3.44.98 \(\int \frac {-180000+640 x+191250 x^2-680 x^3+(-144000 x-71488 x^2-152744 x^3+544 x^4) \log (\frac {16+8 x+17 x^2}{16 x})+(-800+850 x^2+(-640 x-320 x^2-680 x^3) \log (\frac {16+8 x+17 x^2}{16 x})) \log (\log (\frac {16+8 x+17 x^2}{16 x}))}{(16 x+8 x^2+17 x^3) \log (\frac {16+8 x+17 x^2}{16 x})} \, dx\)

Optimal. Leaf size=27 \[ \left (x-5 \left (-225+x-\log \left (\log \left (x+\frac {(4+x)^2}{16 x}\right )\right )\right )\right )^2 \]

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Rubi [A]  time = 0.66, antiderivative size = 23, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, integrand size = 150, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {1594, 6688, 12, 6686} \begin {gather*} \left (-4 x+5 \log \left (\log \left (\frac {17 x}{16}+\frac {1}{x}+\frac {1}{2}\right )\right )+1125\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-180000 + 640*x + 191250*x^2 - 680*x^3 + (-144000*x - 71488*x^2 - 152744*x^3 + 544*x^4)*Log[(16 + 8*x + 1
7*x^2)/(16*x)] + (-800 + 850*x^2 + (-640*x - 320*x^2 - 680*x^3)*Log[(16 + 8*x + 17*x^2)/(16*x)])*Log[Log[(16 +
8*x + 17*x^2)/(16*x)]])/((16*x + 8*x^2 + 17*x^3)*Log[(16 + 8*x + 17*x^2)/(16*x)]),x]

[Out]

(1125 - 4*x + 5*Log[Log[1/2 + x^(-1) + (17*x)/16]])^2

Rule 12

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

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 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-180000+640 x+191250 x^2-680 x^3+\left (-144000 x-71488 x^2-152744 x^3+544 x^4\right ) \log \left (\frac {16+8 x+17 x^2}{16 x}\right )+\left (-800+850 x^2+\left (-640 x-320 x^2-680 x^3\right ) \log \left (\frac {16+8 x+17 x^2}{16 x}\right )\right ) \log \left (\log \left (\frac {16+8 x+17 x^2}{16 x}\right )\right )}{x \left (16+8 x+17 x^2\right ) \log \left (\frac {16+8 x+17 x^2}{16 x}\right )} \, dx\\ &=\int \frac {2 \left (80-85 x^2+4 x \left (16+8 x+17 x^2\right ) \log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )\right ) \left (-1125+4 x-5 \log \left (\log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )\right )\right )}{x \left (16+8 x+17 x^2\right ) \log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )} \, dx\\ &=2 \int \frac {\left (80-85 x^2+4 x \left (16+8 x+17 x^2\right ) \log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )\right ) \left (-1125+4 x-5 \log \left (\log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )\right )\right )}{x \left (16+8 x+17 x^2\right ) \log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )} \, dx\\ &=\left (1125-4 x+5 \log \left (\log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 64, normalized size = 2.37 \begin {gather*} 2 \left (-4500 x+8 x^2+5625 \log \left (\log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )\right )-20 x \log \left (\log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )\right )+\frac {25}{2} \log ^2\left (\log \left (\frac {1}{2}+\frac {1}{x}+\frac {17 x}{16}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-180000 + 640*x + 191250*x^2 - 680*x^3 + (-144000*x - 71488*x^2 - 152744*x^3 + 544*x^4)*Log[(16 + 8
*x + 17*x^2)/(16*x)] + (-800 + 850*x^2 + (-640*x - 320*x^2 - 680*x^3)*Log[(16 + 8*x + 17*x^2)/(16*x)])*Log[Log
[(16 + 8*x + 17*x^2)/(16*x)]])/((16*x + 8*x^2 + 17*x^3)*Log[(16 + 8*x + 17*x^2)/(16*x)]),x]

[Out]

2*(-4500*x + 8*x^2 + 5625*Log[Log[1/2 + x^(-1) + (17*x)/16]] - 20*x*Log[Log[1/2 + x^(-1) + (17*x)/16]] + (25*L
og[Log[1/2 + x^(-1) + (17*x)/16]]^2)/2)

