3.90.100 \(\int \frac {35 x^3+7 x^4+e^{3 x} (5 x^3-2 x^4)+(28 x^3+4 e^{3 x} x^3) \log (\frac {3 e^x x}{28+4 e^{3 x}})}{7+e^{3 x}} \, dx\)

Optimal. Leaf size=24 \[ x^4 \left (1+\log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\right ) \]

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Rubi [A]  time = 0.72, antiderivative size = 26, normalized size of antiderivative = 1.08, number of steps used = 23, number of rules used = 10, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {6742, 2184, 2190, 2531, 6609, 2282, 6589, 14, 43, 2551} \begin {gather*} x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (e^{3 x}+7\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(35*x^3 + 7*x^4 + E^(3*x)*(5*x^3 - 2*x^4) + (28*x^3 + 4*E^(3*x)*x^3)*Log[(3*E^x*x)/(28 + 4*E^(3*x))])/(7 +
E^(3*x)),x]

[Out]

x^4 + x^4*Log[(3*E^x*x)/(4*(7 + E^(3*x)))]

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 2184

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n)
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2190

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 6589

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

Rule 6609

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

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 \left (\frac {21 x^4}{7+e^{3 x}}-x^3 \left (-5+2 x-4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\right )\right ) \, dx\\ &=21 \int \frac {x^4}{7+e^{3 x}} \, dx-\int x^3 \left (-5+2 x-4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\right ) \, dx\\ &=\frac {3 x^5}{5}-3 \int \frac {e^{3 x} x^4}{7+e^{3 x}} \, dx-\int \left (x^3 (-5+2 x)-4 x^3 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\right ) \, dx\\ &=\frac {3 x^5}{5}-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+4 \int x^3 \log \left (1+\frac {e^{3 x}}{7}\right ) \, dx+4 \int x^3 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right ) \, dx-\int x^3 (-5+2 x) \, dx\\ &=\frac {3 x^5}{5}-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+4 \int x^2 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right ) \, dx-\int \left (-5 x^3+2 x^4\right ) \, dx-\int \frac {x^3 \left (7 (1+x)-e^{3 x} (-1+2 x)\right )}{7+e^{3 x}} \, dx\\ &=\frac {5 x^4}{4}+\frac {x^5}{5}-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{3} \int x \text {Li}_3\left (-\frac {e^{3 x}}{7}\right ) \, dx-\int \left (x^3-2 x^4+\frac {21 x^4}{7+e^{3 x}}\right ) \, dx\\ &=x^4+\frac {3 x^5}{5}-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{9} \int \text {Li}_4\left (-\frac {e^{3 x}}{7}\right ) \, dx-21 \int \frac {x^4}{7+e^{3 x}} \, dx\\ &=x^4-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{27} \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {x}{7}\right )}{x} \, dx,x,e^{3 x}\right )+3 \int \frac {e^{3 x} x^4}{7+e^{3 x}} \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )-4 \int x^3 \log \left (1+\frac {e^{3 x}}{7}\right ) \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )-4 \int x^2 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right ) \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{3} \int x \text {Li}_3\left (-\frac {e^{3 x}}{7}\right ) \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} \int \text {Li}_4\left (-\frac {e^{3 x}}{7}\right ) \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{27} \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {x}{7}\right )}{x} \, dx,x,e^{3 x}\right )\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 26, normalized size = 1.08 \begin {gather*} x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(35*x^3 + 7*x^4 + E^(3*x)*(5*x^3 - 2*x^4) + (28*x^3 + 4*E^(3*x)*x^3)*Log[(3*E^x*x)/(28 + 4*E^(3*x))]
)/(7 + E^(3*x)),x]

[Out]

x^4 + x^4*Log[(3*E^x*x)/(4*(7 + E^(3*x)))]

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fricas [A]  time = 0.52, size = 22, normalized size = 0.92 \begin {gather*} x^{4} \log \left (\frac {3 \, x e^{x}}{4 \, {\left (e^{\left (3 \, x\right )} + 7\right )}}\right ) + x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(3*x)+28*x^3)*log(3*x*exp(x)/(4*exp(3*x)+28))+(-2*x^4+5*x^3)*exp(3*x)+7*x^4+35*x^3)/(exp(
3*x)+7),x, algorithm="fricas")

