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

Optimal. Leaf size=17 \[ \frac{a x^4}{4}+\frac{b x^8}{8} \]

[Out]

(a*x^4)/4 + (b*x^8)/8

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Rubi [A]  time = 0.0047609, antiderivative size = 17, 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 = {14} \[ \frac{a x^4}{4}+\frac{b x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^4),x]

[Out]

(a*x^4)/4 + (b*x^8)/8

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^3 \left (a+b x^4\right ) \, dx &=\int \left (a x^3+b x^7\right ) \, dx\\ &=\frac{a x^4}{4}+\frac{b x^8}{8}\\ \end{align*}

Mathematica [A]  time = 0.0009966, size = 17, normalized size = 1. \[ \frac{a x^4}{4}+\frac{b x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^4),x]

[Out]

(a*x^4)/4 + (b*x^8)/8

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Maple [A]  time = 0., size = 14, normalized size = 0.8 \begin{align*}{\frac{a{x}^{4}}{4}}+{\frac{b{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4+1/8*b*x^8

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Maxima [A]  time = 1.01682, size = 19, normalized size = 1.12 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{2}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x^4 + a)^2/b

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Fricas [A]  time = 1.23503, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{8} x^{8} b + \frac{1}{4} x^{4} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*b + 1/4*x^4*a

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Sympy [A]  time = 0.069485, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{4}}{4} + \frac{b x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**4/4 + b*x**8/8

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Giac [A]  time = 1.07424, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{8} \, b x^{8} + \frac{1}{4} \, a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b*x^8 + 1/4*a*x^4