3.87.18 \(\int \frac {84+146 x+60 x^2+(-42-73 x-30 x^2) \log (\frac {1}{31} (28+30 x))+(-45 x-15 x^2) \log (3 x+x^2)}{42 x+59 x^2+15 x^3} \, dx\)

Optimal. Leaf size=26 \[ \left (2-\log \left (1+\frac {1}{31} (-3-x)+x\right )\right ) \log (x+x (2+x)) \]

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Rubi [B]  time = 0.43, antiderivative size = 67, normalized size of antiderivative = 2.58, number of steps used = 16, number of rules used = 8, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1594, 6728, 2494, 2392, 2391, 2394, 2393, 2418} \begin {gather*} \log (14) \log (x)+\left (2+\log \left (\frac {31}{28}\right )\right ) \log (x)+\log \left (\frac {15 (x+3)}{31}\right ) \left (2-\log \left (\frac {2}{31} (15 x+14)\right )\right )+\log \left (\frac {15 (x+3)}{31}\right ) \log (15 x+14)-\log (x (x+3)) \log (15 x+14) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(84 + 146*x + 60*x^2 + (-42 - 73*x - 30*x^2)*Log[(28 + 30*x)/31] + (-45*x - 15*x^2)*Log[3*x + x^2])/(42*x
+ 59*x^2 + 15*x^3),x]

[Out]

(2 + Log[31/28])*Log[x] + Log[14]*Log[x] + Log[(15*(3 + x))/31]*(2 - Log[(2*(14 + 15*x))/31]) + Log[(15*(3 + x
))/31]*Log[14 + 15*x] - Log[x*(3 + x)]*Log[14 + 15*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 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 2392

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

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 2494

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]/((g_.) + (h_.)*(x_)), x_Sym
bol] :> Simp[(Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/h, x] + (-Dist[(b*p*r)/h, Int[Log[g + h*x]/(a
+ b*x), x], x] - Dist[(d*q*r)/h, Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q,
r}, x] && NeQ[b*c - a*d, 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 {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{x \left (42+59 x+15 x^2\right )} \, dx\\ &=\int \left (-\frac {15 \log (x (3+x))}{14+15 x}-\frac {(3+2 x) \left (-2+\log \left (\frac {2}{31} (14+15 x)\right )\right )}{x (3+x)}\right ) \, dx\\ &=-\left (15 \int \frac {\log (x (3+x))}{14+15 x} \, dx\right )-\int \frac {(3+2 x) \left (-2+\log \left (\frac {2}{31} (14+15 x)\right )\right )}{x (3+x)} \, dx\\ &=-\log (x (3+x)) \log (14+15 x)-\int \left (\frac {-2+\log \left (\frac {2}{31} (14+15 x)\right )}{x}+\frac {-2+\log \left (\frac {2}{31} (14+15 x)\right )}{3+x}\right ) \, dx+\int \frac {\log (14+15 x)}{x} \, dx+\int \frac {\log (14+15 x)}{3+x} \, dx\\ &=\log (14) \log (x)+\log \left (\frac {15 (3+x)}{31}\right ) \log (14+15 x)-\log (x (3+x)) \log (14+15 x)-15 \int \frac {\log \left (\frac {15 (3+x)}{31}\right )}{14+15 x} \, dx+\int \frac {\log \left (1+\frac {15 x}{14}\right )}{x} \, dx-\int \frac {-2+\log \left (\frac {2}{31} (14+15 x)\right )}{x} \, dx-\int \frac {-2+\log \left (\frac {2}{31} (14+15 x)\right )}{3+x} \, dx\\ &=\left (2+\log \left (\frac {31}{28}\right )\right ) \log (x)+\log (14) \log (x)+\log \left (\frac {15 (3+x)}{31}\right ) \left (2-\log \left (\frac {2}{31} (14+15 x)\right )\right )+\log \left (\frac {15 (3+x)}{31}\right ) \log (14+15 x)-\log (x (3+x)) \log (14+15 x)-\text {Li}_2\left (-\frac {15 x}{14}\right )+15 \int \frac {\log \left (\frac {15 (3+x)}{31}\right )}{14+15 x} \, dx-\int \frac {\log \left (1+\frac {15 x}{14}\right )}{x} \, dx-\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{31}\right )}{x} \, dx,x,14+15 x\right )\\ &=\left (2+\log \left (\frac {31}{28}\right )\right ) \log (x)+\log (14) \log (x)+\log \left (\frac {15 (3+x)}{31}\right ) \left (2-\log \left (\frac {2}{31} (14+15 x)\right )\right )+\log \left (\frac {15 (3+x)}{31}\right ) \log (14+15 x)-\log (x (3+x)) \log (14+15 x)+\text {Li}_2\left (\frac {1}{31} (-14-15 x)\right )+\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{31}\right )}{x} \, dx,x,14+15 x\right )\\ &=\left (2+\log \left (\frac {31}{28}\right )\right ) \log (x)+\log (14) \log (x)+\log \left (\frac {15 (3+x)}{31}\right ) \left (2-\log \left (\frac {2}{31} (14+15 x)\right )\right )+\log \left (\frac {15 (3+x)}{31}\right ) \log (14+15 x)-\log (x (3+x)) \log (14+15 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 51, normalized size = 1.96 \begin {gather*} -\log \left (\frac {961}{225}\right )-\log \left (\frac {31}{15}\right ) \log \left (\frac {31}{2}\right )+\left (2+\log \left (\frac {31}{2}\right )\right ) \log (x)+\left (2+\log \left (\frac {31}{2}\right )\right ) \log (3+x)-\log (x (3+x)) \log (14+15 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(84 + 146*x + 60*x^2 + (-42 - 73*x - 30*x^2)*Log[(28 + 30*x)/31] + (-45*x - 15*x^2)*Log[3*x + x^2])/
(42*x + 59*x^2 + 15*x^3),x]

