3.2288 \(\int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^4}{4}+\frac{6}{13} a b x^{13/3}+\frac{3}{14} b^2 x^{14/3} \]

[Out]

(a^2*x^4)/4 + (6*a*b*x^(13/3))/13 + (3*b^2*x^(14/3))/14

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Rubi [A]  time = 0.070833, 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 \[ \frac{a^2 x^4}{4}+\frac{6}{13} a b x^{13/3}+\frac{3}{14} b^2 x^{14/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^2*x^3,x]

[Out]

(a^2*x^4)/4 + (6*a*b*x^(13/3))/13 + (3*b^2*x^(14/3))/14

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Rubi in Sympy [A]  time = 10.8763, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{4}}{4} + \frac{6 a b x^{\frac{13}{3}}}{13} + \frac{3 b^{2} x^{\frac{14}{3}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + 6*a*b*x**(13/3)/13 + 3*b**2*x**(14/3)/14

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Mathematica [A]  time = 0.0110125, size = 34, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{6}{13} a b x^{13/3}+\frac{3}{14} b^2 x^{14/3} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^4)/4 + (6*a*b*x^(13/3))/13 + (3*b^2*x^(14/3))/14

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Maple [A]  time = 0.001, size = 25, normalized size = 0.7 \[{\frac{{x}^{4}{a}^{2}}{4}}+{\frac{6\,ab}{13}{x}^{{\frac{13}{3}}}}+{\frac{3\,{b}^{2}}{14}{x}^{{\frac{14}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^2*x^3,x)

[Out]

1/4*x^4*a^2+6/13*a*b*x^(13/3)+3/14*b^2*x^(14/3)

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Maxima [A]  time = 1.44625, size = 270, normalized size = 7.94 \[ \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14}}{14 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a}{13 \, b^{12}} + \frac{55 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{2}}{4 \, b^{12}} - \frac{45 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{3}}{b^{12}} + \frac{99 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{4}}{b^{12}} - \frac{154 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{5}}{b^{12}} + \frac{693 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{6}}{4 \, b^{12}} - \frac{990 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{7}}{7 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{8}}{2 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{9}}{b^{12}} + \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{10}}{4 \, b^{12}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{11}}{b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*(b*x^(1/3) + a)^14/b^12 - 33/13*(b*x^(1/3) + a)^13*a/b^12 + 55/4*(b*x^(1/3)
+ a)^12*a^2/b^12 - 45*(b*x^(1/3) + a)^11*a^3/b^12 + 99*(b*x^(1/3) + a)^10*a^4/b
^12 - 154*(b*x^(1/3) + a)^9*a^5/b^12 + 693/4*(b*x^(1/3) + a)^8*a^6/b^12 - 990/7*
(b*x^(1/3) + a)^7*a^7/b^12 + 165/2*(b*x^(1/3) + a)^6*a^8/b^12 - 33*(b*x^(1/3) +
a)^5*a^9/b^12 + 33/4*(b*x^(1/3) + a)^4*a^10/b^12 - (b*x^(1/3) + a)^3*a^11/b^12

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Fricas [A]  time = 0.212367, size = 32, normalized size = 0.94 \[ \frac{3}{14} \, b^{2} x^{\frac{14}{3}} + \frac{6}{13} \, a b x^{\frac{13}{3}} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*b^2*x^(14/3) + 6/13*a*b*x^(13/3) + 1/4*a^2*x^4

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Sympy [A]  time = 3.39547, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{4}}{4} + \frac{6 a b x^{\frac{13}{3}}}{13} + \frac{3 b^{2} x^{\frac{14}{3}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + 6*a*b*x**(13/3)/13 + 3*b**2*x**(14/3)/14

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GIAC/XCAS [A]  time = 0.254471, size = 32, normalized size = 0.94 \[ \frac{3}{14} \, b^{2} x^{\frac{14}{3}} + \frac{6}{13} \, a b x^{\frac{13}{3}} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*b^2*x^(14/3) + 6/13*a*b*x^(13/3) + 1/4*a^2*x^4