3.7.78 \(\int \frac {(-2+5 x+3 x^2) \log (\frac {19}{2+x})+\log (x) (x-3 x^2+(6 x+3 x^2) \log (\frac {19}{2+x}))}{8 x+4 x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{4} (-1+3 x) \log (x) \log \left (\frac {19}{2+x}\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 0.48, antiderivative size = 87, normalized size of antiderivative = 4.58, number of steps used = 22, number of rules used = 14, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1593, 6742, 2357, 2295, 2317, 2391, 14, 43, 2389, 2394, 2315, 2370, 2411, 2351} \begin {gather*} \frac {1}{4} \text {Li}_2\left (\frac {x}{2}+1\right )+\frac {7 \text {Li}_2\left (-\frac {x}{2}\right )}{4}+\frac {3}{2} \text {Li}_2\left (\frac {x+2}{2}\right )+\frac {7}{4} \log \left (\frac {x}{2}+1\right ) \log (x)+\frac {3}{4} (x+2) \log \left (\frac {19}{x+2}\right ) \log (x)-\frac {7}{4} \log \left (-\frac {x}{2}\right ) \log \left (\frac {19}{x+2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-2 + 5*x + 3*x^2)*Log[19/(2 + x)] + Log[x]*(x - 3*x^2 + (6*x + 3*x^2)*Log[19/(2 + x)]))/(8*x + 4*x^2),x]

[Out]

(7*Log[1 + x/2]*Log[x])/4 - (7*Log[-1/2*x]*Log[19/(2 + x)])/4 + (3*(2 + x)*Log[x]*Log[19/(2 + x)])/4 + PolyLog
[2, 1 + x/2]/4 + (7*PolyLog[2, -1/2*x])/4 + (3*PolyLog[2, (2 + x)/2])/2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2317

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

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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 2370

