\(\int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+(8+52 x+44 x^2-4 x^3-4 x^4) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx\) [10002]

Optimal result

Integrand size = 66, antiderivative size = 21 \[ \int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+\left (8+52 x+44 x^2-4 x^3-4 x^4\right ) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx=\left (-15-x-\frac {8 x}{x+x^2}+2 \log (x)\right )^2 \]

[Out]

(-15-8*x/(x^2+x)+2*ln(x)-x)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {6820, 12, 6874, 1864, 1634, 2404, 2332, 2338, 2351, 31} \[ \int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+\left (8+52 x+44 x^2-4 x^3-4 x^4\right ) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx=x^2+30 x+\frac {224}{x+1}+\frac {64}{(x+1)^2}+4 \log ^2(x)+\frac {32 x \log (x)}{x+1}-4 x \log (x)-92 \log (x) \]

[In]

Int[(-92 - 570*x - 356*x^2 + 24*x^3 + 32*x^4 + 2*x^5 + (8 + 52*x + 44*x^2 - 4*x^3 - 4*x^4)*Log[x])/(x + 3*x^2
+ 3*x^3 + x^4),x]

[Out]

30*x + x^2 + 64/(1 + x)^2 + 224/(1 + x) - 92*Log[x] - 4*x*Log[x] + (32*x*Log[x])/(1 + x) + 4*Log[x]^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 31

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

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2338

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 2351

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

Rule 2404

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

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (2+11 x-x^3\right ) \left (-23-16 x-x^2+2 (1+x) \log (x)\right )}{x (1+x)^3} \, dx \\ & = 2 \int \frac {\left (2+11 x-x^3\right ) \left (-23-16 x-x^2+2 (1+x) \log (x)\right )}{x (1+x)^3} \, dx \\ & = 2 \int \left (\frac {16 \left (-2-11 x+x^3\right )}{(1+x)^3}+\frac {23 \left (-2-11 x+x^3\right )}{x (1+x)^3}+\frac {x \left (-2-11 x+x^3\right )}{(1+x)^3}-\frac {2 \left (-2-11 x+x^3\right ) \log (x)}{x (1+x)^2}\right ) \, dx \\ & = 2 \int \frac {x \left (-2-11 x+x^3\right )}{(1+x)^3} \, dx-4 \int \frac {\left (-2-11 x+x^3\right ) \log (x)}{x (1+x)^2} \, dx+32 \int \frac {-2-11 x+x^3}{(1+x)^3} \, dx+46 \int \frac {-2-11 x+x^3}{x (1+x)^3} \, dx \\ & = 2 \int \left (-3+x-\frac {8}{(1+x)^3}+\frac {16}{(1+x)^2}-\frac {5}{1+x}\right ) \, dx-4 \int \left (\log (x)-\frac {2 \log (x)}{x}-\frac {8 \log (x)}{(1+x)^2}\right ) \, dx+32 \int \left (1+\frac {8}{(1+x)^3}-\frac {8}{(1+x)^2}-\frac {3}{1+x}\right ) \, dx+46 \int \left (-\frac {2}{x}-\frac {8}{(1+x)^3}+\frac {3}{1+x}\right ) \, dx \\ & = 26 x+x^2+\frac {64}{(1+x)^2}+\frac {224}{1+x}-92 \log (x)+32 \log (1+x)-4 \int \log (x) \, dx+8 \int \frac {\log (x)}{x} \, dx+32 \int \frac {\log (x)}{(1+x)^2} \, dx \\ & = 30 x+x^2+\frac {64}{(1+x)^2}+\frac {224}{1+x}-92 \log (x)-4 x \log (x)+\frac {32 x \log (x)}{1+x}+4 \log ^2(x)+32 \log (1+x)-32 \int \frac {1}{1+x} \, dx \\ & = 30 x+x^2+\frac {64}{(1+x)^2}+\frac {224}{1+x}-92 \log (x)-4 x \log (x)+\frac {32 x \log (x)}{1+x}+4 \log ^2(x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(21)=42\).

