3.93.48 \(\int \frac {-36 x^2-2 x^5+(-36 x+10 x^4) \log (5)-20 x^3 \log ^2(5)+20 x^2 \log ^3(5)-10 x \log ^4(5)+2 \log ^5(5)}{-x^5+5 x^4 \log (5)-10 x^3 \log ^2(5)+10 x^2 \log ^3(5)-5 x \log ^4(5)+\log ^5(5)} \, dx\)

Optimal. Leaf size=22 \[ 2 \left (8+x-3 \left (3+\frac {3 x^2}{(x-\log (5))^4}\right )\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 40, normalized size of antiderivative = 1.82, number of steps used = 3, number of rules used = 2, integrand size = 99, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6, 2074} \begin {gather*} 2 x-\frac {18 \log ^2(5)}{(x-\log (5))^4}-\frac {18}{(x-\log (5))^2}-\frac {36 \log (5)}{(x-\log (5))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36*x^2 - 2*x^5 + (-36*x + 10*x^4)*Log[5] - 20*x^3*Log[5]^2 + 20*x^2*Log[5]^3 - 10*x*Log[5]^4 + 2*Log[5]^
5)/(-x^5 + 5*x^4*Log[5] - 10*x^3*Log[5]^2 + 10*x^2*Log[5]^3 - 5*x*Log[5]^4 + Log[5]^5),x]

[Out]

2*x - 18/(x - Log[5])^2 - (36*Log[5])/(x - Log[5])^3 - (18*Log[5]^2)/(x - Log[5])^4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x^5+\left (-36 x+10 x^4\right ) \log (5)-20 x^3 \log ^2(5)-10 x \log ^4(5)+2 \log ^5(5)+x^2 \left (-36+20 \log ^3(5)\right )}{-x^5+5 x^4 \log (5)-10 x^3 \log ^2(5)+10 x^2 \log ^3(5)-5 x \log ^4(5)+\log ^5(5)} \, dx\\ &=\int \left (2+\frac {36}{(x-\log (5))^3}+\frac {108 \log (5)}{(x-\log (5))^4}+\frac {72 \log ^2(5)}{(x-\log (5))^5}\right ) \, dx\\ &=2 x-\frac {18}{(x-\log (5))^2}-\frac {36 \log (5)}{(x-\log (5))^3}-\frac {18 \log ^2(5)}{(x-\log (5))^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 17, normalized size = 0.77 \begin {gather*} 2 \left (x-\frac {9 x^2}{(x-\log (5))^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36*x^2 - 2*x^5 + (-36*x + 10*x^4)*Log[5] - 20*x^3*Log[5]^2 + 20*x^2*Log[5]^3 - 10*x*Log[5]^4 + 2*L
og[5]^5)/(-x^5 + 5*x^4*Log[5] - 10*x^3*Log[5]^2 + 10*x^2*Log[5]^3 - 5*x*Log[5]^4 + Log[5]^5),x]

[Out]

2*(x - (9*x^2)/(x - Log[5])^4)

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fricas [B]  time = 0.47, size = 75, normalized size = 3.41 \begin {gather*} \frac {2 \, {\left (x^{5} - 4 \, x^{4} \log \relax (5) + 6 \, x^{3} \log \relax (5)^{2} - 4 \, x^{2} \log \relax (5)^{3} + x \log \relax (5)^{4} - 9 \, x^{2}\right )}}{x^{4} - 4 \, x^{3} \log \relax (5) + 6 \, x^{2} \log \relax (5)^{2} - 4 \, x \log \relax (5)^{3} + \log \relax (5)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(5)^5-10*x*log(5)^4+20*x^2*log(5)^3-20*x^3*log(5)^2+(10*x^4-36*x)*log(5)-2*x^5-36*x^2)/(log(5)
^5-5*x*log(5)^4+10*x^2*log(5)^3-10*x^3*log(5)^2+5*x^4*log(5)-x^5),x, algorithm="fricas")

[Out]

2*(x^5 - 4*x^4*log(5) + 6*x^3*log(5)^2 - 4*x^2*log(5)^3 + x*log(5)^4 - 9*x^2)/(x^4 - 4*x^3*log(5) + 6*x^2*log(
5)^2 - 4*x*log(5)^3 + log(5)^4)

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giac [A]  time = 0.19, size = 17, normalized size = 0.77 \begin {gather*} 2 \, x - \frac {18 \, x^{2}}{{\left (x - \log \relax (5)\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(5)^5-10*x*log(5)^4+20*x^2*log(5)^3-20*x^3*log(5)^2+(10*x^4-36*x)*log(5)-2*x^5-36*x^2)/(log(5)
^5-5*x*log(5)^4+10*x^2*log(5)^3-10*x^3*log(5)^2+5*x^4*log(5)-x^5),x, algorithm="giac")

