3.604 \(\int x^5 (a+b x^4) \, dx\)

Optimal. Leaf size=17 \[ \frac{a x^6}{6}+\frac{b x^{10}}{10} \]

[Out]

(a*x^6)/6 + (b*x^10)/10

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Rubi [A]  time = 0.0052271, 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} \[ \frac{a x^6}{6}+\frac{b x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^4),x]

[Out]

(a*x^6)/6 + (b*x^10)/10

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

Mathematica [A]  time = 0.0010784, size = 17, normalized size = 1. \[ \frac{a x^6}{6}+\frac{b x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^4),x]

[Out]

(a*x^6)/6 + (b*x^10)/10

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Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \begin{align*}{\frac{{x}^{6}a}{6}}+{\frac{b{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^4+a),x)

[Out]

1/6*x^6*a+1/10*b*x^10

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Maxima [A]  time = 0.961387, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{10} \, b x^{10} + \frac{1}{6} \, a x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a),x, algorithm="maxima")

[Out]

1/10*b*x^10 + 1/6*a*x^6

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Fricas [A]  time = 1.21138, size = 34, normalized size = 2. \begin{align*} \frac{1}{10} x^{10} b + \frac{1}{6} x^{6} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a),x, algorithm="fricas")

[Out]

1/10*x^10*b + 1/6*x^6*a

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Sympy [A]  time = 0.056516, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{6}}{6} + \frac{b x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**4+a),x)

[Out]

a*x**6/6 + b*x**10/10

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Giac [A]  time = 1.09871, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{10} \, b x^{10} + \frac{1}{6} \, a x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a),x, algorithm="giac")

[Out]

1/10*b*x^10 + 1/6*a*x^6