/3} (-3 c+4 x) \end {gather*}

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 = 0.06, size = 17, normalized size = 0.71

method	result	size
gosper	\(-\frac {3 \left (c +x \right )^{\frac {4}{3}} \left (3 c -4 x \right )}{28}\)	\(15\)
derivativedivides	\(-\frac {3 c \left (c +x \right )^{\frac {4}{3}}}{4}+\frac {3 \left (c +x \right )^{\frac {7}{3}}}{7}\)	\(17\)
default	\(-\frac {3 c \left (c +x \right )^{\frac {4}{3}}}{4}+\frac {3 \left (c +x \right )^{\frac {7}{3}}}{7}\)	\(17\)
trager	\(\left (-\frac {9}{28} c^{2}+\frac {3}{28} c x +\frac {3}{7} x^{2}\right ) \left (c +x \right )^{\frac {1}{3}}\)	\(22\)
risch	\(-\frac {3 \left (c +x \right )^{\frac {1}{3}} \left (3 c^{2}-c x -4 x^{2}\right )}{28}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.80, size = 16, normalized size = 0.67 \begin {gather*} \frac {3}{7} \, {\left (c + x\right )}^{\frac {7}{3}} - \frac {3}{4} \, {\left (c + x\right )}^{\frac {4}{3}} c \end {gather*}

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 = 0.65, size = 22, normalized size = 0.92 \begin {gather*} -\frac {3}{28} \, {\left (3 \, c^{2} - c x - 4 \, x^{2}\right )} {\left (c + x\right )}^{\frac {1}{3}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (20) = 40\).
time = 0.51, size = 144, normalized size = 6.00 \begin {gather*} - \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (16) = 32\).
time = 1.13, size = 43, normalized size = 1.79 \begin {gather*} \frac {3}{7} \, {\left (c + x\right )}^{\frac {7}{3}} - \frac {3}{2} \, {\left (c + x\right )}^{\frac {4}{3}} c + 3 \, {\left (c + x\right )}^{\frac {1}{3}} c^{2} + \frac {3}{4} \, {\left ({\left (c + x\right )}^{\frac {4}{3}} - 4 \, {\left (c + x\right )}^{\frac {1}{3}} c\right )} c \end {gather*}

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

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Mupad [B]
time = 0.16, size = 14, normalized size = 0.58 \begin {gather*} -\frac {3\,{\left (c+x\right )}^{4/3}\,\left (3\,c-4\,x\right )}{28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c + x)^(1/3),x)

[Out]

-(3*(c + x)^(4/3)*(3*c - 4*x))/28

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