3.636 \(\int \left (a+b x^4\right )^3 \, dx\)

Optimal. Leaf size=38 \[ a^3 x+\frac{3}{5} a^2 b x^5+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{13}}{13} \]

[Out]

a^3*x + (3*a^2*b*x^5)/5 + (a*b^2*x^9)/3 + (b^3*x^13)/13

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Rubi [A]  time = 0.0276609, antiderivative size = 38, 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 \[ a^3 x+\frac{3}{5} a^2 b x^5+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^3,x]

[Out]

a^3*x + (3*a^2*b*x^5)/5 + (a*b^2*x^9)/3 + (b^3*x^13)/13

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{2} b x^{5}}{5} + \frac{a b^{2} x^{9}}{3} + \frac{b^{3} x^{13}}{13} + \int a^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**3,x)

[Out]

3*a**2*b*x**5/5 + a*b**2*x**9/3 + b**3*x**13/13 + Integral(a**3, x)

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Mathematica [A]  time = 0.0015676, size = 38, normalized size = 1. \[ a^3 x+\frac{3}{5} a^2 b x^5+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^3,x]

[Out]

a^3*x + (3*a^2*b*x^5)/5 + (a*b^2*x^9)/3 + (b^3*x^13)/13

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Maple [A]  time = 0.001, size = 33, normalized size = 0.9 \[{a}^{3}x+{\frac{3\,{a}^{2}b{x}^{5}}{5}}+{\frac{a{b}^{2}{x}^{9}}{3}}+{\frac{{b}^{3}{x}^{13}}{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^3,x)

[Out]

a^3*x+3/5*a^2*b*x^5+1/3*a*b^2*x^9+1/13*b^3*x^13

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Maxima [A]  time = 1.441, size = 43, normalized size = 1.13 \[ \frac{1}{13} \, b^{3} x^{13} + \frac{1}{3} \, a b^{2} x^{9} + \frac{3}{5} \, a^{2} b x^{5} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3,x, algorithm="maxima")

[Out]

1/13*b^3*x^13 + 1/3*a*b^2*x^9 + 3/5*a^2*b*x^5 + a^3*x

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Fricas [A]  time = 0.218377, size = 1, normalized size = 0.03 \[ \frac{1}{13} x^{13} b^{3} + \frac{1}{3} x^{9} b^{2} a + \frac{3}{5} x^{5} b a^{2} + x a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3,x, algorithm="fricas")

[Out]

1/13*x^13*b^3 + 1/3*x^9*b^2*a + 3/5*x^5*b*a^2 + x*a^3

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Sympy [A]  time = 0.091466, size = 34, normalized size = 0.89 \[ a^{3} x + \frac{3 a^{2} b x^{5}}{5} + \frac{a b^{2} x^{9}}{3} + \frac{b^{3} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**3,x)

[Out]

a**3*x + 3*a**2*b*x**5/5 + a*b**2*x**9/3 + b**3*x**13/13

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GIAC/XCAS [A]  time = 0.216401, size = 43, normalized size = 1.13 \[ \frac{1}{13} \, b^{3} x^{13} + \frac{1}{3} \, a b^{2} x^{9} + \frac{3}{5} \, a^{2} b x^{5} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3,x, algorithm="giac")

[Out]

1/13*b^3*x^13 + 1/3*a*b^2*x^9 + 3/5*a^2*b*x^5 + a^3*x