∫ −180x2+60x3+e1 _4 (3+4x)(−36x2+24x3−3x4)+ex(36x2−24x3+3x4) ________________________________________________________________________________________ 25−10ex+e2x+e1

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Rubi [A] time = 3.10, antiderivative size = 47, normalized size of antiderivative = 1.96, number of steps used = 54, number of rules used = 11, integrand size = 104, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6688, 12, 6742, 2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191} \begin {gather*} \frac {3 x^4}{5-\left (1-e^{3/4}\right ) e^x}-\frac {12 x^3}{5-\left (1-e^{3/4}\right ) e^x} \end {gather*}

Int[(-180*x^2 + 60*x^3 + E^((3 + 4*x)/4)*(-36*x^2 + 24*x^3 - 3*x^4) + E^x*(36*x^2 - 24*x^3 + 3*x^4))/(25 - 10*

E^x + E^(2*x) + E^((3 + 4*x)/2) + E^((3 + 4*x)/4)*(10 - 2*E^x)),x]

(-12*x^3)/(5 - (1 - E^(3/4))*E^x) + (3*x^4)/(5 - (1 - E^(3/4))*E^x)

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis

t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)

))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0

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,

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*

((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1))/(b*f*g*n*(p +

1)*Log[F]), x] - Dist[(d*m)/(b*f*g*n*(p + 1)*Log[F]), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1

), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

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

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]

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]

