3.39.64 \(\int \frac {625+(13500+7500 x) \log (2)+(97200+108000 x+37200 x^2) \log ^2(2)+(233280+388800 x+216000 x^2+68800 x^3) \log ^3(2)}{125 x^2+(2700 x^2+1500 x^3) \log (2)+(19440 x^2+21600 x^3+6000 x^4) \log ^2(2)+(46656 x^2+77760 x^3+43200 x^4+8000 x^5) \log ^3(2)} \, dx\) [3864]

Optimal. Leaf size=31 \[ -\frac {5}{x}+\frac {x^2}{\left (x+\frac {9}{5+\frac {5}{4 x \log (2)}}\right )^2} \]

[Out]

x^2/(9/(5/4/x/ln(2)+5)+x)^2-5/x

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Rubi [A]
time = 0.10, antiderivative size = 42, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 1, integrand size = 115, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2099} \begin {gather*} -\frac {5}{x}+\frac {1296 \log ^2(2)}{(20 x \log (2)+5+36 \log (2))^2}-\frac {72 \log (2)}{20 x \log (2)+5+36 \log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(625 + (13500 + 7500*x)*Log[2] + (97200 + 108000*x + 37200*x^2)*Log[2]^2 + (233280 + 388800*x + 216000*x^2
+ 68800*x^3)*Log[2]^3)/(125*x^2 + (2700*x^2 + 1500*x^3)*Log[2] + (19440*x^2 + 21600*x^3 + 6000*x^4)*Log[2]^2
+ (46656*x^2 + 77760*x^3 + 43200*x^4 + 8000*x^5)*Log[2]^3),x]

[Out]

-5/x + (1296*Log[2]^2)/(5 + 36*Log[2] + 20*x*Log[2])^2 - (72*Log[2])/(5 + 36*Log[2] + 20*x*Log[2])

Rule 2099

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 \left (\frac {5}{x^2}-\frac {51840 \log ^3(2)}{(5+36 \log (2)+20 x \log (2))^3}+\frac {1440 \log ^2(2)}{(5+36 \log (2)+20 x \log (2))^2}\right ) \, dx\\ &=-\frac {5}{x}+\frac {1296 \log ^2(2)}{(5+36 \log (2)+20 x \log (2))^2}-\frac {72 \log (2)}{5+36 \log (2)+20 x \log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 38, normalized size = 1.23 \begin {gather*} 5 \left (-\frac {1}{x}-\frac {72 \log (2) (5+18 \log (2)+20 x \log (2))}{5 (5+36 \log (2)+20 x \log (2))^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(625 + (13500 + 7500*x)*Log[2] + (97200 + 108000*x + 37200*x^2)*Log[2]^2 + (233280 + 388800*x + 2160
00*x^2 + 68800*x^3)*Log[2]^3)/(125*x^2 + (2700*x^2 + 1500*x^3)*Log[2] + (19440*x^2 + 21600*x^3 + 6000*x^4)*Log
[2]^2 + (46656*x^2 + 77760*x^3 + 43200*x^4 + 8000*x^5)*Log[2]^3),x]

[Out]

5*(-x^(-1) - (72*Log[2]*(5 + 18*Log[2] + 20*x*Log[2]))/(5*(5 + 36*Log[2] + 20*x*Log[2])^2))

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Maple [A]
time = 0.40, size = 43, normalized size = 1.39

