3.78.100 \(\int \frac {(-34-34 x+e^4 (2 x+2 x^2)) \log (e^5 x)+(34 x+e^4 (x-x^2)) \log ^2(e^5 x)}{x+3 x^2+3 x^3+x^4} \, dx\)

Optimal. Leaf size=21 \[ \frac {\left (-17+e^4 x\right ) \log ^2\left (e^5 x\right )}{(1+x)^2} \]

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Rubi [C]  time = 0.70, antiderivative size = 255, normalized size of antiderivative = 12.14, number of steps used = 29, number of rules used = 14, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {6688, 6742, 44, 37, 2357, 2301, 2319, 2314, 31, 2317, 2391, 2347, 2344, 2318} \begin {gather*} -2 \left (17+e^4\right ) \text {Li}_2(-x)+2 e^4 \text {Li}_2(-x)+34 \text {Li}_2(-x)-\frac {15 e^4 x^2}{2 (x+1)^2}+\frac {10 \left (17+e^4\right )}{x+1}-\frac {170}{x+1}-\frac {5 \left (136+7 e^4\right )}{2 (x+1)^2}-\frac {85}{(x+1)^2}-\frac {e^4 x \log ^2(x)}{x+1}-\frac {\left (17+e^4\right ) \log ^2(x)}{(x+1)^2}+\left (17+e^4\right ) \log ^2(x)-17 \log ^2(x)-\frac {2 \left (17+e^4\right ) x \log (x)}{x+1}+\frac {2 \left (17-4 e^4\right ) x \log (x)}{x+1}-\frac {10 \left (17+e^4\right ) \log (x)}{(x+1)^2}+10 \left (17+e^4\right ) \log (x)-170 \log (x)-2 \left (17+e^4\right ) \log (x) \log (x+1)+2 e^4 \log (x) \log (x+1)+34 \log (x) \log (x+1)-8 \left (17+e^4\right ) \log (x+1)-2 \left (17-4 e^4\right ) \log (x+1)+170 \log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-34 - 34*x + E^4*(2*x + 2*x^2))*Log[E^5*x] + (34*x + E^4*(x - x^2))*Log[E^5*x]^2)/(x + 3*x^2 + 3*x^3 + x
^4),x]

[Out]

-85/(1 + x)^2 - (5*(136 + 7*E^4))/(2*(1 + x)^2) - (15*E^4*x^2)/(2*(1 + x)^2) - 170/(1 + x) + (10*(17 + E^4))/(
1 + x) - 170*Log[x] + 10*(17 + E^4)*Log[x] - (10*(17 + E^4)*Log[x])/(1 + x)^2 + (2*(17 - 4*E^4)*x*Log[x])/(1 +
x) - (2*(17 + E^4)*x*Log[x])/(1 + x) - 17*Log[x]^2 + (17 + E^4)*Log[x]^2 - ((17 + E^4)*Log[x]^2)/(1 + x)^2 -
(E^4*x*Log[x]^2)/(1 + x) + 170*Log[1 + x] - 2*(17 - 4*E^4)*Log[1 + x] - 8*(17 + E^4)*Log[1 + x] + 34*Log[x]*Lo
g[1 + x] + 2*E^4*Log[x]*Log[1 + x] - 2*(17 + E^4)*Log[x]*Log[1 + x] + 34*PolyLog[2, -x] + 2*E^4*PolyLog[2, -x]
- 2*(17 + E^4)*PolyLog[2, -x]

Rule 31

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 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 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2344

