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 = 0.0151333, 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 \[ \frac{3}{7} (c+x)^{7/3}-\frac{3}{4} c (c+x)^{4/3} \]

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

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Rubi in Sympy [A]  time = 1.17599, size = 20, normalized size = 0.83 \[ - \frac{3 c \left (c + x\right )^{\frac{4}{3}}}{4} + \frac{3 \left (c + x\right )^{\frac{7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(c+x)**(1/3),x)

[Out]

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

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Mathematica [A]  time = 0.00801397, size = 25, normalized size = 1.04 \[ \frac{3}{28} \sqrt [3]{c+x} \left (-3 c^2+c x+4 x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(c + x)^(1/3),x]

[Out]

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

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Maple [A]  time = 0.004, size = 15, normalized size = 0.6 \[ -{\frac{9\,c-12\,x}{28} \left ( c+x \right ) ^{{\frac{4}{3}}}} \]

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 = 1.36276, size = 22, normalized size = 0.92 \[ \frac{3}{7} \,{\left (c + x\right )}^{\frac{7}{3}} - \frac{3}{4} \,{\left (c + x\right )}^{\frac{4}{3}} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227295, size = 30, normalized size = 1.25 \[ -\frac{3}{28} \,{\left (3 \, c^{2} - c x - 4 \, x^{2}\right )}{\left (c + x\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.68275, size = 144, normalized size = 6. \[ - \frac{9 c^{\frac{13}{3}} \sqrt [3]{1 + \frac{x}{c}}}{28 c^{2} + 28 c x} + \frac{9 c^{\frac{13}{3}}}{28 c^{2} + 28 c x} - \frac{6 c^{\frac{10}{3}} x \sqrt [3]{1 + \frac{x}{c}}}{28 c^{2} + 28 c x} + \frac{9 c^{\frac{10}{3}} x}{28 c^{2} + 28 c x} + \frac{15 c^{\frac{7}{3}} x^{2} \sqrt [3]{1 + \frac{x}{c}}}{28 c^{2} + 28 c x} + \frac{12 c^{\frac{4}{3}} x^{3} \sqrt [3]{1 + \frac{x}{c}}}{28 c^{2} + 28 c x} \]

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/XCAS [A]  time = 0.212761, size = 22, normalized size = 0.92 \[ \frac{3}{7} \,{\left (c + x\right )}^{\frac{7}{3}} - \frac{3}{4} \,{\left (c + x\right )}^{\frac{4}{3}} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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