[Out]

x^4*log(3/4*x*e^x/(e^(3*x) + 7)) + x^4

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giac [A]  time = 0.20, size = 23, normalized size = 0.96 \begin {gather*} x^{5} + x^{4} \log \left (\frac {3 \, x}{4 \, {\left (e^{\left (3 \, x\right )} + 7\right )}}\right ) + x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(3*x)+28*x^3)*log(3*x*exp(x)/(4*exp(3*x)+28))+(-2*x^4+5*x^3)*exp(3*x)+7*x^4+35*x^3)/(exp(
3*x)+7),x, algorithm="giac")

[Out]

x^5 + x^4*log(3/4*x/(e^(3*x) + 7)) + x^4

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maple [C]  time = 0.23, size = 301, normalized size = 12.54

method	result	size
risch	\(x^{4} \ln \left ({\mathrm e}^{x}\right )-x^{4} \ln \left ({\mathrm e}^{3 x}+7\right )+x^{4} \ln \relax (x )-\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{3}}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3 x}+7}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{2}}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{2}}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{2}}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{2}}{2}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{3}}{2}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3 x}+7}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )}{2}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )}{2}+x^{4} \ln \relax (3)-2 x^{4} \ln \relax (2)+x^{4}\)	\(301\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3*exp(3*x)+28*x^3)*ln(3*x*exp(x)/(4*exp(3*x)+28))+(-2*x^4+5*x^3)*exp(3*x)+7*x^4+35*x^3)/(exp(3*x)+7)
,x,method=_RETURNVERBOSE)

[Out]

x^4*ln(exp(x))-x^4*ln(exp(3*x)+7)+x^4*ln(x)-1/2*I*Pi*x^4*csgn(I*x/(exp(3*x)+7)*exp(x))^3+1/2*I*Pi*x^4*csgn(I/(
exp(3*x)+7))*csgn(I*exp(x)/(exp(3*x)+7))^2+1/2*I*Pi*x^4*csgn(I*exp(x)/(exp(3*x)+7))*csgn(I*x/(exp(3*x)+7)*exp(
x))^2+1/2*I*Pi*x^4*csgn(I*exp(x))*csgn(I*exp(x)/(exp(3*x)+7))^2+1/2*I*Pi*x^4*csgn(I*x)*csgn(I*x/(exp(3*x)+7)*e
xp(x))^2-1/2*I*Pi*x^4*csgn(I*exp(x)/(exp(3*x)+7))^3-1/2*I*Pi*x^4*csgn(I*exp(x))*csgn(I/(exp(3*x)+7))*csgn(I*ex
p(x)/(exp(3*x)+7))-1/2*I*Pi*x^4*csgn(I*x)*csgn(I*exp(x)/(exp(3*x)+7))*csgn(I*x/(exp(3*x)+7)*exp(x))+x^4*ln(3)-
2*x^4*ln(2)+x^4

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maxima [A]  time = 0.47, size = 34, normalized size = 1.42 \begin {gather*} x^{5} + x^{4} {\left (\log \relax (3) - 2 \, \log \relax (2) + 1\right )} + x^{4} \log \relax (x) - x^{4} \log \left (e^{\left (3 \, x\right )} + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(3*x)+28*x^3)*log(3*x*exp(x)/(4*exp(3*x)+28))+(-2*x^4+5*x^3)*exp(3*x)+7*x^4+35*x^3)/(exp(
3*x)+7),x, algorithm="maxima")

[Out]

x^5 + x^4*(log(3) - 2*log(2) + 1) + x^4*log(x) - x^4*log(e^(3*x) + 7)

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mupad [B]  time = 7.20, size = 33, normalized size = 1.38 \begin {gather*} x^4\,\ln \relax (x)-x^4\,\ln \left (4\,{\mathrm {e}}^{3\,x}+28\right )+x^4\,\ln \relax (3)+x^4+x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*x)*(5*x^3 - 2*x^4) + 35*x^3 + 7*x^4 + log((3*x*exp(x))/(4*exp(3*x) + 28))*(4*x^3*exp(3*x) + 28*x^3)
)/(exp(3*x) + 7),x)

[Out]

x^4*log(x) - x^4*log(4*exp(3*x) + 28) + x^4*log(3) + x^4 + x^5

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sympy [A]  time = 0.39, size = 22, normalized size = 0.92 \begin {gather*} x^{4} \log {\left (\frac {3 x e^{x}}{4 e^{3 x} + 28} \right )} + x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3*exp(3*x)+28*x**3)*ln(3*x*exp(x)/(4*exp(3*x)+28))+(-2*x**4+5*x**3)*exp(3*x)+7*x**4+35*x**3)/
(exp(3*x)+7),x)

[Out]

x**4*log(3*x*exp(x)/(4*exp(3*x) + 28)) + x**4

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