3.2281 \(\int \left (a+b \sqrt [3]{x}\right ) x \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^2}{2}+\frac{3}{7} b x^{7/3} \]

[Out]

(a*x^2)/2 + (3*b*x^(7/3))/7

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Rubi [A]  time = 0.0161281, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a x^2}{2}+\frac{3}{7} b x^{7/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))*x,x]

[Out]

(a*x^2)/2 + (3*b*x^(7/3))/7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a \int x\, dx + \frac{3 b x^{\frac{7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Integral(x, x) + 3*b*x**(7/3)/7

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Mathematica [A]  time = 0.00493222, size = 19, normalized size = 1. \[ \frac{a x^2}{2}+\frac{3}{7} b x^{7/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))*x,x]

[Out]

(a*x^2)/2 + (3*b*x^(7/3))/7

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \[{\frac{a{x}^{2}}{2}}+{\frac{3\,b}{7}{x}^{{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))*x,x)

[Out]

1/2*a*x^2+3/7*b*x^(7/3)

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Maxima [A]  time = 1.43855, size = 132, normalized size = 6.95 \[ \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7}}{7 \, b^{6}} - \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a}{2 \, b^{6}} + \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{2}}{b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{3}}{2 \, b^{6}} + \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{4}}{b^{6}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{5}}{2 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/7*(b*x^(1/3) + a)^7/b^6 - 5/2*(b*x^(1/3) + a)^6*a/b^6 + 6*(b*x^(1/3) + a)^5*a^
2/b^6 - 15/2*(b*x^(1/3) + a)^4*a^3/b^6 + 5*(b*x^(1/3) + a)^3*a^4/b^6 - 3/2*(b*x^
(1/3) + a)^2*a^5/b^6

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Fricas [A]  time = 0.210685, size = 18, normalized size = 0.95 \[ \frac{3}{7} \, b x^{\frac{7}{3}} + \frac{1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/7*b*x^(7/3) + 1/2*a*x^2

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Sympy [A]  time = 1.20788, size = 15, normalized size = 0.79 \[ \frac{a x^{2}}{2} + \frac{3 b x^{\frac{7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))*x,x)

[Out]

a*x**2/2 + 3*b*x**(7/3)/7

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GIAC/XCAS [A]  time = 0.248049, size = 18, normalized size = 0.95 \[ \frac{3}{7} \, b x^{\frac{7}{3}} + \frac{1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/7*b*x^(7/3) + 1/2*a*x^2