3.66.74 \(\int \frac {-x^2+2 x \log (5)-\log ^2(5)+(x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+(-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9) \log (5)+(x+216 x^2+216 x^4+72 x^6+8 x^8) \log ^2(5)) \log ^2(x)}{(x^3-2 x^2 \log (5)+x \log ^2(5)) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ 2+x+\left (3+x^2\right )^4-\frac {\log (5)}{-x+\log (5)}+\frac {1}{\log (x)} \]

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Rubi [A]  time = 1.09, antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 6, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {1594, 27, 6742, 1850, 2302, 30} \begin {gather*} x^8+12 x^6+54 x^4+108 x^2+x+\frac {1}{\log (x)}+\frac {\log (5)}{x-\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^2 + 2*x*Log[5] - Log[5]^2 + (x^3 + 216*x^4 + 216*x^6 + 72*x^8 + 8*x^10 + (-x - 2*x^2 - 432*x^3 - 432*x
^5 - 144*x^7 - 16*x^9)*Log[5] + (x + 216*x^2 + 216*x^4 + 72*x^6 + 8*x^8)*Log[5]^2)*Log[x]^2)/((x^3 - 2*x^2*Log
[5] + x*Log[5]^2)*Log[x]^2),x]

[Out]

x + 108*x^2 + 54*x^4 + 12*x^6 + x^8 + Log[5]/(x - Log[5]) + Log[x]^(-1)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{x \left (x^2-2 x \log (5)+\log ^2(5)\right ) \log ^2(x)} \, dx\\ &=\int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{x (x-\log (5))^2 \log ^2(x)} \, dx\\ &=\int \left (\frac {8 x^9+x^2 (1-432 \log (5))-432 x^4 \log (5)-144 x^6 \log (5)-16 x^8 \log (5)-2 x (1-108 \log (5)) \log (5)-(1-\log (5)) \log (5)+72 x^7 \left (1+\frac {\log ^2(5)}{9}\right )+216 x^5 \left (1+\frac {\log ^2(5)}{3}\right )+216 x^3 \left (1+\log ^2(5)\right )}{(x-\log (5))^2}-\frac {1}{x \log ^2(x)}\right ) \, dx\\ &=\int \frac {8 x^9+x^2 (1-432 \log (5))-432 x^4 \log (5)-144 x^6 \log (5)-16 x^8 \log (5)-2 x (1-108 \log (5)) \log (5)-(1-\log (5)) \log (5)+72 x^7 \left (1+\frac {\log ^2(5)}{9}\right )+216 x^5 \left (1+\frac {\log ^2(5)}{3}\right )+216 x^3 \left (1+\log ^2(5)\right )}{(x-\log (5))^2} \, dx-\int \frac {1}{x \log ^2(x)} \, dx\\ &=\int \left (1+216 x+216 x^3+72 x^5+8 x^7-\frac {\log (5)}{(x-\log (5))^2}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=x+108 x^2+54 x^4+12 x^6+x^8+\frac {\log (5)}{x-\log (5)}+\frac {1}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 128, normalized size = 4.92 \begin {gather*} \frac {x-\log (5)+\left (108 x^3+54 x^5+12 x^7+x^9+x^2 (1-108 \log (5))-54 x^4 \log (5)-12 x^6 \log (5)-x^8 \log (5)-x \log (5) \left (2+108 \log (5)+54 \log ^3(5)+12 \log ^5(5)+\log ^7(5)\right )+\log (5) \left (1+\log (5)+108 \log ^2(5)+54 \log ^4(5)+12 \log ^6(5)+\log ^8(5)\right )\right ) \log (x)}{(x-\log (5)) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + 2*x*Log[5] - Log[5]^2 + (x^3 + 216*x^4 + 216*x^6 + 72*x^8 + 8*x^10 + (-x - 2*x^2 - 432*x^3 -
432*x^5 - 144*x^7 - 16*x^9)*Log[5] + (x + 216*x^2 + 216*x^4 + 72*x^6 + 8*x^8)*Log[5]^2)*Log[x]^2)/((x^3 - 2*x
^2*Log[5] + x*Log[5]^2)*Log[x]^2),x]

[Out]

(x - Log[5] + (108*x^3 + 54*x^5 + 12*x^7 + x^9 + x^2*(1 - 108*Log[5]) - 54*x^4*Log[5] - 12*x^6*Log[5] - x^8*Lo
g[5] - x*Log[5]*(2 + 108*Log[5] + 54*Log[5]^3 + 12*Log[5]^5 + Log[5]^7) + Log[5]*(1 + Log[5] + 108*Log[5]^2 +
54*Log[5]^4 + 12*Log[5]^6 + Log[5]^8))*Log[x])/((x - Log[5])*Log[x])

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fricas [B]  time = 0.97, size = 69, normalized size = 2.65 \begin {gather*} \frac {{\left (x^{9} + 12 \, x^{7} + 54 \, x^{5} + 108 \, x^{3} + x^{2} - {\left (x^{8} + 12 \, x^{6} + 54 \, x^{4} + 108 \, x^{2} + x - 1\right )} \log \relax (5)\right )} \log \relax (x) + x - \log \relax (5)}{{\left (x - \log \relax (5)\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^8+72*x^6+216*x^4+216*x^2+x)*log(5)^2+(-16*x^9-144*x^7-432*x^5-432*x^3-2*x^2-x)*log(5)+8*x^10+
72*x^8+216*x^6+216*x^4+x^3)*log(x)^2-log(5)^2+2*x*log(5)-x^2)/(x*log(5)^2-2*x^2*log(5)+x^3)/log(x)^2,x, algori
thm="fricas")

