3.2289 \(\int \left (a+b \sqrt [3]{x}\right )^2 x^2 \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^3}{3}+\frac{3}{5} a b x^{10/3}+\frac{3}{11} b^2 x^{11/3} \]

[Out]

(a^2*x^3)/3 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(11/3))/11

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Rubi [A]  time = 0.0625656, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2 x^3}{3}+\frac{3}{5} a b x^{10/3}+\frac{3}{11} b^2 x^{11/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^2*x^2,x]

[Out]

(a^2*x^3)/3 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(11/3))/11

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Rubi in Sympy [A]  time = 9.29909, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{3}}{3} + \frac{3 a b x^{\frac{10}{3}}}{5} + \frac{3 b^{2} x^{\frac{11}{3}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3/3 + 3*a*b*x**(10/3)/5 + 3*b**2*x**(11/3)/11

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Mathematica [A]  time = 0.012886, size = 34, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{3}{5} a b x^{10/3}+\frac{3}{11} b^2 x^{11/3} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^3)/3 + (3*a*b*x^(10/3))/5 + (3*b^2*x^(11/3))/11

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Maple [A]  time = 0.003, size = 25, normalized size = 0.7 \[{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{3\,ab}{5}{x}^{{\frac{10}{3}}}}+{\frac{3\,{b}^{2}}{11}{x}^{{\frac{11}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^2*x^2,x)

[Out]

1/3*x^3*a^2+3/5*a*b*x^(10/3)+3/11*b^2*x^(11/3)

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Maxima [A]  time = 1.44591, size = 200, normalized size = 5.88 \[ \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11}}{11 \, b^{9}} - \frac{12 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a}{5 \, b^{9}} + \frac{28 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{2}}{3 \, b^{9}} - \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{3}}{b^{9}} + \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{4}}{b^{9}} - \frac{28 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{5}}{b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{6}}{5 \, b^{9}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{7}}{b^{9}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{8}}{b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/11*(b*x^(1/3) + a)^11/b^9 - 12/5*(b*x^(1/3) + a)^10*a/b^9 + 28/3*(b*x^(1/3) +
a)^9*a^2/b^9 - 21*(b*x^(1/3) + a)^8*a^3/b^9 + 30*(b*x^(1/3) + a)^7*a^4/b^9 - 28*
(b*x^(1/3) + a)^6*a^5/b^9 + 84/5*(b*x^(1/3) + a)^5*a^6/b^9 - 6*(b*x^(1/3) + a)^4
*a^7/b^9 + (b*x^(1/3) + a)^3*a^8/b^9

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Fricas [A]  time = 0.21574, size = 32, normalized size = 0.94 \[ \frac{3}{11} \, b^{2} x^{\frac{11}{3}} + \frac{3}{5} \, a b x^{\frac{10}{3}} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/11*b^2*x^(11/3) + 3/5*a*b*x^(10/3) + 1/3*a^2*x^3

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Sympy [A]  time = 2.00397, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{3}}{3} + \frac{3 a b x^{\frac{10}{3}}}{5} + \frac{3 b^{2} x^{\frac{11}{3}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3/3 + 3*a*b*x**(10/3)/5 + 3*b**2*x**(11/3)/11

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GIAC/XCAS [A]  time = 0.263706, size = 32, normalized size = 0.94 \[ \frac{3}{11} \, b^{2} x^{\frac{11}{3}} + \frac{3}{5} \, a b x^{\frac{10}{3}} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/11*b^2*x^(11/3) + 3/5*a*b*x^(10/3) + 1/3*a^2*x^3