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fricas [B]  time = 0.68, size = 54, normalized size = 2.00 \begin {gather*} 16 \, x^{2} - 10 \, {\left (4 \, x - 1125\right )} \log \left (\log \left (\frac {17 \, x^{2} + 8 \, x + 16}{16 \, x}\right )\right ) + 25 \, \log \left (\log \left (\frac {17 \, x^{2} + 8 \, x + 16}{16 \, x}\right )\right )^{2} - 9000 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-680*x^3-320*x^2-640*x)*log(1/16*(17*x^2+8*x+16)/x)+850*x^2-800)*log(log(1/16*(17*x^2+8*x+16)/x))
+(544*x^4-152744*x^3-71488*x^2-144000*x)*log(1/16*(17*x^2+8*x+16)/x)-680*x^3+191250*x^2+640*x-180000)/(17*x^3+
8*x^2+16*x)/log(1/16*(17*x^2+8*x+16)/x),x, algorithm="fricas")

[Out]

16*x^2 - 10*(4*x - 1125)*log(log(1/16*(17*x^2 + 8*x + 16)/x)) + 25*log(log(1/16*(17*x^2 + 8*x + 16)/x))^2 - 90
00*x

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giac [B]  time = 0.34, size = 97, normalized size = 3.59 \begin {gather*} 16 \, x^{2} - 25 \, \log \left (\log \left (17 \, x^{2} + 8 \, x + 16\right ) - \log \left (16 \, x\right )\right )^{2} - 10 \, {\left (4 \, x - 5 \, \log \left (\log \left (17 \, x^{2} + 8 \, x + 16\right ) - \log \left (16 \, x\right )\right )\right )} \log \left (\log \left (\frac {17 \, x^{2} + 8 \, x + 16}{16 \, x}\right )\right ) - 9000 \, x + 11250 \, \log \left (\log \left (17 \, x^{2} + 8 \, x + 16\right ) - \log \left (16 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-680*x^3-320*x^2-640*x)*log(1/16*(17*x^2+8*x+16)/x)+850*x^2-800)*log(log(1/16*(17*x^2+8*x+16)/x))
+(544*x^4-152744*x^3-71488*x^2-144000*x)*log(1/16*(17*x^2+8*x+16)/x)-680*x^3+191250*x^2+640*x-180000)/(17*x^3+
8*x^2+16*x)/log(1/16*(17*x^2+8*x+16)/x),x, algorithm="giac")

[Out]

16*x^2 - 25*log(log(17*x^2 + 8*x + 16) - log(16*x))^2 - 10*(4*x - 5*log(log(17*x^2 + 8*x + 16) - log(16*x)))*l
og(log(1/16*(17*x^2 + 8*x + 16)/x)) - 9000*x + 11250*log(log(17*x^2 + 8*x + 16) - log(16*x))

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-680 x^{3}-320 x^{2}-640 x \right ) \ln \left (\frac {17 x^{2}+8 x +16}{16 x}\right )+850 x^{2}-800\right ) \ln \left (\ln \left (\frac {17 x^{2}+8 x +16}{16 x}\right )\right )+\left (544 x^{4}-152744 x^{3}-71488 x^{2}-144000 x \right ) \ln \left (\frac {17 x^{2}+8 x +16}{16 x}\right )-680 x^{3}+191250 x^{2}+640 x -180000}{\left (17 x^{3}+8 x^{2}+16 x \right ) \ln \left (\frac {17 x^{2}+8 x +16}{16 x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-680*x^3-320*x^2-640*x)*ln(1/16*(17*x^2+8*x+16)/x)+850*x^2-800)*ln(ln(1/16*(17*x^2+8*x+16)/x))+(544*x^4
-152744*x^3-71488*x^2-144000*x)*ln(1/16*(17*x^2+8*x+16)/x)-680*x^3+191250*x^2+640*x-180000)/(17*x^3+8*x^2+16*x
)/ln(1/16*(17*x^2+8*x+16)/x),x)

[Out]

int((((-680*x^3-320*x^2-640*x)*ln(1/16*(17*x^2+8*x+16)/x)+850*x^2-800)*ln(ln(1/16*(17*x^2+8*x+16)/x))+(544*x^4
-152744*x^3-71488*x^2-144000*x)*ln(1/16*(17*x^2+8*x+16)/x)-680*x^3+191250*x^2+640*x-180000)/(17*x^3+8*x^2+16*x
)/ln(1/16*(17*x^2+8*x+16)/x),x)

