3.638 \(\int (a+b x^4)^3 \, dx\)

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

[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 = 0.0109337, 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, Rules used = {194} \[ \frac{3}{5} a^2 b x^5+a^3 x+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{13}}{13} \]

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a+b x^4\right )^3 \, dx &=\int \left (a^3+3 a^2 b x^4+3 a b^2 x^8+b^3 x^{12}\right ) \, dx\\ &=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}\\ \end{align*}

Mathematica [A]  time = 0.0008407, size = 38, normalized size = 1. \[ \frac{3}{5} a^2 b x^5+a^3 x+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{13}}{13} \]

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., size = 33, normalized size = 0.9 \begin{align*}{a}^{3}x+{\frac{3\,{a}^{2}b{x}^{5}}{5}}+{\frac{a{b}^{2}{x}^{9}}{3}}+{\frac{{b}^{3}{x}^{13}}{13}} \end{align*}

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 = 0.99049, size = 43, normalized size = 1.13 \begin{align*} \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 \end{align*}

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 = 1.26296, size = 74, normalized size = 1.95 \begin{align*} \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} \end{align*}

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 = 0.092543, size = 34, normalized size = 0.89 \begin{align*} a^{3} x + \frac{3 a^{2} b x^{5}}{5} + \frac{a b^{2} x^{9}}{3} + \frac{b^{3} x^{13}}{13} \end{align*}

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 [A]  time = 1.08443, size = 43, normalized size = 1.13 \begin{align*} \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 \end{align*}

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