3.29.4 \(\int \frac {20000+7000 x-3100 x^2-1940 x^3-340 x^4-32 x^5-12 x^6+(-4 x^5-4 x^6) \log (8-2 x-x^2)}{-8 x^5+2 x^6+x^7} \, dx\)

Optimal. Leaf size=29 \[ \left (-4+\frac {5 (1+x)}{x}\right )^4-(3+\log ((2-x) (4+x)))^2 \]

________________________________________________________________________________________

Rubi [C]  time = 0.74, antiderivative size = 135, normalized size of antiderivative = 4.66, number of steps used = 38, number of rules used = 14, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1594, 6728, 36, 31, 72, 2528, 2524, 2418, 2394, 2393, 2391, 2390, 12, 2301} \begin {gather*} 2 \text {Li}_2\left (\frac {2-x}{6}\right )+2 \text {Li}_2\left (\frac {x+4}{6}\right )+\frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}-2 \log \left (-x^2-2 x+8\right ) \log (x-2)-2 \log (x+4) \log \left (-x^2-2 x+8\right )+\frac {20}{x}+\log ^2(x-2)+\log ^2(x+4)+2 \log \left (\frac {x+4}{6}\right ) \log (x-2)-6 \log (2-x)+2 \log \left (\frac {2-x}{6}\right ) \log (x+4)-6 \log (x+4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20000 + 7000*x - 3100*x^2 - 1940*x^3 - 340*x^4 - 32*x^5 - 12*x^6 + (-4*x^5 - 4*x^6)*Log[8 - 2*x - x^2])/(
-8*x^5 + 2*x^6 + x^7),x]

[Out]

625/x^4 + 500/x^3 + 150/x^2 + 20/x - 6*Log[2 - x] + Log[-2 + x]^2 + 2*Log[-2 + x]*Log[(4 + x)/6] - 6*Log[4 + x
] + 2*Log[(2 - x)/6]*Log[4 + x] + Log[4 + x]^2 - 2*Log[-2 + x]*Log[8 - 2*x - x^2] - 2*Log[4 + x]*Log[8 - 2*x -
x^2] + 2*PolyLog[2, (2 - x)/6] + 2*PolyLog[2, (4 + x)/6]