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,

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

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x^2 \left (20 (-3+x)+\left (1-e^{3/4}\right ) e^x \left (12-8 x+x^2\right )\right )}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2} \, dx\\ &=3 \int \frac {x^2 \left (20 (-3+x)+\left (1-e^{3/4}\right ) e^x \left (12-8 x+x^2\right )\right )}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2} \, dx\\ &=3 \int \left (\frac {5 (-4+x) x^3}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2}+\frac {x^2 \left (-12+8 x-x^2\right )}{5-\left (1-e^{3/4}\right ) e^x}\right ) \, dx\\ &=3 \int \frac {x^2 \left (-12+8 x-x^2\right )}{5-\left (1-e^{3/4}\right ) e^x} \, dx+15 \int \frac {(-4+x) x^3}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2} \, dx\\ &=3 \int \left (\frac {12 x^2}{-5+\left (1-e^{3/4}\right ) e^x}+\frac {8 x^3}{5-\left (1-e^{3/4}\right ) e^x}+\frac {x^4}{-5+\left (1-e^{3/4}\right ) e^x}\right ) \, dx+15 \int \left (-\frac {4 x^3}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2}+\frac {x^4}{\left (5-\left (1-e^{3/4}\right ) e^x\right )^2}\right ) \, dx\\ &=3 \int \frac {x^4}{-5+\left (1-e^{3/4}\right ) e^x} \, dx+15 \int \frac {x^4}{\left (5+\left (-1+e^{3/4}\right ) e^x\right )^2} \, dx+24 \int \frac {x^3}{5+\left (-1+e^{3/4}\right ) e^x} \, dx+36 \int \frac {x^2}{-5+\left (1-e^{3/4}\right ) e^x} \, dx-60 \int \frac {x^3}{\left (5+\left (-1+e^{3/4}\right ) e^x\right )^2} \, dx\\ &=-\frac {12 x^3}{5}+\frac {6 x^4}{5}-\frac {3 x^5}{25}+3 \int \frac {x^4}{5+\left (-1+e^{3/4}\right ) e^x} \, dx-12 \int \frac {x^3}{5+\left (-1+e^{3/4}\right ) e^x} \, dx+\frac {1}{5} \left (3 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^4}{-5+\left (1-e^{3/4}\right ) e^x} \, dx+\left (3 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^4}{\left (5+\left (-1+e^{3/4}\right ) e^x\right )^2} \, dx+\frac {1}{5} \left (24 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^3}{5+\left (-1+e^{3/4}\right ) e^x} \, dx+\frac {1}{5} \left (36 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^2}{-5+\left (1-e^{3/4}\right ) e^x} \, dx-\left (12 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^3}{\left (5+\left (-1+e^{3/4}\right ) e^x\right )^2} \, dx\\ &=-\frac {12 x^3}{5}-\frac {12 x^3}{5-\left (1-e^{3/4}\right ) e^x}+\frac {3 x^4}{5}+\frac {3 x^4}{5-\left (1-e^{3/4}\right ) e^x}+\frac {36}{5} x^2 \log \left (1-\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {24}{5} x^3 \log \left (1-\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {3}{5} x^4 \log \left (1-\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {12}{5} \int x^3 \log \left (1-\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx-12 \int \frac {x^3}{5+\left (-1+e^{3/4}\right ) e^x} \, dx-\frac {72}{5} \int x \log \left (1-\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx+\frac {72}{5} \int x^2 \log \left (1+\frac {1}{5} \left (-1+e^{3/4}\right ) e^x\right ) \, dx+36 \int \frac {x^2}{5+\left (-1+e^{3/4}\right ) e^x} \, dx+\frac {1}{5} \left (3 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^4}{5+\left (-1+e^{3/4}\right ) e^x} \, dx-\frac {1}{5} \left (12 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^3}{5+\left (-1+e^{3/4}\right ) e^x} \, dx\\ &=-\frac {12 x^3}{5-\left (1-e^{3/4}\right ) e^x}+\frac {3 x^4}{5-\left (1-e^{3/4}\right ) e^x}+\frac {36}{5} x^2 \log \left (1-\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {12}{5} x^3 \log \left (1-\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {72}{5} x \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {72}{5} x^2 \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {12}{5} x^3 \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {12}{5} \int x^3 \log \left (1+\frac {1}{5} \left (-1+e^{3/4}\right ) e^x\right ) \, dx-\frac {36}{5} \int x^2 \log \left (1+\frac {1}{5} \left (-1+e^{3/4}\right ) e^x\right ) \, dx-\frac {36}{5} \int x^2 \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx-\frac {72}{5} \int \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx+\frac {144}{5} \int x \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx-\frac {1}{5} \left (12 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^3}{5+\left (-1+e^{3/4}\right ) e^x} \, dx+\frac {1}{5} \left (36 \left (1-e^{3/4}\right )\right ) \int \frac {e^x x^2}{5+\left (-1+e^{3/4}\right ) e^x} \, dx\\ &=-\frac {12 x^3}{5-\left (1-e^{3/4}\right ) e^x}+\frac {3 x^4}{5-\left (1-e^{3/4}\right ) e^x}+\frac {72}{5} x \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {36}{5} x^2 \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {144}{5} x \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {36}{5} x^2 \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {36}{5} \int x^2 \log \left (1+\frac {1}{5} \left (-1+e^{3/4}\right ) e^x\right ) \, dx+\frac {36}{5} \int x^2 \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx+\frac {72}{5} \int x \log \left (1+\frac {1}{5} \left (-1+e^{3/4}\right ) e^x\right ) \, dx-\frac {72}{5} \int x \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx+\frac {72}{5} \int x \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx-\frac {72}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) x\right )}{x} \, dx,x,e^x\right )-\frac {144}{5} \int \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx\\ &=-\frac {12 x^3}{5-\left (1-e^{3/4}\right ) e^x}+\frac {3 x^4}{5-\left (1-e^{3/4}\right ) e^x}-\frac {72}{5} \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {72}{5} x \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {72}{5} x \text {Li}_4\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {72}{5} \int \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx-\frac {72}{5} \int x \text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx+\frac {72}{5} \int \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx-\frac {72}{5} \int x \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx-\frac {72}{5} \int \text {Li}_4\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx-\frac {144}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) x\right )}{x} \, dx,x,e^x\right )\\ &=-\frac {12 x^3}{5-\left (1-e^{3/4}\right ) e^x}+\frac {3 x^4}{5-\left (1-e^{3/4}\right ) e^x}-\frac {72}{5} \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {144}{5} \text {Li}_4\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {72}{5} \int \text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx+\frac {72}{5} \int \text {Li}_4\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right ) \, dx+\frac {72}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {1}{5} \left (1-e^{3/4}\right ) x\right )}{x} \, dx,x,e^x\right )+\frac {72}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) x\right )}{x} \, dx,x,e^x\right )-\frac {72}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (\frac {1}{5} \left (1-e^{3/4}\right ) x\right )}{x} \, dx,x,e^x\right )\\ &=-\frac {12 x^3}{5-\left (1-e^{3/4}\right ) e^x}+\frac {3 x^4}{5-\left (1-e^{3/4}\right ) e^x}-\frac {72}{5} \text {Li}_4\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )-\frac {72}{5} \text {Li}_5\left (\frac {1}{5} \left (1-e^{3/4}\right ) e^x\right )+\frac {72}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {1}{5} \left (1-e^{3/4}\right ) x\right )}{x} \, dx,x,e^x\right )+\frac {72}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (\frac {1}{5} \left (1-e^{3/4}\right ) x\right )}{x} \, dx,x,e^x\right )\\ &=-\frac {12 x^3}{5-\left (1-e^{3/4}\right ) e^x}+\frac {3 x^4}{5-\left (1-e^{3/4}\right ) e^x}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.37, size = 24, normalized size = 1.00 \begin {gather*} \frac {3 (-4+x) x^3}{5-e^x+e^{\frac {3}{4}+x}} \end {gather*}

