3.1439 \(\int x^8 (a+b x^7)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^9}{9}+\frac{1}{8} a b x^{16}+\frac{b^2 x^{23}}{23} \]

[Out]

(a^2*x^9)/9 + (a*b*x^16)/8 + (b^2*x^23)/23

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Rubi [A]  time = 0.0115207, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{a^2 x^9}{9}+\frac{1}{8} a b x^{16}+\frac{b^2 x^{23}}{23} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^7)^2,x]

[Out]

(a^2*x^9)/9 + (a*b*x^16)/8 + (b^2*x^23)/23

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^8 \left (a+b x^7\right )^2 \, dx &=\int \left (a^2 x^8+2 a b x^{15}+b^2 x^{22}\right ) \, dx\\ &=\frac{a^2 x^9}{9}+\frac{1}{8} a b x^{16}+\frac{b^2 x^{23}}{23}\\ \end{align*}

Mathematica [A]  time = 0.0007884, size = 30, normalized size = 1. \[ \frac{a^2 x^9}{9}+\frac{1}{8} a b x^{16}+\frac{b^2 x^{23}}{23} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^7)^2,x]

[Out]

(a^2*x^9)/9 + (a*b*x^16)/8 + (b^2*x^23)/23

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{9}}{9}}+{\frac{ab{x}^{16}}{8}}+{\frac{{b}^{2}{x}^{23}}{23}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x^7+a)^2,x)

[Out]

1/9*a^2*x^9+1/8*a*b*x^16+1/23*b^2*x^23

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Maxima [A]  time = 0.948745, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{23} \, b^{2} x^{23} + \frac{1}{8} \, a b x^{16} + \frac{1}{9} \, a^{2} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a)^2,x, algorithm="maxima")

[Out]

1/23*b^2*x^23 + 1/8*a*b*x^16 + 1/9*a^2*x^9

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Fricas [A]  time = 1.50807, size = 59, normalized size = 1.97 \begin{align*} \frac{1}{23} x^{23} b^{2} + \frac{1}{8} x^{16} b a + \frac{1}{9} x^{9} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a)^2,x, algorithm="fricas")

[Out]

1/23*x^23*b^2 + 1/8*x^16*b*a + 1/9*x^9*a^2

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Sympy [A]  time = 0.060665, size = 24, normalized size = 0.8 \begin{align*} \frac{a^{2} x^{9}}{9} + \frac{a b x^{16}}{8} + \frac{b^{2} x^{23}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x**7+a)**2,x)

[Out]

a**2*x**9/9 + a*b*x**16/8 + b**2*x**23/23

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Giac [A]  time = 1.16269, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{23} \, b^{2} x^{23} + \frac{1}{8} \, a b x^{16} + \frac{1}{9} \, a^{2} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a)^2,x, algorithm="giac")

[Out]

1/23*b^2*x^23 + 1/8*a*b*x^16 + 1/9*a^2*x^9