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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20000+7000 x-3100 x^2-1940 x^3-340 x^4-32 x^5-12 x^6+\left (-4 x^5-4 x^6\right ) \log \left (8-2 x-x^2\right )}{x^5 \left (-8+2 x+x^2\right )} \, dx\\ &=\int \left (-\frac {32}{(-2+x) (4+x)}+\frac {20000}{(-2+x) x^5 (4+x)}+\frac {7000}{(-2+x) x^4 (4+x)}-\frac {3100}{(-2+x) x^3 (4+x)}-\frac {1940}{(-2+x) x^2 (4+x)}-\frac {340}{(-2+x) x (4+x)}-\frac {12 x}{(-2+x) (4+x)}-\frac {4 (1+x) \log \left (8-2 x-x^2\right )}{(-2+x) (4+x)}\right ) \, dx\\ &=-\left (4 \int \frac {(1+x) \log \left (8-2 x-x^2\right )}{(-2+x) (4+x)} \, dx\right )-12 \int \frac {x}{(-2+x) (4+x)} \, dx-32 \int \frac {1}{(-2+x) (4+x)} \, dx-340 \int \frac {1}{(-2+x) x (4+x)} \, dx-1940 \int \frac {1}{(-2+x) x^2 (4+x)} \, dx-3100 \int \frac {1}{(-2+x) x^3 (4+x)} \, dx+7000 \int \frac {1}{(-2+x) x^4 (4+x)} \, dx+20000 \int \frac {1}{(-2+x) x^5 (4+x)} \, dx\\ &=-\left (4 \int \left (\frac {\log \left (8-2 x-x^2\right )}{2 (-2+x)}+\frac {\log \left (8-2 x-x^2\right )}{2 (4+x)}\right ) \, dx\right )-\frac {16}{3} \int \frac {1}{-2+x} \, dx+\frac {16}{3} \int \frac {1}{4+x} \, dx-12 \int \left (\frac {1}{3 (-2+x)}+\frac {2}{3 (4+x)}\right ) \, dx-340 \int \left (\frac {1}{12 (-2+x)}-\frac {1}{8 x}+\frac {1}{24 (4+x)}\right ) \, dx-1940 \int \left (\frac {1}{24 (-2+x)}-\frac {1}{8 x^2}-\frac {1}{32 x}-\frac {1}{96 (4+x)}\right ) \, dx-3100 \int \left (\frac {1}{48 (-2+x)}-\frac {1}{8 x^3}-\frac {1}{32 x^2}-\frac {3}{128 x}+\frac {1}{384 (4+x)}\right ) \, dx+7000 \int \left (\frac {1}{96 (-2+x)}-\frac {1}{8 x^4}-\frac {1}{32 x^3}-\frac {3}{128 x^2}-\frac {5}{512 x}-\frac {1}{1536 (4+x)}\right ) \, dx+20000 \int \left (\frac {1}{192 (-2+x)}-\frac {1}{8 x^5}-\frac {1}{32 x^4}-\frac {3}{128 x^3}-\frac {5}{512 x^2}-\frac {11}{2048 x}+\frac {1}{6144 (4+x)}\right ) \, dx\\ &=\frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}+\frac {20}{x}-6 \log (2-x)-6 \log (4+x)-2 \int \frac {\log \left (8-2 x-x^2\right )}{-2+x} \, dx-2 \int \frac {\log \left (8-2 x-x^2\right )}{4+x} \, dx\\ &=\frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}+\frac {20}{x}-6 \log (2-x)-6 \log (4+x)-2 \log (-2+x) \log \left (8-2 x-x^2\right )-2 \log (4+x) \log \left (8-2 x-x^2\right )+2 \int \frac {(-2-2 x) \log (-2+x)}{8-2 x-x^2} \, dx+2 \int \frac {(-2-2 x) \log (4+x)}{8-2 x-x^2} \, dx\\ &=\frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}+\frac {20}{x}-6 \log (2-x)-6 \log (4+x)-2 \log (-2+x) \log \left (8-2 x-x^2\right )-2 \log (4+x) \log \left (8-2 x-x^2\right )+2 \int \left (-\frac {2 \log (-2+x)}{-8-2 x}-\frac {2 \log (-2+x)}{4-2 x}\right ) \, dx+2 \int \left (-\frac {2 \log (4+x)}{-8-2 x}-\frac {2 \log (4+x)}{4-2 x}\right ) \, dx\\ &=\frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}+\frac {20}{x}-6 \log (2-x)-6 \log (4+x)-2 \log (-2+x) \log \left (8-2 x-x^2\right )-2 \log (4+x) \log \left (8-2 x-x^2\right )-4 \int \frac {\log (-2+x)}{-8-2 x} \, dx-4 \int \frac {\log (-2+x)}{4-2 x} \, dx-4 \int \frac {\log (4+x)}{-8-2 x} \, dx-4 \int \frac {\log (4+x)}{4-2 x} \, dx\\ &=\frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}+\frac {20}{x}-6 \log (2-x)+2 \log (-2+x) \log \left (\frac {4+x}{6}\right )-6 \log (4+x)+2 \log \left (\frac {2-x}{6}\right ) \log (4+x)-2 \log (-2+x) \log \left (8-2 x-x^2\right )-2 \log (4+x) \log \left (8-2 x-x^2\right )-2 \int \frac {\log \left (\frac {1}{12} (4-2 x)\right )}{4+x} \, dx-2 \int \frac {\log \left (\frac {1}{12} (8+2 x)\right )}{-2+x} \, dx-4 \operatorname {Subst}\left (\int -\frac {\log (x)}{2 x} \, dx,x,-2+x\right )-4 \operatorname {Subst}\left (\int -\frac {\log (x)}{2 x} \, dx,x,4+x\right )\\ &=\frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}+\frac {20}{x}-6 \log (2-x)+2 \log (-2+x) \log \left (\frac {4+x}{6}\right )-6 \log (4+x)+2 \log \left (\frac {2-x}{6}\right ) \log (4+x)-2 \log (-2+x) \log \left (8-2 x-x^2\right )-2 \log (4+x) \log \left (8-2 x-x^2\right )-2 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{6}\right )}{x} \, dx,x,4+x\right )-2 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{6}\right )}{x} \, dx,x,-2+x\right )+2 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+x\right )+2 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,4+x\right )\\ &=\frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}+\frac {20}{x}-6 \log (2-x)+\log ^2(-2+x)+2 \log (-2+x) \log \left (\frac {4+x}{6}\right )-6 \log (4+x)+2 \log \left (\frac {2-x}{6}\right ) \log (4+x)+\log ^2(4+x)-2 \log (-2+x) \log \left (8-2 x-x^2\right )-2 \log (4+x) \log \left (8-2 x-x^2\right )+2 \text {Li}_2\left (\frac {2-x}{6}\right )+2 \text {Li}_2\left (\frac {4+x}{6}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.12, size = 91, normalized size = 3.14 \begin {gather*} \frac {625}{x^4}+\frac {500}{x^3}+\frac {150}{x^2}+\frac {20}{x}+2 \log (6) \log (4-2 x)+\log ^2(-2+x)-6 \log (4+x)+\log ^2(4+x)-2 \log (4+x) \log \left (8-2 x-x^2\right )-2 \log (-2+x) \left (3-\log (4+x)+\log \left (-6 \left (-8+2 x+x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20000 + 7000*x - 3100*x^2 - 1940*x^3 - 340*x^4 - 32*x^5 - 12*x^6 + (-4*x^5 - 4*x^6)*Log[8 - 2*x - x
^2])/(-8*x^5 + 2*x^6 + x^7),x]