method	result	size
default	\(-\frac {72 \ln \left (2\right )}{20 x \ln \left (2\right )+36 \ln \left (2\right )+5}+\frac {1296 \ln \left (2\right )^{2}}{\left (20 x \ln \left (2\right )+36 \ln \left (2\right )+5\right )^{2}}-\frac {5}{x}\)	\(43\)
norman	\(\frac {-3440 x^{2} \ln \left (2\right )^{2}-\frac {\left (3398400 \ln \left (2\right )^{4}+544000 \ln \left (2\right )^{3}\right ) x}{400 \ln \left (2\right )^{2}}-6480 \ln \left (2\right )^{2}-1800 \ln \left (2\right )-125}{x \left (20 x \ln \left (2\right )+36 \ln \left (2\right )+5\right )^{2}}\)	\(59\)
gosper	\(-\frac {3440 x^{2} \ln \left (2\right )^{2}+8496 x \ln \left (2\right )^{2}+6480 \ln \left (2\right )^{2}+1360 x \ln \left (2\right )+1800 \ln \left (2\right )+125}{x \left (400 x^{2} \ln \left (2\right )^{2}+1440 x \ln \left (2\right )^{2}+1296 \ln \left (2\right )^{2}+200 x \ln \left (2\right )+360 \ln \left (2\right )+25\right )}\)	\(74\)
risch	\(\frac {-3440 x^{2} \ln \left (2\right )^{2}+400 \left (-\frac {531 \ln \left (2\right )^{2}}{25}-\frac {17 \ln \left (2\right )}{5}\right ) x -6480 \ln \left (2\right )^{2}-1800 \ln \left (2\right )-125}{x \left (400 x^{2} \ln \left (2\right )^{2}+1440 x \ln \left (2\right )^{2}+1296 \ln \left (2\right )^{2}+200 x \ln \left (2\right )+360 \ln \left (2\right )+25\right )}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((68800*x^3+216000*x^2+388800*x+233280)*ln(2)^3+(37200*x^2+108000*x+97200)*ln(2)^2+(7500*x+13500)*ln(2)+62
5)/((8000*x^5+43200*x^4+77760*x^3+46656*x^2)*ln(2)^3+(6000*x^4+21600*x^3+19440*x^2)*ln(2)^2+(1500*x^3+2700*x^2
)*ln(2)+125*x^2),x,method=_RETURNVERBOSE)

[Out]

-72*ln(2)/(20*x*ln(2)+36*ln(2)+5)+1296*ln(2)^2/(20*x*ln(2)+36*ln(2)+5)^2-5/x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (31) = 62\).
time = 0.30, size = 79, normalized size = 2.55 \begin {gather*} -\frac {3440 \, x^{2} \log \left (2\right )^{2} + 16 \, {\left (531 \, \log \left (2\right )^{2} + 85 \, \log \left (2\right )\right )} x + 6480 \, \log \left (2\right )^{2} + 1800 \, \log \left (2\right ) + 125}{400 \, x^{3} \log \left (2\right )^{2} + 40 \, {\left (36 \, \log \left (2\right )^{2} + 5 \, \log \left (2\right )\right )} x^{2} + {\left (1296 \, \log \left (2\right )^{2} + 360 \, \log \left (2\right ) + 25\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((68800*x^3+216000*x^2+388800*x+233280)*log(2)^3+(37200*x^2+108000*x+97200)*log(2)^2+(7500*x+13500)*
log(2)+625)/((8000*x^5+43200*x^4+77760*x^3+46656*x^2)*log(2)^3+(6000*x^4+21600*x^3+19440*x^2)*log(2)^2+(1500*x
^3+2700*x^2)*log(2)+125*x^2),x, algorithm="maxima")

[Out]

-(3440*x^2*log(2)^2 + 16*(531*log(2)^2 + 85*log(2))*x + 6480*log(2)^2 + 1800*log(2) + 125)/(400*x^3*log(2)^2 +
40*(36*log(2)^2 + 5*log(2))*x^2 + (1296*log(2)^2 + 360*log(2) + 25)*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (31) = 62\).
time = 0.36, size = 68, normalized size = 2.19 \begin {gather*} -\frac {16 \, {\left (215 \, x^{2} + 531 \, x + 405\right )} \log \left (2\right )^{2} + 40 \, {\left (34 \, x + 45\right )} \log \left (2\right ) + 125}{16 \, {\left (25 \, x^{3} + 90 \, x^{2} + 81 \, x\right )} \log \left (2\right )^{2} + 40 \, {\left (5 \, x^{2} + 9 \, x\right )} \log \left (2\right ) + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((68800*x^3+216000*x^2+388800*x+233280)*log(2)^3+(37200*x^2+108000*x+97200)*log(2)^2+(7500*x+13500)*
log(2)+625)/((8000*x^5+43200*x^4+77760*x^3+46656*x^2)*log(2)^3+(6000*x^4+21600*x^3+19440*x^2)*log(2)^2+(1500*x
^3+2700*x^2)*log(2)+125*x^2),x, algorithm="fricas")