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

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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 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 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 {(5+\log (x)) \left (-34+\left (136+7 e^4\right ) x-3 e^4 x^2+\left (34-e^4 (-1+x)\right ) x \log (x)\right )}{x (1+x)^3} \, dx\\ &=\int \left (\frac {5 \left (136+7 e^4\right )}{(1+x)^3}-\frac {170}{x (1+x)^3}-\frac {15 e^4 x}{(1+x)^3}+\frac {2 \left (-17+3 \left (51+2 e^4\right ) x-4 e^4 x^2\right ) \log (x)}{x (1+x)^3}-\frac {\left (-34-e^4+e^4 x\right ) \log ^2(x)}{(1+x)^3}\right ) \, dx\\ &=-\frac {5 \left (136+7 e^4\right )}{2 (1+x)^2}+2 \int \frac {\left (-17+3 \left (51+2 e^4\right ) x-4 e^4 x^2\right ) \log (x)}{x (1+x)^3} \, dx-170 \int \frac {1}{x (1+x)^3} \, dx-\left (15 e^4\right ) \int \frac {x}{(1+x)^3} \, dx-\int \frac {\left (-34-e^4+e^4 x\right ) \log ^2(x)}{(1+x)^3} \, dx\\ &=-\frac {5 \left (136+7 e^4\right )}{2 (1+x)^2}-\frac {15 e^4 x^2}{2 (1+x)^2}+2 \int \left (-\frac {17 \log (x)}{x}+\frac {10 \left (17+e^4\right ) \log (x)}{(1+x)^3}+\frac {\left (17-4 e^4\right ) \log (x)}{(1+x)^2}+\frac {17 \log (x)}{1+x}\right ) \, dx-170 \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^3}-\frac {1}{(1+x)^2}\right ) \, dx-\int \left (-\frac {2 \left (17+e^4\right ) \log ^2(x)}{(1+x)^3}+\frac {e^4 \log ^2(x)}{(1+x)^2}\right ) \, dx\\ &=-\frac {85}{(1+x)^2}-\frac {5 \left (136+7 e^4\right )}{2 (1+x)^2}-\frac {15 e^4 x^2}{2 (1+x)^2}-\frac {170}{1+x}-170 \log (x)+170 \log (1+x)-34 \int \frac {\log (x)}{x} \, dx+34 \int \frac {\log (x)}{1+x} \, dx-e^4 \int \frac {\log ^2(x)}{(1+x)^2} \, dx+\left (2 \left (17-4 e^4\right )\right ) \int \frac {\log (x)}{(1+x)^2} \, dx+\left (2 \left (17+e^4\right )\right ) \int \frac {\log ^2(x)}{(1+x)^3} \, dx+\left (20 \left (17+e^4\right )\right ) \int \frac {\log (x)}{(1+x)^3} \, dx\\ &=-\frac {85}{(1+x)^2}-\frac {5 \left (136+7 e^4\right )}{2 (1+x)^2}-\frac {15 e^4 x^2}{2 (1+x)^2}-\frac {170}{1+x}-170 \log (x)-\frac {10 \left (17+e^4\right ) \log (x)}{(1+x)^2}+\frac {2 \left (17-4 e^4\right ) x \log (x)}{1+x}-17 \log ^2(x)-\frac {\left (17+e^4\right ) \log ^2(x)}{(1+x)^2}-\frac {e^4 x \log ^2(x)}{1+x}+170 \log (1+x)+34 \log (x) \log (1+x)-34 \int \frac {\log (1+x)}{x} \, dx+\left (2 e^4\right ) \int \frac {\log (x)}{1+x} \, dx-\left (2 \left (17-4 e^4\right )\right ) \int \frac {1}{1+x} \, dx+\left (2 \left (17+e^4\right )\right ) \int \frac {\log (x)}{x (1+x)^2} \, dx+\left (10 \left (17+e^4\right )\right ) \int \frac {1}{x (1+x)^2} \, dx\\ &=-\frac {85}{(1+x)^2}-\frac {5 \left (136+7 e^4\right )}{2 (1+x)^2}-\frac {15 e^4 x^2}{2 (1+x)^2}-\frac {170}{1+x}-170 \log (x)-\frac {10 \left (17+e^4\right ) \log (x)}{(1+x)^2}+\frac {2 \left (17-4 e^4\right ) x \log (x)}{1+x}-17 \log ^2(x)-\frac {\left (17+e^4\right ) \log ^2(x)}{(1+x)^2}-\frac {e^4 x \log ^2(x)}{1+x}+170 \log (1+x)-2 \left (17-4 e^4\right ) \log (1+x)+34 \log (x) \log (1+x)+2 e^4 \log (x) \log (1+x)+34 \text {Li}_2(-x)-\left (2 e^4\right ) \int \frac {\log (1+x)}{x} \, dx-\left (2 \left (17+e^4\right )\right ) \int \frac {\log (x)}{(1+x)^2} \, dx+\left (2 \left (17+e^4\right )\right ) \int \frac {\log (x)}{x (1+x)} \, dx+\left (10 \left (17+e^4\right )\right ) \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx\\ &=-\frac {85}{(1+x)^2}-\frac {5 \left (136+7 e^4\right )}{2 (1+x)^2}-\frac {15 e^4 x^2}{2 (1+x)^2}-\frac {170}{1+x}+\frac {10 \left (17+e^4\right )}{1+x}-170 \log (x)+10 \left (17+e^4\right ) \log (x)-\frac {10 \left (17+e^4\right ) \log (x)}{(1+x)^2}+\frac {2 \left (17-4 e^4\right ) x \log (x)}{1+x}-\frac {2 \left (17+e^4\right ) x \log (x)}{1+x}-17 \log ^2(x)-\frac {\left (17+e^4\right ) \log ^2(x)}{(1+x)^2}-\frac {e^4 x \log ^2(x)}{1+x}+170 \log (1+x)-2 \left (17-4 e^4\right ) \log (1+x)-10 \left (17+e^4\right ) \log (1+x)+34 \log (x) \log (1+x)+2 e^4 \log (x) \log (1+x)+34 \text {Li}_2(-x)+2 e^4 \text {Li}_2(-x)+\left (2 \left (17+e^4\right )\right ) \int \frac {1}{1+x} \, dx+\left (2 \left (17+e^4\right )\right ) \int \frac {\log (x)}{x} \, dx-\left (2 \left (17+e^4\right )\right ) \int \frac {\log (x)}{1+x} \, dx\\ &=-\frac {85}{(1+x)^2}-\frac {5 \left (136+7 e^4\right )}{2 (1+x)^2}-\frac {15 e^4 x^2}{2 (1+x)^2}-\frac {170}{1+x}+\frac {10 \left (17+e^4\right )}{1+x}-170 \log (x)+10 \left (17+e^4\right ) \log (x)-\frac {10 \left (17+e^4\right ) \log (x)}{(1+x)^2}+\frac {2 \left (17-4 e^4\right ) x \log (x)}{1+x}-\frac {2 \left (17+e^4\right ) x \log (x)}{1+x}-17 \log ^2(x)+\left (17+e^4\right ) \log ^2(x)-\frac {\left (17+e^4\right ) \log ^2(x)}{(1+x)^2}-\frac {e^4 x \log ^2(x)}{1+x}+170 \log (1+x)-2 \left (17-4 e^4\right ) \log (1+x)-8 \left (17+e^4\right ) \log (1+x)+34 \log (x) \log (1+x)+2 e^4 \log (x) \log (1+x)-2 \left (17+e^4\right ) \log (x) \log (1+x)+34 \text {Li}_2(-x)+2 e^4 \text {Li}_2(-x)+\left (2 \left (17+e^4\right )\right ) \int \frac {\log (1+x)}{x} \, dx\\ &=-\frac {85}{(1+x)^2}-\frac {5 \left (136+7 e^4\right )}{2 (1+x)^2}-\frac {15 e^4 x^2}{2 (1+x)^2}-\frac {170}{1+x}+\frac {10 \left (17+e^4\right )}{1+x}-170 \log (x)+10 \left (17+e^4\right ) \log (x)-\frac {10 \left (17+e^4\right ) \log (x)}{(1+x)^2}+\frac {2 \left (17-4 e^4\right ) x \log (x)}{1+x}-\frac {2 \left (17+e^4\right ) x \log (x)}{1+x}-17 \log ^2(x)+\left (17+e^4\right ) \log ^2(x)-\frac {\left (17+e^4\right ) \log ^2(x)}{(1+x)^2}-\frac {e^4 x \log ^2(x)}{1+x}+170 \log (1+x)-2 \left (17-4 e^4\right ) \log (1+x)-8 \left (17+e^4\right ) \log (1+x)+34 \log (x) \log (1+x)+2 e^4 \log (x) \log (1+x)-2 \left (17+e^4\right ) \log (x) \log (1+x)+34 \text {Li}_2(-x)+2 e^4 \text {Li}_2(-x)-2 \left (17+e^4\right ) \text {Li}_2(-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 19, normalized size = 0.90 \begin {gather*} \frac {\left (-17+e^4 x\right ) (5+\log (x))^2}{(1+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-34 - 34*x + E^4*(2*x + 2*x^2))*Log[E^5*x] + (34*x + E^4*(x - x^2))*Log[E^5*x]^2)/(x + 3*x^2 + 3*x
^3 + x^4),x]