[Out]

625/x^4 + 500/x^3 + 150/x^2 + 20/x + 2*Log[6]*Log[4 - 2*x] + Log[-2 + x]^2 - 6*Log[4 + x] + Log[4 + x]^2 - 2*L
og[4 + x]*Log[8 - 2*x - x^2] - 2*Log[-2 + x]*(3 - Log[4 + x] + Log[-6*(-8 + 2*x + x^2)])

________________________________________________________________________________________

fricas [A]  time = 1.06, size = 53, normalized size = 1.83 \begin {gather*} -\frac {x^{4} \log \left (-x^{2} - 2 \, x + 8\right )^{2} + 6 \, x^{4} \log \left (-x^{2} - 2 \, x + 8\right ) - 20 \, x^{3} - 150 \, x^{2} - 500 \, x - 625}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^6-4*x^5)*log(-x^2-2*x+8)-12*x^6-32*x^5-340*x^4-1940*x^3-3100*x^2+7000*x+20000)/(x^7+2*x^6-8*x
^5),x, algorithm="fricas")

[Out]

-(x^4*log(-x^2 - 2*x + 8)^2 + 6*x^4*log(-x^2 - 2*x + 8) - 20*x^3 - 150*x^2 - 500*x - 625)/x^4

________________________________________________________________________________________

giac [A]  time = 0.24, size = 47, normalized size = 1.62 \begin {gather*} -\log \left (-x^{2} - 2 \, x + 8\right )^{2} + \frac {5 \, {\left (4 \, x^{3} + 30 \, x^{2} + 100 \, x + 125\right )}}{x^{4}} - 6 \, \log \left (x^{2} + 2 \, x - 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^6-4*x^5)*log(-x^2-2*x+8)-12*x^6-32*x^5-340*x^4-1940*x^3-3100*x^2+7000*x+20000)/(x^7+2*x^6-8*x
^5),x, algorithm="giac")

[Out]

-log(-x^2 - 2*x + 8)^2 + 5*(4*x^3 + 30*x^2 + 100*x + 125)/x^4 - 6*log(x^2 + 2*x - 8)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 51, normalized size = 1.76

