3.4.41 \(\int x (5+x^2)^8 \, dx\) [341]

Optimal. Leaf size=11 \[ \frac {1}{18} \left (5+x^2\right )^9 \]

[Out]

1/18*(x^2+5)^9

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {267} \begin {gather*} \frac {1}{18} \left (x^2+5\right )^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(5 + x^2)^8,x]

[Out]

(5 + x^2)^9/18

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 (5+x^2\right )^8 \, dx &=\frac {1}{18} \left (5+x^2\right )^9\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {1}{18} \left (5+x^2\right )^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(5 + x^2)^8,x]

[Out]

(5 + x^2)^9/18

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(11)=22\).
time = 1.81, size = 45, normalized size = 4.09 \begin {gather*} \frac {x^2 \left (3515625+2812500 x^2+1312500 x^4+393750 x^6+78750 x^8+10500 x^{10}+900 x^{12}+45 x^{14}+x^{16}\right )}{18} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x*(5 + x^2)^8,x]')

[Out]

x ^ 2 (3515625 + 2812500 x ^ 2 + 1312500 x ^ 4 + 393750 x ^ 6 + 78750 x ^ 8 + 10500 x ^ 10 + 900 x ^ 12 + 45 x
^ 14 + x ^ 16) / 18

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Maple [A]
time = 0.05, size = 10, normalized size = 0.91

method	result	size
default	\(\frac {\left (x^{2}+5\right )^{9}}{18}\)	\(10\)
gosper	\(\frac {390625}{2} x^{2}+156250 x^{4}+\frac {218750}{3} x^{6}+21875 x^{8}+4375 x^{10}+\frac {1750}{3} x^{12}+50 x^{14}+\frac {5}{2} x^{16}+\frac {1}{18} x^{18}\)	\(47\)
norman	\(\frac {390625}{2} x^{2}+156250 x^{4}+\frac {218750}{3} x^{6}+21875 x^{8}+4375 x^{10}+\frac {1750}{3} x^{12}+50 x^{14}+\frac {5}{2} x^{16}+\frac {1}{18} x^{18}\)	\(47\)
risch	\(\frac {1}{18} x^{18}+\frac {5}{2} x^{16}+50 x^{14}+\frac {1750}{3} x^{12}+4375 x^{10}+21875 x^{8}+\frac {218750}{3} x^{6}+156250 x^{4}+\frac {390625}{2} x^{2}+\frac {1953125}{18}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2+5)^8,x,method=_RETURNVERBOSE)

[Out]

1/18*(x^2+5)^9

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Maxima [A]
time = 0.26, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{18} \, {\left (x^{2} + 5\right )}^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2+5)^8,x, algorithm="maxima")

[Out]

1/18*(x^2 + 5)^9

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (9) = 18\).
time = 0.33, size = 46, normalized size = 4.18 \begin {gather*} \frac {1}{18} \, x^{18} + \frac {5}{2} \, x^{16} + 50 \, x^{14} + \frac {1750}{3} \, x^{12} + 4375 \, x^{10} + 21875 \, x^{8} + \frac {218750}{3} \, x^{6} + 156250 \, x^{4} + \frac {390625}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2+5)^8,x, algorithm="fricas")

[Out]

1/18*x^18 + 5/2*x^16 + 50*x^14 + 1750/3*x^12 + 4375*x^10 + 21875*x^8 + 218750/3*x^6 + 156250*x^4 + 390625/2*x^
2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (7) = 14\)
time = 0.03, size = 51, normalized size = 4.64 \begin {gather*} \frac {x^{18}}{18} + \frac {5 x^{16}}{2} + 50 x^{14} + \frac {1750 x^{12}}{3} + 4375 x^{10} + 21875 x^{8} + \frac {218750 x^{6}}{3} + 156250 x^{4} + \frac {390625 x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x**2+5)**8,x)

[Out]

x**18/18 + 5*x**16/2 + 50*x**14 + 1750*x**12/3 + 4375*x**10 + 21875*x**8 + 218750*x**6/3 + 156250*x**4 + 39062
5*x**2/2

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Giac [A]
time = 0.00, size = 12, normalized size = 1.09 \begin {gather*} \frac {\left (x^{2}+5\right )^{9}}{2\cdot 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2+5)^8,x)

[Out]

1/18*(x^2 + 5)^9

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Mupad [B]
time = 0.21, size = 9, normalized size = 0.82 \begin {gather*} \frac {{\left (x^2+5\right )}^9}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2 + 5)^8,x)

[Out]

(x^2 + 5)^9/18

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