[Out]

((-17 + E^4*x)*(5 + Log[x])^2)/(1 + x)^2

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fricas [A]  time = 0.79, size = 24, normalized size = 1.14 \begin {gather*} \frac {{\left (x e^{4} - 17\right )} \log \left (x e^{5}\right )^{2}}{x^{2} + 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*exp(4)+34*x)*log(x*exp(5))^2+((2*x^2+2*x)*exp(4)-34*x-34)*log(x*exp(5)))/(x^4+3*x^3+3*x^2
+x),x, algorithm="fricas")

[Out]

(x*e^4 - 17)*log(x*e^5)^2/(x^2 + 2*x + 1)

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giac [B]  time = 0.16, size = 43, normalized size = 2.05 \begin {gather*} \frac {x e^{4} \log \relax (x)^{2} + 10 \, x e^{4} \log \relax (x) + 25 \, x e^{4} - 17 \, \log \relax (x)^{2} - 170 \, \log \relax (x) - 425}{x^{2} + 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*exp(4)+34*x)*log(x*exp(5))^2+((2*x^2+2*x)*exp(4)-34*x-34)*log(x*exp(5)))/(x^4+3*x^3+3*x^2
+x),x, algorithm="giac")

[Out]

(x*e^4*log(x)^2 + 10*x*e^4*log(x) + 25*x*e^4 - 17*log(x)^2 - 170*log(x) - 425)/(x^2 + 2*x + 1)

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maple [A]  time = 0.29, size = 25, normalized size = 1.19

method	result	size
risch	\(\frac {\left (x \,{\mathrm e}^{4}-17\right ) \ln \left (x \,{\mathrm e}^{5}\right )^{2}}{x^{2}+2 x +1}\)	\(25\)
norman	\(\frac {x \,{\mathrm e}^{4} \ln \left (x \,{\mathrm e}^{5}\right )^{2}-17 \ln \left (x \,{\mathrm e}^{5}\right )^{2}}{\left (x +1\right )^{2}}\)	\(28\)
derivativedivides	error in convert/parfrac: numeric exception: division by zero\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2+x)*exp(4)+34*x)*ln(x*exp(5))^2+((2*x^2+2*x)*exp(4)-34*x-34)*ln(x*exp(5)))/(x^4+3*x^3+3*x^2+x),x,me
thod=_RETURNVERBOSE)

[Out]

(x*exp(4)-17)/(x^2+2*x+1)*ln(x*exp(5))^2

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maxima [B]  time = 0.43, size = 39, normalized size = 1.86 \begin {gather*} \frac {{\left (x e^{4} - 17\right )} \log \relax (x)^{2} + 25 \, x e^{4} + 10 \, {\left (x e^{4} - 17\right )} \log \relax (x) - 425}{x^{2} + 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*exp(4)+34*x)*log(x*exp(5))^2+((2*x^2+2*x)*exp(4)-34*x-34)*log(x*exp(5)))/(x^4+3*x^3+3*x^2
+x),x, algorithm="maxima")

[Out]

((x*e^4 - 17)*log(x)^2 + 25*x*e^4 + 10*(x*e^4 - 17)*log(x) - 425)/(x^2 + 2*x + 1)

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mupad [B]  time = 5.26, size = 19, normalized size = 0.90 \begin {gather*} \frac {{\ln \left (x\,{\mathrm {e}}^5\right )}^2\,\left (x\,{\mathrm {e}}^4-17\right )}{{\left (x+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x*exp(5))*(34*x - exp(4)*(2*x + 2*x^2) + 34) - log(x*exp(5))^2*(34*x + exp(4)*(x - x^2)))/(x + 3*x^2
+ 3*x^3 + x^4),x)

[Out]

(log(x*exp(5))^2*(x*exp(4) - 17))/(x + 1)^2

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sympy [A]  time = 0.22, size = 22, normalized size = 1.05 \begin {gather*} \frac {\left (x e^{4} - 17\right ) \log {\left (x e^{5} \right )}^{2}}{x^{2} + 2 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2+x)*exp(4)+34*x)*ln(x*exp(5))**2+((2*x**2+2*x)*exp(4)-34*x-34)*ln(x*exp(5)))/(x**4+3*x**3+3*
x**2+x),x)

[Out]

(x*exp(4) - 17)*log(x*exp(5))**2/(x**2 + 2*x + 1)

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