\(\int \frac {64 x}{1344+368 x^2+25 x^4+(2944+400 x^2) \log (5)+1600 \log ^2(5)} \, dx\) [7840]

Optimal result

Integrand size = 33, antiderivative size = 22 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (-\frac {25}{8}+\frac {2}{4+\frac {x^2}{2}+4 \log (5)}\right ) \]

[Out]

ln(2/(1/2*x^2+4*ln(5)+4)-25/8)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {12, 2014, 1121, 630, 31} \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (25 x^2+8 (21+25 \log (5))\right )-\log \left (x^2+8 (1+\log (5))\right ) \]

[In]

Int[(64*x)/(1344 + 368*x^2 + 25*x^4 + (2944 + 400*x^2)*Log[5] + 1600*Log[5]^2),x]

[Out]

-Log[x^2 + 8*(1 + Log[5])] + Log[25*x^2 + 8*(21 + 25*Log[5])]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 630

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 2014

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] &&
TrinomialQ[u, x] &&  !TrinomialMatchQ[u, x]

Rubi steps \begin{align*} \text {integral}& = 64 \int \frac {x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx \\ & = 64 \int \frac {x}{25 x^4+64 (1+\log (5)) (21+25 \log (5))+16 x^2 (23+25 \log (5))} \, dx \\ & = 32 \text {Subst}\left (\int \frac {1}{25 x^2+64 (1+\log (5)) (21+25 \log (5))+16 x (23+25 \log (5))} \, dx,x,x^2\right ) \\ & = -\left (25 \text {Subst}\left (\int \frac {1}{25 x+200 (1+\log (5))} \, dx,x,x^2\right )\right )+25 \text {Subst}\left (\int \frac {1}{25 x+8 (21+25 \log (5))} \, dx,x,x^2\right ) \\ & = -\log \left (x^2+8 (1+\log (5))\right )+\log \left (25 x^2+8 (21+25 \log (5))\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=64 \left (-\frac {1}{64} \log \left (8+x^2+8 \log (5)\right )+\frac {1}{64} \log \left (168+25 x^2+200 \log (5)\right )\right ) \]

[In]

Integrate[(64*x)/(1344 + 368*x^2 + 25*x^4 + (2944 + 400*x^2)*Log[5] + 1600*Log[5]^2),x]

[Out]

64*(-1/64*Log[8 + x^2 + 8*Log[5]] + Log[168 + 25*x^2 + 200*Log[5]]/64)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09

method	result	size
parallelrisch	\(-\ln \left (x^{2}+8 \ln \left (5\right )+8\right )+\ln \left (x^{2}+8 \ln \left (5\right )+\frac {168}{25}\right )\)	\(24\)
default	\(\ln \left (25 x^{2}+200 \ln \left (5\right )+168\right )-\ln \left (x^{2}+8 \ln \left (5\right )+8\right )\)	\(26\)
norman	\(\ln \left (25 x^{2}+200 \ln \left (5\right )+168\right )-\ln \left (x^{2}+8 \ln \left (5\right )+8\right )\)	\(26\)
risch	\(-\ln \left (-x^{2}-8 \ln \left (5\right )-8\right )+\ln \left (25 x^{2}+200 \ln \left (5\right )+168\right )\)	\(28\)

[In]

int(64*x/(1600*ln(5)^2+(400*x^2+2944)*ln(5)+25*x^4+368*x^2+1344),x,method=_RETURNVERBOSE)

[Out]

-ln(x^2+8*ln(5)+8)+ln(x^2+8*ln(5)+168/25)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (25 \, x^{2} + 200 \, \log \left (5\right ) + 168\right ) - \log \left (x^{2} + 8 \, \log \left (5\right ) + 8\right ) \]

[In]

integrate(64*x/(1600*log(5)^2+(400*x^2+2944)*log(5)+25*x^4+368*x^2+1344),x, algorithm="fricas")

[Out]

log(25*x^2 + 200*log(5) + 168) - log(x^2 + 8*log(5) + 8)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log {\left (x^{2} + \frac {168}{25} + 8 \log {\left (5 \right )} \right )} - \log {\left (x^{2} + 8 + 8 \log {\left (5 \right )} \right )} \]

[In]

integrate(64*x/(1600*ln(5)**2+(400*x**2+2944)*ln(5)+25*x**4+368*x**2+1344),x)

[Out]

log(x**2 + 168/25 + 8*log(5)) - log(x**2 + 8 + 8*log(5))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (25 \, x^{2} + 200 \, \log \left (5\right ) + 168\right ) - \log \left (x^{2} + 8 \, \log \left (5\right ) + 8\right ) \]

[In]

integrate(64*x/(1600*log(5)^2+(400*x^2+2944)*log(5)+25*x^4+368*x^2+1344),x, algorithm="maxima")

[Out]

log(25*x^2 + 200*log(5) + 168) - log(x^2 + 8*log(5) + 8)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (25 \, x^{2} + 200 \, \log \left (5\right ) + 168\right ) - \log \left (x^{2} + 8 \, \log \left (5\right ) + 8\right ) \]

[In]

integrate(64*x/(1600*log(5)^2+(400*x^2+2944)*log(5)+25*x^4+368*x^2+1344),x, algorithm="giac")

[Out]

log(25*x^2 + 200*log(5) + 168) - log(x^2 + 8*log(5) + 8)

Mupad [B] (verification not implemented)

Time = 13.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=2\,\mathrm {atanh}\left (\frac {320000\,x^2}{58880000\,\ln \left (5\right )+2\,x^2\,\left (2000000\,\ln \left (5\right )+1840000\right )[phone]\,{\ln \left (5\right )}^2+26880000}\right ) \]

[In]

int((64*x)/(log(5)*(400*x^2 + 2944) + 1600*log(5)^2 + 368*x^2 + 25*x^4 + 1344),x)

[Out]

2*atanh((320000*x^2)/(58880000*log(5) + 2*x^2*(2000000*log(5) + 1840000) + 32000000*log(5)^2 + 26880000))