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x
^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 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 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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 {\left (-2+5 x+3 x^2\right ) \log \left (\frac {19}{2+x}\right )+\log (x) \left (x-3 x^2+\left (6 x+3 x^2\right ) \log \left (\frac {19}{2+x}\right )\right )}{x (8+4 x)} \, dx\\ &=\int \left (\frac {(1-3 x) \log (x)}{4 (2+x)}+\frac {(-1+3 x+3 x \log (x)) \log \left (\frac {19}{2+x}\right )}{4 x}\right ) \, dx\\ &=\frac {1}{4} \int \frac {(1-3 x) \log (x)}{2+x} \, dx+\frac {1}{4} \int \frac {(-1+3 x+3 x \log (x)) \log \left (\frac {19}{2+x}\right )}{x} \, dx\\ &=\frac {1}{4} \int \left (-3 \log (x)+\frac {7 \log (x)}{2+x}\right ) \, dx+\frac {1}{4} \int \left (3 \log \left (\frac {19}{2+x}\right )-\frac {\log \left (\frac {19}{2+x}\right )}{x}+3 \log (x) \log \left (\frac {19}{2+x}\right )\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\log \left (\frac {19}{2+x}\right )}{x} \, dx\right )-\frac {3}{4} \int \log (x) \, dx+\frac {3}{4} \int \log \left (\frac {19}{2+x}\right ) \, dx+\frac {3}{4} \int \log (x) \log \left (\frac {19}{2+x}\right ) \, dx+\frac {7}{4} \int \frac {\log (x)}{2+x} \, dx\\ &=\frac {3 x}{4}+\frac {7}{4} \log \left (1+\frac {x}{2}\right ) \log (x)-\frac {1}{4} \log \left (-\frac {x}{2}\right ) \log \left (\frac {19}{2+x}\right )+\frac {3}{4} (2+x) \log (x) \log \left (\frac {19}{2+x}\right )-\frac {1}{4} \int \frac {\log \left (-\frac {x}{2}\right )}{2+x} \, dx-\frac {3}{4} \int \left (1+\frac {(2+x) \log \left (\frac {19}{2+x}\right )}{x}\right ) \, dx+\frac {3}{4} \operatorname {Subst}\left (\int \log \left (\frac {19}{x}\right ) \, dx,x,2+x\right )-\frac {7}{4} \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx\\ &=\frac {3 x}{4}+\frac {7}{4} \log \left (1+\frac {x}{2}\right ) \log (x)+\frac {3}{4} (2+x) \log \left (\frac {19}{2+x}\right )-\frac {1}{4} \log \left (-\frac {x}{2}\right ) \log \left (\frac {19}{2+x}\right )+\frac {3}{4} (2+x) \log (x) \log \left (\frac {19}{2+x}\right )+\frac {1}{4} \text {Li}_2\left (1+\frac {x}{2}\right )+\frac {7 \text {Li}_2\left (-\frac {x}{2}\right )}{4}-\frac {3}{4} \int \frac {(2+x) \log \left (\frac {19}{2+x}\right )}{x} \, dx\\ &=\frac {3 x}{4}+\frac {7}{4} \log \left (1+\frac {x}{2}\right ) \log (x)+\frac {3}{4} (2+x) \log \left (\frac {19}{2+x}\right )-\frac {1}{4} \log \left (-\frac {x}{2}\right ) \log \left (\frac {19}{2+x}\right )+\frac {3}{4} (2+x) \log (x) \log \left (\frac {19}{2+x}\right )+\frac {1}{4} \text {Li}_2\left (1+\frac {x}{2}\right )+\frac {7 \text {Li}_2\left (-\frac {x}{2}\right )}{4}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {x \log \left (\frac {19}{x}\right )}{-2+x} \, dx,x,2+x\right )\\ &=\frac {3 x}{4}+\frac {7}{4} \log \left (1+\frac {x}{2}\right ) \log (x)+\frac {3}{4} (2+x) \log \left (\frac {19}{2+x}\right )-\frac {1}{4} \log \left (-\frac {x}{2}\right ) \log \left (\frac {19}{2+x}\right )+\frac {3}{4} (2+x) \log (x) \log \left (\frac {19}{2+x}\right )+\frac {1}{4} \text {Li}_2\left (1+\frac {x}{2}\right )+\frac {7 \text {Li}_2\left (-\frac {x}{2}\right )}{4}-\frac {3}{4} \operatorname {Subst}\left (\int \left (\log \left (\frac {19}{x}\right )+\frac {2 \log \left (\frac {19}{x}\right )}{-2+x}\right ) \, dx,x,2+x\right )\\ &=\frac {3 x}{4}+\frac {7}{4} \log \left (1+\frac {x}{2}\right ) \log (x)+\frac {3}{4} (2+x) \log \left (\frac {19}{2+x}\right )-\frac {1}{4} \log \left (-\frac {x}{2}\right ) \log \left (\frac {19}{2+x}\right )+\frac {3}{4} (2+x) \log (x) \log \left (\frac {19}{2+x}\right )+\frac {1}{4} \text {Li}_2\left (1+\frac {x}{2}\right )+\frac {7 \text {Li}_2\left (-\frac {x}{2}\right )}{4}-\frac {3}{4} \operatorname {Subst}\left (\int \log \left (\frac {19}{x}\right ) \, dx,x,2+x\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {\log \left (\frac {19}{x}\right )}{-2+x} \, dx,x,2+x\right )\\ &=\frac {7}{4} \log \left (1+\frac {x}{2}\right ) \log (x)-\frac {7}{4} \log \left (-\frac {x}{2}\right ) \log \left (\frac {19}{2+x}\right )+\frac {3}{4} (2+x) \log (x) \log \left (\frac {19}{2+x}\right )+\frac {1}{4} \text {Li}_2\left (1+\frac {x}{2}\right )+\frac {7 \text {Li}_2\left (-\frac {x}{2}\right )}{4}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,2+x\right )\\ &=\frac {7}{4} \log \left (1+\frac {x}{2}\right ) \log (x)-\frac {7}{4} \log \left (-\frac {x}{2}\right ) \log \left (\frac {19}{2+x}\right )+\frac {3}{4} (2+x) \log (x) \log \left (\frac {19}{2+x}\right )+\frac {1}{4} \text {Li}_2\left (1+\frac {x}{2}\right )+\frac {7 \text {Li}_2\left (-\frac {x}{2}\right )}{4}+\frac {3}{2} \text {Li}_2\left (\frac {2+x}{2}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{4} (-1+3 x) \log (x) \log \left (\frac {19}{2+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2 + 5*x + 3*x^2)*Log[19/(2 + x)] + Log[x]*(x - 3*x^2 + (6*x + 3*x^2)*Log[19/(2 + x)]))/(8*x + 4*x
^2),x]

