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

Optimal. Leaf size=43 \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^2}{2}+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{14}}{14} \]

[Out]

(a^3*x^2)/2 + (a^2*b*x^6)/2 + (3*a*b^2*x^10)/10 + (b^3*x^14)/14

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Rubi [A]  time = 0.013336, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {270} \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^2}{2}+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^4)^3,x]

[Out]

(a^3*x^2)/2 + (a^2*b*x^6)/2 + (3*a*b^2*x^10)/10 + (b^3*x^14)/14

Rule 270

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

Rubi steps

\begin{align*} \int x \left (a+b x^4\right )^3 \, dx &=\int \left (a^3 x+3 a^2 b x^5+3 a b^2 x^9+b^3 x^{13}\right ) \, dx\\ &=\frac{a^3 x^2}{2}+\frac{1}{2} a^2 b x^6+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{14}}{14}\\ \end{align*}

Mathematica [A]  time = 0.001576, size = 43, normalized size = 1. \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^2}{2}+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^4)^3,x]

[Out]

(a^3*x^2)/2 + (a^2*b*x^6)/2 + (3*a*b^2*x^10)/10 + (b^3*x^14)/14

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Maple [A]  time = 0., size = 36, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{{a}^{2}b{x}^{6}}{2}}+{\frac{3\,a{b}^{2}{x}^{10}}{10}}+{\frac{{b}^{3}{x}^{14}}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^4+a)^3,x)

[Out]

1/2*x^2*a^3+1/2*a^2*b*x^6+3/10*a*b^2*x^10+1/14*b^3*x^14

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Maxima [A]  time = 0.966163, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{14} \, b^{3} x^{14} + \frac{3}{10} \, a b^{2} x^{10} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{2} \, a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/10*a*b^2*x^10 + 1/2*a^2*b*x^6 + 1/2*a^3*x^2

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Fricas [A]  time = 1.22438, size = 85, normalized size = 1.98 \begin{align*} \frac{1}{14} x^{14} b^{3} + \frac{3}{10} x^{10} b^{2} a + \frac{1}{2} x^{6} b a^{2} + \frac{1}{2} x^{2} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*x^14*b^3 + 3/10*x^10*b^2*a + 1/2*x^6*b*a^2 + 1/2*x^2*a^3

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Sympy [A]  time = 0.070538, size = 37, normalized size = 0.86 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{a^{2} b x^{6}}{2} + \frac{3 a b^{2} x^{10}}{10} + \frac{b^{3} x^{14}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**4+a)**3,x)

[Out]

a**3*x**2/2 + a**2*b*x**6/2 + 3*a*b**2*x**10/10 + b**3*x**14/14

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Giac [A]  time = 1.10131, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{14} \, b^{3} x^{14} + \frac{3}{10} \, a b^{2} x^{10} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{2} \, a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/10*a*b^2*x^10 + 1/2*a^2*b*x^6 + 1/2*a^3*x^2