3.15.39 \(\int x^8 (a+b x^7)^2 \, dx\) [1439]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, 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 = {276} \begin {gather*} \frac {a^2 x^9}{9}+\frac {1}{8} a b x^{16}+\frac {b^2 x^{23}}{23} \end {gather*}

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 276

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*}

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^9}{9}+\frac {1}{8} a b x^{16}+\frac {b^2 x^{23}}{23} \end {gather*}

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 = 0.17, size = 25, normalized size = 0.83

method	result	size
gosper	\(\frac {1}{9} a^{2} x^{9}+\frac {1}{8} a b \,x^{16}+\frac {1}{23} b^{2} x^{23}\)	\(25\)
default	\(\frac {1}{9} a^{2} x^{9}+\frac {1}{8} a b \,x^{16}+\frac {1}{23} b^{2} x^{23}\)	\(25\)
norman	\(\frac {1}{9} a^{2} x^{9}+\frac {1}{8} a b \,x^{16}+\frac {1}{23} b^{2} x^{23}\)	\(25\)
risch	\(\frac {1}{9} a^{2} x^{9}+\frac {1}{8} a b \,x^{16}+\frac {1}{23} b^{2} x^{23}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{23} \, b^{2} x^{23} + \frac {1}{8} \, a b x^{16} + \frac {1}{9} \, a^{2} x^{9} \end {gather*}

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 = 0.35, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{23} \, b^{2} x^{23} + \frac {1}{8} \, a b x^{16} + \frac {1}{9} \, a^{2} x^{9} \end {gather*}

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

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Sympy [A]
time = 0.01, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^{2} x^{9}}{9} + \frac {a b x^{16}}{8} + \frac {b^{2} x^{23}}{23} \end {gather*}

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 = 1.81, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{23} \, b^{2} x^{23} + \frac {1}{8} \, a b x^{16} + \frac {1}{9} \, a^{2} x^{9} \end {gather*}

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

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Mupad [B]
time = 0.04, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^9}{9}+\frac {a\,b\,x^{16}}{8}+\frac {b^2\,x^{23}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x^7)^2,x)

[Out]

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

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