3.1436 \(\int x^8 (a+b x^7) \, dx\)

Optimal. Leaf size=17 \[ \frac{a x^9}{9}+\frac{b x^{16}}{16} \]

[Out]

(a*x^9)/9 + (b*x^16)/16

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Rubi [A]  time = 0.0049247, 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} \[ \frac{a x^9}{9}+\frac{b x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^7),x]

[Out]

(a*x^9)/9 + (b*x^16)/16

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^8 \left (a+b x^7\right ) \, dx &=\int \left (a x^8+b x^{15}\right ) \, dx\\ &=\frac{a x^9}{9}+\frac{b x^{16}}{16}\\ \end{align*}

Mathematica [A]  time = 0.0011724, size = 17, normalized size = 1. \[ \frac{a x^9}{9}+\frac{b x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^7),x]

[Out]

(a*x^9)/9 + (b*x^16)/16

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Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \begin{align*}{\frac{a{x}^{9}}{9}}+{\frac{b{x}^{16}}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x^7+a),x)

[Out]

1/9*a*x^9+1/16*b*x^16

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Maxima [A]  time = 0.949015, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{16} \, b x^{16} + \frac{1}{9} \, a x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a),x, algorithm="maxima")

[Out]

1/16*b*x^16 + 1/9*a*x^9

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Fricas [A]  time = 1.47321, size = 34, normalized size = 2. \begin{align*} \frac{1}{16} x^{16} b + \frac{1}{9} x^{9} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a),x, algorithm="fricas")

[Out]

1/16*x^16*b + 1/9*x^9*a

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Sympy [A]  time = 0.052927, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{9}}{9} + \frac{b x^{16}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x**7+a),x)

[Out]

a*x**9/9 + b*x**16/16

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Giac [A]  time = 1.10248, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{16} \, b x^{16} + \frac{1}{9} \, a x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a),x, algorithm="giac")

[Out]

1/16*b*x^16 + 1/9*a*x^9