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+\left (8+52 x+44 x^2-4 x^3-4 x^4\right ) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx=2 \left (15 x+\frac {x^2}{2}+\frac {32}{(1+x)^2}+\frac {16 (7-\log (x))}{1+x}-30 \log (x)-2 x \log (x)+2 \log ^2(x)\right ) \]

[In]

Integrate[(-92 - 570*x - 356*x^2 + 24*x^3 + 32*x^4 + 2*x^5 + (8 + 52*x + 44*x^2 - 4*x^3 - 4*x^4)*Log[x])/(x +
3*x^2 + 3*x^3 + x^4),x]

[Out]

2*(15*x + x^2/2 + 32/(1 + x)^2 + (16*(7 - Log[x]))/(1 + x) - 30*Log[x] - 2*x*Log[x] + 2*Log[x]^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).

Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24

method	result	size
default	\(x^{2}+30 x -92 \ln \left (x \right )+\frac {64}{\left (1+x \right )^{2}}+\frac {224}{1+x}-4 x \ln \left (x \right )+4 \ln \left (x \right )^{2}+\frac {32 \ln \left (x \right ) x}{1+x}\)	\(47\)
parts	\(x^{2}+30 x -92 \ln \left (x \right )+\frac {64}{\left (1+x \right )^{2}}+\frac {224}{1+x}-4 x \ln \left (x \right )+4 \ln \left (x \right )^{2}+\frac {32 \ln \left (x \right ) x}{1+x}\)	\(47\)
norman	\(\frac {x^{4}-92 \ln \left (x \right )+132 x -156 x \ln \left (x \right )-68 x^{2} \ln \left (x \right )+32 x^{3}+4 \ln \left (x \right )^{2}+8 x \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )^{2}-4 x^{3} \ln \left (x \right )+227}{\left (1+x \right )^{2}}\)	\(65\)
risch	\(4 \ln \left (x \right )^{2}-\frac {4 \left (x^{2}+x +8\right ) \ln \left (x \right )}{1+x}-\frac {-x^{4}+60 x^{2} \ln \left (x \right )-32 x^{3}+120 x \ln \left (x \right )-61 x^{2}+60 \ln \left (x \right )-254 x -288}{\left (1+x \right )^{2}}\)	\(66\)
parallelrisch	\(\frac {x^{4}-92 \ln \left (x \right )+132 x -156 x \ln \left (x \right )-68 x^{2} \ln \left (x \right )+32 x^{3}+4 \ln \left (x \right )^{2}+8 x \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )^{2}-4 x^{3} \ln \left (x \right )+227}{x^{2}+2 x +1}\)	\(70\)

[In]

int(((-4*x^4-4*x^3+44*x^2+52*x+8)*ln(x)+2*x^5+32*x^4+24*x^3-356*x^2-570*x-92)/(x^4+3*x^3+3*x^2+x),x,method=_RE
TURNVERBOSE)

[Out]

x^2+30*x-92*ln(x)+64/(1+x)^2+224/(1+x)-4*x*ln(x)+4*ln(x)^2+32*ln(x)*x/(1+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.86 \[ \int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+\left (8+52 x+44 x^2-4 x^3-4 x^4\right ) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx=\frac {x^{4} + 32 \, x^{3} + 4 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 61 \, x^{2} - 4 \, {\left (x^{3} + 17 \, x^{2} + 39 \, x + 23\right )} \log \left (x\right ) + 254 \, x + 288}{x^{2} + 2 \, x + 1} \]

[In]

integrate(((-4*x^4-4*x^3+44*x^2+52*x+8)*log(x)+2*x^5+32*x^4+24*x^3-356*x^2-570*x-92)/(x^4+3*x^3+3*x^2+x),x, al
gorithm="fricas")

[Out]

(x^4 + 32*x^3 + 4*(x^2 + 2*x + 1)*log(x)^2 + 61*x^2 - 4*(x^3 + 17*x^2 + 39*x + 23)*log(x) + 254*x + 288)/(x^2
+ 2*x + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (17) = 34\).

Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+\left (8+52 x+44 x^2-4 x^3-4 x^4\right ) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx=x^{2} + 30 x + \frac {224 x + 288}{x^{2} + 2 x + 1} + 4 \log {\left (x \right )}^{2} - 60 \log {\left (x \right )} + \frac {\left (- 4 x^{2} - 4 x - 32\right ) \log {\left (x \right )}}{x + 1} \]

[In]

integrate(((-4*x**4-4*x**3+44*x**2+52*x+8)*ln(x)+2*x**5+32*x**4+24*x**3-356*x**2-570*x-92)/(x**4+3*x**3+3*x**2
+x),x)

[Out]

x**2 + 30*x + (224*x + 288)/(x**2 + 2*x + 1) + 4*log(x)**2 - 60*log(x) + (-4*x**2 - 4*x - 32)*log(x)/(x + 1)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (19) = 38\).

Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 6.52 \[ \int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+\left (8+52 x+44 x^2-4 x^3-4 x^4\right ) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx=x^{2} + 26 \, x + \frac {8 \, x + 7}{x^{2} + 2 \, x + 1} - \frac {16 \, {\left (6 \, x + 5\right )}}{x^{2} + 2 \, x + 1} + \frac {12 \, {\left (4 \, x + 3\right )}}{x^{2} + 2 \, x + 1} - \frac {46 \, {\left (2 \, x + 3\right )}}{x^{2} + 2 \, x + 1} + \frac {178 \, {\left (2 \, x + 1\right )}}{x^{2} + 2 \, x + 1} + \frac {4 \, {\left ({\left (x + 1\right )} \log \left (x\right )^{2} + x^{2} - {\left (x^{2} + x + 8\right )} \log \left (x\right ) + x\right )}}{x + 1} + \frac {285}{x^{2} + 2 \, x + 1} - 60 \, \log \left (x\right ) \]

[In]

integrate(((-4*x^4-4*x^3+44*x^2+52*x+8)*log(x)+2*x^5+32*x^4+24*x^3-356*x^2-570*x-92)/(x^4+3*x^3+3*x^2+x),x, al
gorithm="maxima")

[Out]

x^2 + 26*x + (8*x + 7)/(x^2 + 2*x + 1) - 16*(6*x + 5)/(x^2 + 2*x + 1) + 12*(4*x + 3)/(x^2 + 2*x + 1) - 46*(2*x
+ 3)/(x^2 + 2*x + 1) + 178*(2*x + 1)/(x^2 + 2*x + 1) + 4*((x + 1)*log(x)^2 + x^2 - (x^2 + x + 8)*log(x) + x)/
(x + 1) + 285/(x^2 + 2*x + 1) - 60*log(x)

Giac [F]

\[ \int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+\left (8+52 x+44 x^2-4 x^3-4 x^4\right ) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx=\int { \frac {2 \, {\left (x^{5} + 16 \, x^{4} + 12 \, x^{3} - 178 \, x^{2} - 2 \, {\left (x^{4} + x^{3} - 11 \, x^{2} - 13 \, x - 2\right )} \log \left (x\right ) - 285 \, x - 46\right )}}{x^{4} + 3 \, x^{3} + 3 \, x^{2} + x} \,d x } \]

[In]

integrate(((-4*x^4-4*x^3+44*x^2+52*x+8)*log(x)+2*x^5+32*x^4+24*x^3-356*x^2-570*x-92)/(x^4+3*x^3+3*x^2+x),x, al
gorithm="giac")

[Out]

integrate(2*(x^5 + 16*x^4 + 12*x^3 - 178*x^2 - 2*(x^4 + x^3 - 11*x^2 - 13*x - 2)*log(x) - 285*x - 46)/(x^4 + 3
*x^3 + 3*x^2 + x), x)

Mupad [B] (verification not implemented)

Time = 16.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.10 \[ \int \frac {-92-570 x-356 x^2+24 x^3+32 x^4+2 x^5+\left (8+52 x+44 x^2-4 x^3-4 x^4\right ) \log (x)}{x+3 x^2+3 x^3+x^4} \, dx=4\,{\ln \left (x\right )}^2-60\,\ln \left (x\right )-x\,\left (4\,\ln \left (x\right )-30\right )-\frac {32\,\ln \left (x\right )+x\,\left (32\,\ln \left (x\right )-224\right )-288}{{\left (x+1\right )}^2}+x^2 \]

[In]

int((log(x)*(52*x + 44*x^2 - 4*x^3 - 4*x^4 + 8) - 570*x - 356*x^2 + 24*x^3 + 32*x^4 + 2*x^5 - 92)/(x + 3*x^2 +
3*x^3 + x^4),x)

[Out]

4*log(x)^2 - 60*log(x) - x*(4*log(x) - 30) - (32*log(x) + x*(32*log(x) - 224) - 288)/(x + 1)^2 + x^2