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maxima [B]  time = 0.49, size = 62, normalized size = 2.30 \begin {gather*} 16 \, x^{2} - 10 \, {\left (4 \, x - 1125\right )} \log \left (-4 \, \log \relax (2) + \log \left (17 \, x^{2} + 8 \, x + 16\right ) - \log \relax (x)\right ) + 25 \, \log \left (-4 \, \log \relax (2) + \log \left (17 \, x^{2} + 8 \, x + 16\right ) - \log \relax (x)\right )^{2} - 9000 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-680*x^3-320*x^2-640*x)*log(1/16*(17*x^2+8*x+16)/x)+850*x^2-800)*log(log(1/16*(17*x^2+8*x+16)/x))
+(544*x^4-152744*x^3-71488*x^2-144000*x)*log(1/16*(17*x^2+8*x+16)/x)-680*x^3+191250*x^2+640*x-180000)/(17*x^3+
8*x^2+16*x)/log(1/16*(17*x^2+8*x+16)/x),x, algorithm="maxima")

[Out]

16*x^2 - 10*(4*x - 1125)*log(-4*log(2) + log(17*x^2 + 8*x + 16) - log(x)) + 25*log(-4*log(2) + log(17*x^2 + 8*
x + 16) - log(x))^2 - 9000*x

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mupad [B]  time = 3.63, size = 66, normalized size = 2.44 \begin {gather*} 16\,x^2-40\,x\,\ln \left (\ln \left (\frac {\frac {17\,x^2}{16}+\frac {x}{2}+1}{x}\right )\right )-9000\,x+25\,{\ln \left (\ln \left (\frac {\frac {17\,x^2}{16}+\frac {x}{2}+1}{x}\right )\right )}^2+11250\,\ln \left (\ln \left (\frac {\frac {17\,x^2}{16}+\frac {x}{2}+1}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((x/2 + (17*x^2)/16 + 1)/x)*(144000*x + 71488*x^2 + 152744*x^3 - 544*x^4) - 640*x + log(log((x/2 + (1
7*x^2)/16 + 1)/x))*(log((x/2 + (17*x^2)/16 + 1)/x)*(640*x + 320*x^2 + 680*x^3) - 850*x^2 + 800) - 191250*x^2 +
680*x^3 + 180000)/(log((x/2 + (17*x^2)/16 + 1)/x)*(16*x + 8*x^2 + 17*x^3)),x)

[Out]

11250*log(log((x/2 + (17*x^2)/16 + 1)/x)) - 9000*x - 40*x*log(log((x/2 + (17*x^2)/16 + 1)/x)) + 16*x^2 + 25*lo
g(log((x/2 + (17*x^2)/16 + 1)/x))^2

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sympy [B]  time = 0.58, size = 66, normalized size = 2.44 \begin {gather*} 16 x^{2} - 40 x \log {\left (\log {\left (\frac {\frac {17 x^{2}}{16} + \frac {x}{2} + 1}{x} \right )} \right )} - 9000 x + 25 \log {\left (\log {\left (\frac {\frac {17 x^{2}}{16} + \frac {x}{2} + 1}{x} \right )} \right )}^{2} + 11250 \log {\left (\log {\left (\frac {\frac {17 x^{2}}{16} + \frac {x}{2} + 1}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-680*x**3-320*x**2-640*x)*ln(1/16*(17*x**2+8*x+16)/x)+850*x**2-800)*ln(ln(1/16*(17*x**2+8*x+16)/x
))+(544*x**4-152744*x**3-71488*x**2-144000*x)*ln(1/16*(17*x**2+8*x+16)/x)-680*x**3+191250*x**2+640*x-180000)/(
17*x**3+8*x**2+16*x)/ln(1/16*(17*x**2+8*x+16)/x),x)

[Out]

16*x**2 - 40*x*log(log((17*x**2/16 + x/2 + 1)/x)) - 9000*x + 25*log(log((17*x**2/16 + x/2 + 1)/x))**2 + 11250*
log(log((17*x**2/16 + x/2 + 1)/x))

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