3.1.91 \(\int x (a+b x^2)^8 \, dx\) [91]

Optimal. Leaf size=16 \[ \frac {\left (a+b x^2\right )^9}{18 b} \]

[Out]

1/18*(b*x^2+a)^9/b

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \begin {gather*} \frac {\left (a+b x^2\right )^9}{18 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^8,x]

[Out]

(a + b*x^2)^9/(18*b)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \left (a+b x^2\right )^8 \, dx &=\frac {\left (a+b x^2\right )^9}{18 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^9}{18 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^8,x]

[Out]

(a + b*x^2)^9/(18*b)

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Maple [A]
time = 0.05, size = 15, normalized size = 0.94

method	result	size
default	\(\frac {\left (b \,x^{2}+a \right )^{9}}{18 b}\)	\(15\)
gosper	\(\frac {1}{2} a^{8} x^{2}+2 a^{7} b \,x^{4}+\frac {14}{3} a^{6} b^{2} x^{6}+7 a^{5} b^{3} x^{8}+7 a^{4} b^{4} x^{10}+\frac {14}{3} a^{3} b^{5} x^{12}+2 a^{2} b^{6} x^{14}+\frac {1}{2} a \,b^{7} x^{16}+\frac {1}{18} b^{8} x^{18}\)	\(91\)
norman	\(\frac {1}{2} a^{8} x^{2}+2 a^{7} b \,x^{4}+\frac {14}{3} a^{6} b^{2} x^{6}+7 a^{5} b^{3} x^{8}+7 a^{4} b^{4} x^{10}+\frac {14}{3} a^{3} b^{5} x^{12}+2 a^{2} b^{6} x^{14}+\frac {1}{2} a \,b^{7} x^{16}+\frac {1}{18} b^{8} x^{18}\)	\(91\)
risch	\(\frac {b^{8} x^{18}}{18}+\frac {a \,b^{7} x^{16}}{2}+2 a^{2} b^{6} x^{14}+\frac {14 a^{3} b^{5} x^{12}}{3}+7 a^{4} b^{4} x^{10}+7 a^{5} b^{3} x^{8}+\frac {14 a^{6} b^{2} x^{6}}{3}+2 a^{7} b \,x^{4}+\frac {a^{8} x^{2}}{2}+\frac {a^{9}}{18 b}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(b*x^2+a)^9/b

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Maxima [A]
time = 0.28, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{9}}{18 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(b*x^2 + a)^9/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (14) = 28\).
time = 1.05, size = 90, normalized size = 5.62 \begin {gather*} \frac {1}{18} \, b^{8} x^{18} + \frac {1}{2} \, a b^{7} x^{16} + 2 \, a^{2} b^{6} x^{14} + \frac {14}{3} \, a^{3} b^{5} x^{12} + 7 \, a^{4} b^{4} x^{10} + 7 \, a^{5} b^{3} x^{8} + \frac {14}{3} \, a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{4} + \frac {1}{2} \, a^{8} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*b^8*x^18 + 1/2*a*b^7*x^16 + 2*a^2*b^6*x^14 + 14/3*a^3*b^5*x^12 + 7*a^4*b^4*x^10 + 7*a^5*b^3*x^8 + 14/3*a^
6*b^2*x^6 + 2*a^7*b*x^4 + 1/2*a^8*x^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (10) = 20\).
time = 0.02, size = 99, normalized size = 6.19 \begin {gather*} \frac {a^{8} x^{2}}{2} + 2 a^{7} b x^{4} + \frac {14 a^{6} b^{2} x^{6}}{3} + 7 a^{5} b^{3} x^{8} + 7 a^{4} b^{4} x^{10} + \frac {14 a^{3} b^{5} x^{12}}{3} + 2 a^{2} b^{6} x^{14} + \frac {a b^{7} x^{16}}{2} + \frac {b^{8} x^{18}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x**2/2 + 2*a**7*b*x**4 + 14*a**6*b**2*x**6/3 + 7*a**5*b**3*x**8 + 7*a**4*b**4*x**10 + 14*a**3*b**5*x**12/
3 + 2*a**2*b**6*x**14 + a*b**7*x**16/2 + b**8*x**18/18

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Giac [A]
time = 1.35, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{9}}{18 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(b*x^2 + a)^9/b

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Mupad [B]
time = 4.61, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b\,x^2+a\right )}^9}{18\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^8,x)

[Out]

(a + b*x^2)^9/(18*b)

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