[Out]

((-1 + 3*x)*Log[x]*Log[19/(2 + x)])/4

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{4} \, {\left (3 \, x - 1\right )} \log \relax (x) \log \left (\frac {19}{x + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+6*x)*log(19/(2+x))-3*x^2+x)*log(x)+(3*x^2+5*x-2)*log(19/(2+x)))/(4*x^2+8*x),x, algorithm="f
ricas")

[Out]

1/4*(3*x - 1)*log(x)*log(19/(x + 2))

________________________________________________________________________________________

giac [A]  time = 0.42, size = 30, normalized size = 1.58 \begin {gather*} \frac {3}{4} \, x \log \left (19\right ) \log \relax (x) - \frac {1}{4} \, {\left (3 \, x \log \relax (x) - \log \relax (x)\right )} \log \left (x + 2\right ) - \frac {1}{4} \, \log \left (19\right ) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+6*x)*log(19/(2+x))-3*x^2+x)*log(x)+(3*x^2+5*x-2)*log(19/(2+x)))/(4*x^2+8*x),x, algorithm="g
iac")

[Out]

3/4*x*log(19)*log(x) - 1/4*(3*x*log(x) - log(x))*log(x + 2) - 1/4*log(19)*log(x)

________________________________________________________________________________________

maple [A]  time = 0.24, size = 27, normalized size = 1.42

method	result	size
norman	\(-\frac {\ln \relax (x ) \ln \left (\frac {19}{2+x}\right )}{4}+\frac {3 \ln \relax (x ) \ln \left (\frac {19}{2+x}\right ) x}{4}\)	\(27\)
risch	\(\left (-\frac {3 x \ln \relax (x )}{4}+\frac {\ln \relax (x )}{4}\right ) \ln \left (2+x \right )+\frac {3 x \ln \relax (x ) \ln \left (19\right )}{4}-\frac {\ln \relax (x ) \ln \left (19\right )}{4}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x^2+6*x)*ln(19/(2+x))-3*x^2+x)*ln(x)+(3*x^2+5*x-2)*ln(19/(2+x)))/(4*x^2+8*x),x,method=_RETURNVERBOSE)

[Out]

-1/4*ln(x)*ln(19/(2+x))+3/4*ln(x)*ln(19/(2+x))*x

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 28, normalized size = 1.47 \begin {gather*} -\frac {1}{4} \, {\left (3 \, x - 1\right )} \log \left (x + 2\right ) \log \relax (x) + \frac {1}{4} \, {\left (3 \, x \log \left (19\right ) - \log \left (19\right )\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+6*x)*log(19/(2+x))-3*x^2+x)*log(x)+(3*x^2+5*x-2)*log(19/(2+x)))/(4*x^2+8*x),x, algorithm="m
axima")

[Out]

-1/4*(3*x - 1)*log(x + 2)*log(x) + 1/4*(3*x*log(19) - log(19))*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.76, size = 17, normalized size = 0.89 \begin {gather*} \ln \relax (x)\,\left (\frac {3\,x}{4}-\frac {1}{4}\right )\,\left (\ln \left (19\right )+\ln \left (\frac {1}{x+2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(x + log(19/(x + 2))*(6*x + 3*x^2) - 3*x^2) + log(19/(x + 2))*(5*x + 3*x^2 - 2))/(8*x + 4*x^2),x)

[Out]

log(x)*((3*x)/4 - 1/4)*(log(19) + log(1/(x + 2)))

________________________________________________________________________________________

sympy [A]  time = 0.39, size = 19, normalized size = 1.00 \begin {gather*} \left (\frac {3 x \log {\relax (x )}}{4} - \frac {\log {\relax (x )}}{4}\right ) \log {\left (\frac {19}{x + 2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x**2+6*x)*ln(19/(2+x))-3*x**2+x)*ln(x)+(3*x**2+5*x-2)*ln(19/(2+x)))/(4*x**2+8*x),x)

[Out]

(3*x*log(x)/4 - log(x)/4)*log(19/(x + 2))

________________________________________________________________________________________