3.20.14 \(\int \frac {(37500-5625 x-4375 x^2) \log (6)}{-25000 x^5+25000 x^6-10000 x^7+2000 x^8-200 x^9+8 x^{10}} \, dx\)

Optimal. Leaf size=24 \[ 2+\frac {(3+x) \log (6)}{8 \left (-x+\frac {x^2}{5}\right )^4} \]

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Rubi [B]  time = 0.11, antiderivative size = 86, normalized size of antiderivative = 3.58, number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 2074} \begin {gather*} \frac {3 \log (6)}{8 x^4}+\frac {17 \log (6)}{40 x^3}+\frac {\log (6)}{4 x^2}+\frac {11 \log (6)}{100 x}+\frac {11 \log (6)}{100 (5-x)}+\frac {3 \log (6)}{10 (5-x)^2}+\frac {27 \log (6)}{40 (5-x)^3}+\frac {\log (6)}{(5-x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((37500 - 5625*x - 4375*x^2)*Log[6])/(-25000*x^5 + 25000*x^6 - 10000*x^7 + 2000*x^8 - 200*x^9 + 8*x^10),x]

[Out]

Log[6]/(5 - x)^4 + (27*Log[6])/(40*(5 - x)^3) + (3*Log[6])/(10*(5 - x)^2) + (11*Log[6])/(100*(5 - x)) + (3*Log
[6])/(8*x^4) + (17*Log[6])/(40*x^3) + Log[6]/(4*x^2) + (11*Log[6])/(100*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\log (6) \int \frac {37500-5625 x-4375 x^2}{-25000 x^5+25000 x^6-10000 x^7+2000 x^8-200 x^9+8 x^{10}} \, dx\\ &=\log (6) \int \left (-\frac {4}{(-5+x)^5}+\frac {81}{40 (-5+x)^4}-\frac {3}{5 (-5+x)^3}+\frac {11}{100 (-5+x)^2}-\frac {3}{2 x^5}-\frac {51}{40 x^4}-\frac {1}{2 x^3}-\frac {11}{100 x^2}\right ) \, dx\\ &=\frac {\log (6)}{(5-x)^4}+\frac {27 \log (6)}{40 (5-x)^3}+\frac {3 \log (6)}{10 (5-x)^2}+\frac {11 \log (6)}{100 (5-x)}+\frac {3 \log (6)}{8 x^4}+\frac {17 \log (6)}{40 x^3}+\frac {\log (6)}{4 x^2}+\frac {11 \log (6)}{100 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.71 \begin {gather*} \frac {625 (3+x) \log (6)}{8 (-5+x)^4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((37500 - 5625*x - 4375*x^2)*Log[6])/(-25000*x^5 + 25000*x^6 - 10000*x^7 + 2000*x^8 - 200*x^9 + 8*x^
10),x]

[Out]

(625*(3 + x)*Log[6])/(8*(-5 + x)^4*x^4)

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fricas [A]  time = 0.86, size = 33, normalized size = 1.38 \begin {gather*} \frac {625 \, {\left (x + 3\right )} \log \relax (6)}{8 \, {\left (x^{8} - 20 \, x^{7} + 150 \, x^{6} - 500 \, x^{5} + 625 \, x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4375*x^2-5625*x+37500)*log(6)/(8*x^10-200*x^9+2000*x^8-10000*x^7+25000*x^6-25000*x^5),x, algorithm
="fricas")

[Out]

625/8*(x + 3)*log(6)/(x^8 - 20*x^7 + 150*x^6 - 500*x^5 + 625*x^4)

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giac [A]  time = 0.34, size = 16, normalized size = 0.67 \begin {gather*} \frac {625 \, {\left (x + 3\right )} \log \relax (6)}{8 \, {\left (x^{2} - 5 \, x\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4375*x^2-5625*x+37500)*log(6)/(8*x^10-200*x^9+2000*x^8-10000*x^7+25000*x^6-25000*x^5),x, algorithm
="giac")

[Out]

625/8*(x + 3)*log(6)/(x^2 - 5*x)^4

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maple [A]  time = 0.05, size = 20, normalized size = 0.83

method	result	size
norman	\(\frac {\frac {625 x \ln \relax (6)}{8}+\frac {1875 \ln \relax (6)}{8}}{x^{4} \left (x -5\right )^{4}}\)	\(20\)
gosper	\(\frac {625 \left (3+x \right ) \ln \relax (6)}{8 x^{4} \left (x^{4}-20 x^{3}+150 x^{2}-500 x +625\right )}\)	\(31\)
risch	\(\frac {\left (\ln \relax (2)+\ln \relax (3)\right ) \left (\frac {625 x}{8}+\frac {1875}{8}\right )}{x^{4} \left (x^{4}-20 x^{3}+150 x^{2}-500 x +625\right )}\)	\(35\)
default	\(\frac {625 \ln \relax (6) \left (\frac {3}{625 x^{4}}+\frac {17}{3125 x^{3}}+\frac {2}{625 x^{2}}+\frac {22}{15625 x}+\frac {8}{625 \left (x -5\right )^{4}}-\frac {27}{3125 \left (x -5\right )^{3}}+\frac {12}{3125 \left (x -5\right )^{2}}-\frac {22}{15625 \left (x -5\right )}\right )}{8}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4375*x^2-5625*x+37500)*ln(6)/(8*x^10-200*x^9+2000*x^8-10000*x^7+25000*x^6-25000*x^5),x,method=_RETURNVER
BOSE)

[Out]

(625/8*x*ln(6)+1875/8*ln(6))/x^4/(x-5)^4

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maxima [A]  time = 0.45, size = 33, normalized size = 1.38 \begin {gather*} \frac {625 \, {\left (x + 3\right )} \log \relax (6)}{8 \, {\left (x^{8} - 20 \, x^{7} + 150 \, x^{6} - 500 \, x^{5} + 625 \, x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4375*x^2-5625*x+37500)*log(6)/(8*x^10-200*x^9+2000*x^8-10000*x^7+25000*x^6-25000*x^5),x, algorithm
="maxima")

[Out]

625/8*(x + 3)*log(6)/(x^8 - 20*x^7 + 150*x^6 - 500*x^5 + 625*x^4)

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mupad [B]  time = 1.14, size = 39, normalized size = 1.62 \begin {gather*} \frac {1875\,\ln \relax (6)+625\,x\,\ln \relax (6)}{8\,x^8-160\,x^7+1200\,x^6-4000\,x^5+5000\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(6)*(5625*x + 4375*x^2 - 37500))/(25000*x^5 - 25000*x^6 + 10000*x^7 - 2000*x^8 + 200*x^9 - 8*x^10),x)

[Out]

(1875*log(6) + 625*x*log(6))/(5000*x^4 - 4000*x^5 + 1200*x^6 - 160*x^7 + 8*x^8)

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sympy [B]  time = 0.29, size = 39, normalized size = 1.62 \begin {gather*} - \frac {- 625 x \log {\relax (6 )} - 1875 \log {\relax (6 )}}{8 x^{8} - 160 x^{7} + 1200 x^{6} - 4000 x^{5} + 5000 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4375*x**2-5625*x+37500)*ln(6)/(8*x**10-200*x**9+2000*x**8-10000*x**7+25000*x**6-25000*x**5),x)

[Out]

-(-625*x*log(6) - 1875*log(6))/(8*x**8 - 160*x**7 + 1200*x**6 - 4000*x**5 + 5000*x**4)

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