[Out]

-Log[961/225] - Log[31/15]*Log[31/2] + (2 + Log[31/2])*Log[x] + (2 + Log[31/2])*Log[3 + x] - Log[x*(3 + x)]*Lo
g[14 + 15*x]

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fricas [A]  time = 0.64, size = 18, normalized size = 0.69 \begin {gather*} -{\left (\log \left (\frac {30}{31} \, x + \frac {28}{31}\right ) - 2\right )} \log \left (x^{2} + 3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x^2-45*x)*log(x^2+3*x)+(-30*x^2-73*x-42)*log(30/31*x+28/31)+60*x^2+146*x+84)/(15*x^3+59*x^2+42
*x),x, algorithm="fricas")

[Out]

-(log(30/31*x + 28/31) - 2)*log(x^2 + 3*x)

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giac [B]  time = 0.22, size = 41, normalized size = 1.58 \begin {gather*} -\log \left (x^{2} + 3 \, x\right ) \log \left (15 \, x + 14\right ) + {\left (\log \left (31\right ) - \log \relax (2) + 2\right )} \log \left (x + 3\right ) + {\left (\log \left (31\right ) - \log \relax (2) + 2\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x^2-45*x)*log(x^2+3*x)+(-30*x^2-73*x-42)*log(30/31*x+28/31)+60*x^2+146*x+84)/(15*x^3+59*x^2+42
*x),x, algorithm="giac")

[Out]

-log(x^2 + 3*x)*log(15*x + 14) + (log(31) - log(2) + 2)*log(x + 3) + (log(31) - log(2) + 2)*log(x)

