3.2290 \(\int \left (a+b \sqrt [3]{x}\right )^2 x \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]

[Out]

(a^2*x^2)/2 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(8/3))/8

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Rubi [A]  time = 0.0542323, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^2*x,x]

[Out]

(a^2*x^2)/2 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(8/3))/8

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Rubi in Sympy [A]  time = 8.03824, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{2}}{2} + \frac{6 a b x^{\frac{7}{3}}}{7} + \frac{3 b^{2} x^{\frac{8}{3}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**2*x,x)

[Out]

a**2*x**2/2 + 6*a*b*x**(7/3)/7 + 3*b**2*x**(8/3)/8

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Mathematica [A]  time = 0.0102203, size = 34, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{6}{7} a b x^{7/3}+\frac{3}{8} b^2 x^{8/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^2*x,x]

[Out]

(a^2*x^2)/2 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(8/3))/8

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Maple [A]  time = 0.002, size = 25, normalized size = 0.7 \[{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{6\,ab}{7}{x}^{{\frac{7}{3}}}}+{\frac{3\,{b}^{2}}{8}{x}^{{\frac{8}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^2*x,x)

[Out]

1/2*a^2*x^2+6/7*a*b*x^(7/3)+3/8*b^2*x^(8/3)

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Maxima [A]  time = 1.44189, size = 132, normalized size = 3.88 \[ \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8}}{8 \, b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a}{7 \, b^{6}} + \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{2}}{b^{6}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{3}}{b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{4}}{4 \, b^{6}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{5}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^2*x,x, algorithm="maxima")

[Out]

3/8*(b*x^(1/3) + a)^8/b^6 - 15/7*(b*x^(1/3) + a)^7*a/b^6 + 5*(b*x^(1/3) + a)^6*a
^2/b^6 - 6*(b*x^(1/3) + a)^5*a^3/b^6 + 15/4*(b*x^(1/3) + a)^4*a^4/b^6 - (b*x^(1/
3) + a)^3*a^5/b^6

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Fricas [A]  time = 0.212494, size = 32, normalized size = 0.94 \[ \frac{3}{8} \, b^{2} x^{\frac{8}{3}} + \frac{6}{7} \, a b x^{\frac{7}{3}} + \frac{1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^2*x,x, algorithm="fricas")

[Out]

3/8*b^2*x^(8/3) + 6/7*a*b*x^(7/3) + 1/2*a^2*x^2

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Sympy [A]  time = 1.30568, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{2}}{2} + \frac{6 a b x^{\frac{7}{3}}}{7} + \frac{3 b^{2} x^{\frac{8}{3}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**2*x,x)

[Out]

a**2*x**2/2 + 6*a*b*x**(7/3)/7 + 3*b**2*x**(8/3)/8

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GIAC/XCAS [A]  time = 0.258597, size = 32, normalized size = 0.94 \[ \frac{3}{8} \, b^{2} x^{\frac{8}{3}} + \frac{6}{7} \, a b x^{\frac{7}{3}} + \frac{1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^2*x,x, algorithm="giac")

[Out]

3/8*b^2*x^(8/3) + 6/7*a*b*x^(7/3) + 1/2*a^2*x^2