Integrate[(-180*x^2 + 60*x^3 + E^((3 + 4*x)/4)*(-36*x^2 + 24*x^3 - 3*x^4) + E^x*(36*x^2 - 24*x^3 + 3*x^4))/(25

- 10*E^x + E^(2*x) + E^((3 + 4*x)/2) + E^((3 + 4*x)/4)*(10 - 2*E^x)),x]

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fricas [A] time = 0.73, size = 29, normalized size = 1.21 \begin {gather*} \frac {3 \, {\left (x^{4} - 4 \, x^{3}\right )} e^{\frac {3}{4}}}{{\left (e^{\frac {3}{4}} - 1\right )} e^{\left (x + \frac {3}{4}\right )} + 5 \, e^{\frac {3}{4}}} \end {gather*}

integrate(((-3*x^4+24*x^3-36*x^2)*exp(3/4+x)+(3*x^4-24*x^3+36*x^2)*exp(x)+60*x^3-180*x^2)/(exp(3/4+x)^2+(-2*ex

p(x)+10)*exp(3/4+x)+exp(x)^2-10*exp(x)+25),x, algorithm="fricas")

3*(x^4 - 4*x^3)*e^(3/4)/((e^(3/4) - 1)*e^(x + 3/4) + 5*e^(3/4))

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giac [A] time = 0.15, size = 23, normalized size = 0.96 \begin {gather*} \frac {3 \, {\left (x^{4} - 4 \, x^{3}\right )}}{e^{\left (x + \frac {3}{4}\right )} - e^{x} + 5} \end {gather*}

integrate(((-3*x^4+24*x^3-36*x^2)*exp(3/4+x)+(3*x^4-24*x^3+36*x^2)*exp(x)+60*x^3-180*x^2)/(exp(3/4+x)^2+(-2*ex

p(x)+10)*exp(3/4+x)+exp(x)^2-10*exp(x)+25),x, algorithm="giac")

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method	result	size

risch	\(\frac {3 x^{3} \left (x -4\right )}{5-{\mathrm e}^{x}+{\mathrm e}^{\frac {3}{4}+x}}\)	\(21\)
norman	\(\frac {3 x^{4}-12 x^{3}}{{\mathrm e}^{\frac {3}{4}} {\mathrm e}^{x}-{\mathrm e}^{x}+5}\)	\(26\)

int(((-3*x^4+24*x^3-36*x^2)*exp(3/4+x)+(3*x^4-24*x^3+36*x^2)*exp(x)+60*x^3-180*x^2)/(exp(3/4+x)^2+(-2*exp(x)+1

0)*exp(3/4+x)+exp(x)^2-10*exp(x)+25),x,method=_RETURNVERBOSE)

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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

integrate(((-3*x^4+24*x^3-36*x^2)*exp(3/4+x)+(3*x^4-24*x^3+36*x^2)*exp(x)+60*x^3-180*x^2)/(exp(3/4+x)^2+(-2*ex

p(x)+10)*exp(3/4+x)+exp(x)^2-10*exp(x)+25),x, algorithm="maxima")

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B] time = 2.20, size = 19, normalized size = 0.79 \begin {gather*} \frac {3\,x^3\,\left (x-4\right )}{{\mathrm {e}}^x\,\left ({\mathrm {e}}^{3/4}-1\right )+5} \end {gather*}

int((exp(x)*(36*x^2 - 24*x^3 + 3*x^4) - exp(x + 3/4)*(36*x^2 - 24*x^3 + 3*x^4) - 180*x^2 + 60*x^3)/(exp(2*x) +

exp(2*x + 3/2) - 10*exp(x) - exp(x + 3/4)*(2*exp(x) - 10) + 25),x)

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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

integrate(((-3*x**4+24*x**3-36*x**2)*exp(3/4+x)+(3*x**4-24*x**3+36*x**2)*exp(x)+60*x**3-180*x**2)/(exp(3/4+x)*

*2+(-2*exp(x)+10)*exp(3/4+x)+exp(x)**2-10*exp(x)+25),x)

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3.37.1 \(\int \frac {-180 x^2+60 x^3+e^{\frac {1}{4} (3+4 x)} (-36 x^2+24 x^3-3 x^4)+e^x (36 x^2-24 x^3+3 x^4)}{25-10 e^x+e^{2 x}+e^{\frac {1}{2} (3+4 x)}+e^{\frac {1}{4} (3+4 x)} (10-2 e^x)} \, dx\)