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

Optimal. Leaf size=17 \[ \frac{a x^5}{5}+\frac{b x^9}{9} \]

[Out]

(a*x^5)/5 + (b*x^9)/9

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Rubi [A]  time = 0.0052259, 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} \[ \frac{a x^5}{5}+\frac{b x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^4),x]

[Out]

(a*x^5)/5 + (b*x^9)/9

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^4 \left (a+b x^4\right ) \, dx &=\int \left (a x^4+b x^8\right ) \, dx\\ &=\frac{a x^5}{5}+\frac{b x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.0010005, size = 17, normalized size = 1. \[ \frac{a x^5}{5}+\frac{b x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^4),x]

[Out]

(a*x^5)/5 + (b*x^9)/9

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Maple [A]  time = 0., size = 14, normalized size = 0.8 \begin{align*}{\frac{a{x}^{5}}{5}}+{\frac{b{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*x^5+1/9*b*x^9

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Maxima [A]  time = 0.983462, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{9} \, b x^{9} + \frac{1}{5} \, a x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*x^9 + 1/5*a*x^5

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Fricas [A]  time = 1.27614, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{9} x^{9} b + \frac{1}{5} x^{5} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*b + 1/5*x^5*a

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Sympy [A]  time = 0.059591, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{5}}{5} + \frac{b x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**5/5 + b*x**9/9

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Giac [A]  time = 1.08069, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{9} \, b x^{9} + \frac{1}{5} \, a x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*x^9 + 1/5*a*x^5