[Out]

-(16*(215*x^2 + 531*x + 405)*log(2)^2 + 40*(34*x + 45)*log(2) + 125)/(16*(25*x^3 + 90*x^2 + 81*x)*log(2)^2 + 4
0*(5*x^2 + 9*x)*log(2) + 25*x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (20) = 40\).
time = 0.94, size = 78, normalized size = 2.52 \begin {gather*} \frac {- 3440 x^{2} \log {\left (2 \right )}^{2} + x \left (- 8496 \log {\left (2 \right )}^{2} - 1360 \log {\left (2 \right )}\right ) - 6480 \log {\left (2 \right )}^{2} - 1800 \log {\left (2 \right )} - 125}{400 x^{3} \log {\left (2 \right )}^{2} + x^{2} \cdot \left (200 \log {\left (2 \right )} + 1440 \log {\left (2 \right )}^{2}\right ) + x \left (25 + 360 \log {\left (2 \right )} + 1296 \log {\left (2 \right )}^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((68800*x**3+216000*x**2+388800*x+233280)*ln(2)**3+(37200*x**2+108000*x+97200)*ln(2)**2+(7500*x+1350
0)*ln(2)+625)/((8000*x**5+43200*x**4+77760*x**3+46656*x**2)*ln(2)**3+(6000*x**4+21600*x**3+19440*x**2)*ln(2)**
2+(1500*x**3+2700*x**2)*ln(2)+125*x**2),x)

[Out]

(-3440*x**2*log(2)**2 + x*(-8496*log(2)**2 - 1360*log(2)) - 6480*log(2)**2 - 1800*log(2) - 125)/(400*x**3*log(
2)**2 + x**2*(200*log(2) + 1440*log(2)**2) + x*(25 + 360*log(2) + 1296*log(2)**2))

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Giac [A]
time = 0.41, size = 39, normalized size = 1.26 \begin {gather*} -\frac {72 \, {\left (20 \, x \log \left (2\right )^{2} + 18 \, \log \left (2\right )^{2} + 5 \, \log \left (2\right )\right )}}{{\left (20 \, x \log \left (2\right ) + 36 \, \log \left (2\right ) + 5\right )}^{2}} - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((68800*x^3+216000*x^2+388800*x+233280)*log(2)^3+(37200*x^2+108000*x+97200)*log(2)^2+(7500*x+13500)*
log(2)+625)/((8000*x^5+43200*x^4+77760*x^3+46656*x^2)*log(2)^3+(6000*x^4+21600*x^3+19440*x^2)*log(2)^2+(1500*x
^3+2700*x^2)*log(2)+125*x^2),x, algorithm="giac")

[Out]

-72*(20*x*log(2)^2 + 18*log(2)^2 + 5*log(2))/(20*x*log(2) + 36*log(2) + 5)^2 - 5/x

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Mupad [B]
time = 2.39, size = 42, normalized size = 1.35 \begin {gather*} \frac {1296\,{\ln \left (2\right )}^2}{{\left (36\,\ln \left (2\right )+20\,x\,\ln \left (2\right )+5\right )}^2}-\frac {5}{x}-\frac {72\,\ln \left (2\right )}{36\,\ln \left (2\right )+20\,x\,\ln \left (2\right )+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)*(7500*x + 13500) + log(2)^2*(108000*x + 37200*x^2 + 97200) + log(2)^3*(388800*x + 216000*x^2 + 688
00*x^3 + 233280) + 625)/(log(2)^2*(19440*x^2 + 21600*x^3 + 6000*x^4) + log(2)^3*(46656*x^2 + 77760*x^3 + 43200
*x^4 + 8000*x^5) + log(2)*(2700*x^2 + 1500*x^3) + 125*x^2),x)

[Out]

(1296*log(2)^2)/(36*log(2) + 20*x*log(2) + 5)^2 - 5/x - (72*log(2))/(36*log(2) + 20*x*log(2) + 5)

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