method	result	size
risch	\(-\ln \left (-x^{2}-2 x +8\right )^{2}-\frac {6 \ln \left (x^{2}+2 x -8\right ) x^{4}-20 x^{3}-150 x^{2}-500 x -625}{x^{4}}\)	\(51\)
norman	\(\frac {625-6 x^{4} \ln \left (-x^{2}-2 x +8\right )+500 x +150 x^{2}+20 x^{3}-x^{4} \ln \left (-x^{2}-2 x +8\right )^{2}}{x^{4}}\)	\(54\)
default	\(-2 \ln \left (x -2\right ) \ln \left (-x^{2}-2 x +8\right )+\ln \left (x -2\right )^{2}+2 \ln \left (x -2\right ) \ln \left (\frac {2}{3}+\frac {x}{6}\right )-2 \ln \left (4+x \right ) \ln \left (-x^{2}-2 x +8\right )+\ln \left (4+x \right )^{2}+2 \left (\ln \left (4+x \right )-\ln \left (\frac {2}{3}+\frac {x}{6}\right )\right ) \ln \left (-\frac {x}{6}+\frac {1}{3}\right )+\frac {625}{x^{4}}+\frac {500}{x^{3}}+\frac {150}{x^{2}}+\frac {20}{x}-6 \ln \left (x -2\right )-6 \ln \left (4+x \right )\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^6-4*x^5)*ln(-x^2-2*x+8)-12*x^6-32*x^5-340*x^4-1940*x^3-3100*x^2+7000*x+20000)/(x^7+2*x^6-8*x^5),x,m
ethod=_RETURNVERBOSE)

[Out]

-ln(-x^2-2*x+8)^2-(6*ln(x^2+2*x-8)*x^4-20*x^3-150*x^2-500*x-625)/x^4

________________________________________________________________________________________

maxima [B]  time = 0.46, size = 91, normalized size = 3.14 \begin {gather*} -\log \left (x + 4\right )^{2} - 2 \, \log \left (x + 4\right ) \log \left (-x + 2\right ) - \log \left (-x + 2\right )^{2} - \frac {775 \, {\left (x + 2\right )}}{8 \, x^{2}} - \frac {485}{2 \, x} + \frac {875 \, {\left (9 \, x^{2} + 6 \, x + 16\right )}}{48 \, x^{3}} + \frac {625 \, {\left (15 \, x^{3} + 18 \, x^{2} + 16 \, x + 48\right )}}{48 \, x^{4}} - 6 \, \log \left (x + 4\right ) - 6 \, \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^6-4*x^5)*log(-x^2-2*x+8)-12*x^6-32*x^5-340*x^4-1940*x^3-3100*x^2+7000*x+20000)/(x^7+2*x^6-8*x
^5),x, algorithm="maxima")

[Out]

-log(x + 4)^2 - 2*log(x + 4)*log(-x + 2) - log(-x + 2)^2 - 775/8*(x + 2)/x^2 - 485/2/x + 875/48*(9*x^2 + 6*x +
16)/x^3 + 625/48*(15*x^3 + 18*x^2 + 16*x + 48)/x^4 - 6*log(x + 4) - 6*log(x - 2)

________________________________________________________________________________________

mupad [B]  time = 0.28, size = 46, normalized size = 1.59 \begin {gather*} \frac {20\,x^3+150\,x^2+500\,x+625}{x^4}-{\ln \left (-x^2-2\,x+8\right )}^2-6\,\ln \left (x^2+2\,x-8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(8 - x^2 - 2*x)*(4*x^5 + 4*x^6) - 7000*x + 3100*x^2 + 1940*x^3 + 340*x^4 + 32*x^5 + 12*x^6 - 20000)/(
2*x^6 - 8*x^5 + x^7),x)

[Out]

(500*x + 150*x^2 + 20*x^3 + 625)/x^4 - log(8 - x^2 - 2*x)^2 - 6*log(2*x + x^2 - 8)

________________________________________________________________________________________

sympy [B]  time = 0.21, size = 44, normalized size = 1.52 \begin {gather*} - \log {\left (- x^{2} - 2 x + 8 \right )}^{2} - 6 \log {\left (x^{2} + 2 x - 8 \right )} - \frac {- 20 x^{3} - 150 x^{2} - 500 x - 625}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**6-4*x**5)*ln(-x**2-2*x+8)-12*x**6-32*x**5-340*x**4-1940*x**3-3100*x**2+7000*x+20000)/(x**7+2
*x**6-8*x**5),x)

[Out]

-log(-x**2 - 2*x + 8)**2 - 6*log(x**2 + 2*x - 8) - (-20*x**3 - 150*x**2 - 500*x - 625)/x**4

________________________________________________________________________________________