[Out]

((x^9 + 12*x^7 + 54*x^5 + 108*x^3 + x^2 - (x^8 + 12*x^6 + 54*x^4 + 108*x^2 + x - 1)*log(5))*log(x) + x - log(5
))/((x - log(5))*log(x))

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giac [A]  time = 0.19, size = 35, normalized size = 1.35 \begin {gather*} x^{8} + 12 \, x^{6} + 54 \, x^{4} + 108 \, x^{2} + x + \frac {\log \relax (5)}{x - \log \relax (5)} + \frac {1}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^8+72*x^6+216*x^4+216*x^2+x)*log(5)^2+(-16*x^9-144*x^7-432*x^5-432*x^3-2*x^2-x)*log(5)+8*x^10+
72*x^8+216*x^6+216*x^4+x^3)*log(x)^2-log(5)^2+2*x*log(5)-x^2)/(x*log(5)^2-2*x^2*log(5)+x^3)/log(x)^2,x, algori
thm="giac")

[Out]

x^8 + 12*x^6 + 54*x^4 + 108*x^2 + x + log(5)/(x - log(5)) + 1/log(x)

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maple [A]  time = 0.08, size = 36, normalized size = 1.38

method	result	size
default	\(x^{8}+12 x^{6}+54 x^{4}+108 x^{2}+x +\frac {\ln \relax (5)}{-\ln \relax (5)+x}+\frac {1}{\ln \relax (x )}\)	\(36\)
risch	\(\frac {x^{8} \ln \relax (5)-x^{9}+12 x^{6} \ln \relax (5)-12 x^{7}+54 x^{4} \ln \relax (5)-54 x^{5}+108 x^{2} \ln \relax (5)-108 x^{3}+x \ln \relax (5)-x^{2}-\ln \relax (5)}{\ln \relax (5)-x}+\frac {1}{\ln \relax (x )}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^8+72*x^6+216*x^4+216*x^2+x)*ln(5)^2+(-16*x^9-144*x^7-432*x^5-432*x^3-2*x^2-x)*ln(5)+8*x^10+72*x^8+2
16*x^6+216*x^4+x^3)*ln(x)^2-ln(5)^2+2*x*ln(5)-x^2)/(x*ln(5)^2-2*x^2*ln(5)+x^3)/ln(x)^2,x,method=_RETURNVERBOSE
)

[Out]

x^8+12*x^6+54*x^4+108*x^2+x+ln(5)/(-ln(5)+x)+1/ln(x)

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maxima [B]  time = 0.50, size = 80, normalized size = 3.08 \begin {gather*} \frac {{\left (x^{9} - x^{8} \log \relax (5) + 12 \, x^{7} - 12 \, x^{6} \log \relax (5) + 54 \, x^{5} - 54 \, x^{4} \log \relax (5) + 108 \, x^{3} - x^{2} {\left (108 \, \log \relax (5) - 1\right )} - x \log \relax (5) + \log \relax (5)\right )} \log \relax (x) + x - \log \relax (5)}{{\left (x - \log \relax (5)\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^8+72*x^6+216*x^4+216*x^2+x)*log(5)^2+(-16*x^9-144*x^7-432*x^5-432*x^3-2*x^2-x)*log(5)+8*x^10+
72*x^8+216*x^6+216*x^4+x^3)*log(x)^2-log(5)^2+2*x*log(5)-x^2)/(x*log(5)^2-2*x^2*log(5)+x^3)/log(x)^2,x, algori
thm="maxima")

[Out]

((x^9 - x^8*log(5) + 12*x^7 - 12*x^6*log(5) + 54*x^5 - 54*x^4*log(5) + 108*x^3 - x^2*(108*log(5) - 1) - x*log(
5) + log(5))*log(x) + x - log(5))/((x - log(5))*log(x))

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mupad [B]  time = 4.57, size = 35, normalized size = 1.35 \begin {gather*} x+\frac {1}{\ln \relax (x)}+\frac {\ln \relax (5)}{x-\ln \relax (5)}+108\,x^2+54\,x^4+12\,x^6+x^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*log(5) + log(x)^2*(x^3 - log(5)*(x + 2*x^2 + 432*x^3 + 432*x^5 + 144*x^7 + 16*x^9) + 216*x^4 + 216*x^
6 + 72*x^8 + 8*x^10 + log(5)^2*(x + 216*x^2 + 216*x^4 + 72*x^6 + 8*x^8)) - log(5)^2 - x^2)/(log(x)^2*(x*log(5)
^2 - 2*x^2*log(5) + x^3)),x)

[Out]

x + 1/log(x) + log(5)/(x - log(5)) + 108*x^2 + 54*x^4 + 12*x^6 + x^8

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sympy [A]  time = 0.28, size = 32, normalized size = 1.23 \begin {gather*} x^{8} + 12 x^{6} + 54 x^{4} + 108 x^{2} + x + \frac {1}{\log {\relax (x )}} + \frac {\log {\relax (5 )}}{x - \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**8+72*x**6+216*x**4+216*x**2+x)*ln(5)**2+(-16*x**9-144*x**7-432*x**5-432*x**3-2*x**2-x)*ln(5)
+8*x**10+72*x**8+216*x**6+216*x**4+x**3)*ln(x)**2-ln(5)**2+2*x*ln(5)-x**2)/(x*ln(5)**2-2*x**2*ln(5)+x**3)/ln(x
)**2,x)

[Out]

x**8 + 12*x**6 + 54*x**4 + 108*x**2 + x + 1/log(x) + log(5)/(x - log(5))

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