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maple [B]  time = 0.12, size = 45, normalized size = 1.73

method	result	size
default	\(\ln \left (31\right ) \ln \left (\left (3+x \right ) x \right )-\ln \left (15 x +14\right ) \ln \left (x^{2}+3 x \right )+2 \ln \left (\left (3+x \right ) x \right )-\ln \relax (2) \ln \left (\left (3+x \right ) x \right )\)	\(45\)
risch	\(2 \ln \left (3+x \right )+2 \ln \relax (x )-\frac {i \pi \ln \left (15 x +14\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (3+x \right )\right )^{2}}{2}+\frac {i \pi \ln \left (15 x +14\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (3+x \right )\right ) \mathrm {csgn}\left (i \left (3+x \right )\right )}{2}+\frac {i \pi \ln \left (15 x +14\right ) \mathrm {csgn}\left (i x \left (3+x \right )\right )^{3}}{2}-\frac {i \pi \ln \left (15 x +14\right ) \mathrm {csgn}\left (i x \left (3+x \right )\right )^{2} \mathrm {csgn}\left (i \left (3+x \right )\right )}{2}-\ln \relax (x ) \ln \left (\frac {15 x}{14}+1\right )-\left (\ln \left (3+x \right )-\ln \left (\frac {15 x}{31}+\frac {45}{31}\right )\right ) \ln \left (-\frac {14}{31}-\frac {15 x}{31}\right )-\left (\ln \left (\frac {30 x}{31}+\frac {28}{31}\right )-\ln \left (\frac {15 x}{14}+1\right )\right ) \ln \left (-\frac {15 x}{14}\right )-\ln \left (\frac {30 x}{31}+\frac {28}{31}\right ) \ln \left (\frac {15 x}{31}+\frac {45}{31}\right )\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-15*x^2-45*x)*ln(x^2+3*x)+(-30*x^2-73*x-42)*ln(30/31*x+28/31)+60*x^2+146*x+84)/(15*x^3+59*x^2+42*x),x,me
thod=_RETURNVERBOSE)

[Out]

ln(31)*ln((3+x)*x)-ln(15*x+14)*ln(x^2+3*x)+2*ln((3+x)*x)-ln(2)*ln((3+x)*x)

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maxima [B]  time = 0.50, size = 65, normalized size = 2.50 \begin {gather*} -\frac {1}{31} \, {\left (31 \, \log \left (x + 3\right ) + 31 \, \log \relax (x) - 90\right )} \log \left (15 \, x + 14\right ) + \frac {1}{31} \, {\left (31 \, \log \left (31\right ) - 31 \, \log \relax (2) + 34\right )} \log \left (x + 3\right ) + {\left (\log \left (31\right ) - \log \relax (2)\right )} \log \relax (x) - \frac {90}{31} \, \log \left (15 \, x + 14\right ) + \frac {28}{31} \, \log \left (x + 3\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x^2-45*x)*log(x^2+3*x)+(-30*x^2-73*x-42)*log(30/31*x+28/31)+60*x^2+146*x+84)/(15*x^3+59*x^2+42
*x),x, algorithm="maxima")

[Out]

-1/31*(31*log(x + 3) + 31*log(x) - 90)*log(15*x + 14) + 1/31*(31*log(31) - 31*log(2) + 34)*log(x + 3) + (log(3
1) - log(2))*log(x) - 90/31*log(15*x + 14) + 28/31*log(x + 3) + 2*log(x)

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mupad [B]  time = 5.59, size = 18, normalized size = 0.69 \begin {gather*} -\ln \left (x^2+3\,x\right )\,\left (\ln \left (\frac {30\,x}{31}+\frac {28}{31}\right )-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((146*x - log(3*x + x^2)*(45*x + 15*x^2) - log((30*x)/31 + 28/31)*(73*x + 30*x^2 + 42) + 60*x^2 + 84)/(42*x
+ 59*x^2 + 15*x^3),x)

[Out]

-log(3*x + x^2)*(log((30*x)/31 + 28/31) - 2)

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sympy [A]  time = 0.38, size = 27, normalized size = 1.04 \begin {gather*} - \log {\left (\frac {30 x}{31} + \frac {28}{31} \right )} \log {\left (x^{2} + 3 x \right )} + 2 \log {\left (x^{2} + 3 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x**2-45*x)*ln(x**2+3*x)+(-30*x**2-73*x-42)*ln(30/31*x+28/31)+60*x**2+146*x+84)/(15*x**3+59*x**
2+42*x),x)

[Out]

-log(30*x/31 + 28/31)*log(x**2 + 3*x) + 2*log(x**2 + 3*x)

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