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

Optimal. Leaf size=21 \[ \left (-15-x-\frac {8 x}{x+x^2}+2 \log (x)\right )^2 \]

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Rubi [B]  time = 0.46, antiderivative size = 46, normalized size of antiderivative = 2.19, number of steps used = 16, number of rules used = 10, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {6688, 12, 6742, 1850, 1620, 2357, 2295, 2301, 2314, 31} \begin {gather*} 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) \end {gather*}

Antiderivative was successfully verified.

[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 1620

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 1850

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 2295

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

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 2314

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 2357

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 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 48, normalized size = 2.29 \begin {gather*} 2 \left (15 x+\frac {x^2}{2}+\frac {16 (9+7 x)}{(1+x)^2}-\frac {2 \left (23+16 x+x^2\right ) \log (x)}{1+x}+2 \log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[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 + (16*(9 + 7*x))/(1 + x)^2 - (2*(23 + 16*x + x^2)*Log[x])/(1 + x) + 2*Log[x]^2)

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fricas [B]  time = 2.43, size = 60, normalized size = 2.86 \begin {gather*} \frac {x^{4} + 32 \, x^{3} + 4 \, {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} + 61 \, x^{2} - 4 \, {\left (x^{3} + 17 \, x^{2} + 39 \, x + 23\right )} \log \relax (x) + 254 \, x + 288}{x^{2} + 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \relax (x) - 285 \, x - 46\right )}}{x^{4} + 3 \, x^{3} + 3 \, x^{2} + x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.07, size = 47, normalized size = 2.24

method	result	size
default	\(x^{2}+30 x -92 \ln \relax (x )+\frac {64}{\left (x +1\right )^{2}}+\frac {224}{x +1}-4 x \ln \relax (x )+4 \ln \relax (x )^{2}+\frac {32 \ln \relax (x ) x}{x +1}\)	\(47\)
risch	\(4 \ln \relax (x )^{2}-\frac {4 \left (x^{2}+x +8\right ) \ln \relax (x )}{x +1}-\frac {-x^{4}+60 x^{2} \ln \relax (x )-32 x^{3}+120 x \ln \relax (x )-61 x^{2}+60 \ln \relax (x )-254 x -288}{\left (x +1\right )^{2}}\)	\(66\)
norman	\(\frac {x^{4}-66 x^{2}-14 \ln \relax (x )+10 x^{2} \ln \relax (x )+32 x^{3}+4 \ln \relax (x )^{2}+8 x \ln \relax (x )^{2}+4 x^{2} \ln \relax (x )^{2}-4 x^{3} \ln \relax (x )+161}{\left (x +1\right )^{2}}-78 \ln \relax (x )\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[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/(x+1)^2+224/(x+1)-4*x*ln(x)+4*ln(x)^2+32*ln(x)*x/(x+1)

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maxima [B]  time = 0.52, size = 137, normalized size = 6.52 \begin {gather*} 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 \relax (x)^{2} + x^{2} - {\left (x^{2} + x + 8\right )} \log \relax (x) + x\right )}}{x + 1} + \frac {285}{x^{2} + 2 \, x + 1} - 60 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 7.36, size = 44, normalized size = 2.10 \begin {gather*} 4\,{\ln \relax (x)}^2-60\,\ln \relax (x)-x\,\left (4\,\ln \relax (x)-30\right )-\frac {32\,\ln \relax (x)+x\,\left (32\,\ln \relax (x)-224\right )-288}{{\left (x+1\right )}^2}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.21, size = 49, normalized size = 2.33 \begin {gather*} x^{2} + 30 x + \frac {224 x + 288}{x^{2} + 2 x + 1} + 4 \log {\relax (x )}^{2} - 60 \log {\relax (x )} + \frac {\left (- 4 x^{2} - 4 x - 32\right ) \log {\relax (x )}}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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