[Out]

2*x - 18*x^2/(x - log(5))^4

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maple [B]  time = 0.09, size = 41, normalized size = 1.86

method	result	size
default	\(2 x -\frac {36 \ln \relax (5)}{\left (-\ln \relax (5)+x \right )^{3}}-\frac {18 \ln \relax (5)^{2}}{\left (-\ln \relax (5)+x \right )^{4}}-\frac {18}{\left (-\ln \relax (5)+x \right )^{2}}\)	\(41\)
risch	\(2 x -\frac {18 x^{2}}{\ln \relax (5)^{4}-4 \ln \relax (5)^{3} x +6 x^{2} \ln \relax (5)^{2}-4 x^{3} \ln \relax (5)+x^{4}}\)	\(43\)
norman	\(\frac {-20 x^{3} \ln \relax (5)^{2}-30 x \ln \relax (5)^{4}+\left (40 \ln \relax (5)^{3}-18\right ) x^{2}+2 x^{5}+8 \ln \relax (5)^{5}}{\left (\ln \relax (5)-x \right )^{4}}\)	\(50\)
gosper	\(\frac {8 \ln \relax (5)^{5}-30 x \ln \relax (5)^{4}+40 x^{2} \ln \relax (5)^{3}-20 x^{3} \ln \relax (5)^{2}+2 x^{5}-18 x^{2}}{\ln \relax (5)^{4}-4 \ln \relax (5)^{3} x +6 x^{2} \ln \relax (5)^{2}-4 x^{3} \ln \relax (5)+x^{4}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(5)^5-10*x*ln(5)^4+20*x^2*ln(5)^3-20*x^3*ln(5)^2+(10*x^4-36*x)*ln(5)-2*x^5-36*x^2)/(ln(5)^5-5*x*ln(5)
^4+10*x^2*ln(5)^3-10*x^3*ln(5)^2+5*x^4*ln(5)-x^5),x,method=_RETURNVERBOSE)

[Out]

2*x-36*ln(5)/(-ln(5)+x)^3-18*ln(5)^2/(-ln(5)+x)^4-18/(-ln(5)+x)^2

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maxima [B]  time = 0.35, size = 42, normalized size = 1.91 \begin {gather*} 2 \, x - \frac {18 \, x^{2}}{x^{4} - 4 \, x^{3} \log \relax (5) + 6 \, x^{2} \log \relax (5)^{2} - 4 \, x \log \relax (5)^{3} + \log \relax (5)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(5)^5-10*x*log(5)^4+20*x^2*log(5)^3-20*x^3*log(5)^2+(10*x^4-36*x)*log(5)-2*x^5-36*x^2)/(log(5)
^5-5*x*log(5)^4+10*x^2*log(5)^3-10*x^3*log(5)^2+5*x^4*log(5)-x^5),x, algorithm="maxima")

[Out]

2*x - 18*x^2/(x^4 - 4*x^3*log(5) + 6*x^2*log(5)^2 - 4*x*log(5)^3 + log(5)^4)

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mupad [B]  time = 0.17, size = 17, normalized size = 0.77 \begin {gather*} 2\,x-\frac {18\,x^2}{{\left (x-\ln \relax (5)\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*x^3*log(5)^2 - 20*x^2*log(5)^3 + log(5)*(36*x - 10*x^4) + 10*x*log(5)^4 - 2*log(5)^5 + 36*x^2 + 2*x^5
)/(10*x^2*log(5)^3 - 10*x^3*log(5)^2 - 5*x*log(5)^4 + 5*x^4*log(5) + log(5)^5 - x^5),x)

[Out]

2*x - (18*x^2)/(x - log(5))^4

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sympy [B]  time = 0.29, size = 42, normalized size = 1.91 \begin {gather*} - \frac {18 x^{2}}{x^{4} - 4 x^{3} \log {\relax (5 )} + 6 x^{2} \log {\relax (5 )}^{2} - 4 x \log {\relax (5 )}^{3} + \log {\relax (5 )}^{4}} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(5)**5-10*x*ln(5)**4+20*x**2*ln(5)**3-20*x**3*ln(5)**2+(10*x**4-36*x)*ln(5)-2*x**5-36*x**2)/(ln
(5)**5-5*x*ln(5)**4+10*x**2*ln(5)**3-10*x**3*ln(5)**2+5*x**4*ln(5)-x**5),x)

[Out]

-18*x**2/(x**4 - 4*x**3*log(5) + 6*x**2*log(5)**2 - 4*x*log(5)**3 + log(5)**4) + 2*x

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