\(\int \frac {300+690 x+204 x^2+18 x^3+e^x (-500-2150 x-1215 x^2-240 x^3-15 x^4)+(-300-120 x-12 x^2+e^x (500+700 x+220 x^2+20 x^3)) \log (x)}{100+40 x+4 x^2} \, dx\) [10242]

Optimal result

Integrand size = 84, antiderivative size = 26 \[ \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{100+40 x+4 x^2} \, dx=\left (-3+5 e^x\right ) x \left (-2-\frac {3 x}{4}-\frac {x}{5+x}+\log (x)\right ) \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(26)=52\).

Time = 1.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.96, number of steps used = 58, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {27, 12, 6874, 45, 2351, 31, 2384, 2354, 2438, 2393, 2332, 6820, 2208, 2209, 2230, 2225, 2207, 2634} \[ \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{100+40 x+4 x^2} \, dx=-\frac {15}{4} e^x x^2+\frac {9 x^2}{4}+\frac {3 x^2 \log (x)}{x+5}-15 e^x x+9 x+25 e^x-\frac {125 e^x}{x+5}+\frac {75}{x+5}+\frac {15 x \log (x)}{x+5}-6 x \log (x)-5 e^x \log (x)+5 e^x (x+1) \log (x)-30 \log \left (\frac {x}{5}+1\right ) (\log (x)+1)+15 \log \left (\frac {x}{5}+1\right ) (2 \log (x)+1)+15 \log (x+5) \]

[In]

Int[(300 + 690*x + 204*x^2 + 18*x^3 + E^x*(-500 - 2150*x - 1215*x^2 - 240*x^3 - 15*x^4) + (-300 - 120*x - 12*x
^2 + E^x*(500 + 700*x + 220*x^2 + 20*x^3))*Log[x])/(100 + 40*x + 4*x^2),x]

[Out]

25*E^x + 9*x - 15*E^x*x + (9*x^2)/4 - (15*E^x*x^2)/4 + 75/(5 + x) - (125*E^x)/(5 + x) - 5*E^x*Log[x] - 6*x*Log
[x] + 5*E^x*(1 + x)*Log[x] + (15*x*Log[x])/(5 + x) + (3*x^2*Log[x])/(5 + x) - 30*Log[1 + x/5]*(1 + Log[x]) + 1
5*Log[1 + x/5]*(1 + 2*Log[x]) + 15*Log[5 + x]

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

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

Rule 45

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 2207

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

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2230

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !TrueQ[$UseGamma]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2351

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 2354

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 2384

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

Rule 2393

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 2438

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 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{4 (5+x)^2} \, dx \\ & = \frac {1}{4} \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{(5+x)^2} \, dx \\ & = \frac {1}{4} \int \left (\frac {300}{(5+x)^2}+\frac {690 x}{(5+x)^2}+\frac {204 x^2}{(5+x)^2}+\frac {18 x^3}{(5+x)^2}-\frac {300 \log (x)}{(5+x)^2}-\frac {120 x \log (x)}{(5+x)^2}-\frac {12 x^2 \log (x)}{(5+x)^2}-\frac {5 e^x \left (100+430 x+243 x^2+48 x^3+3 x^4-100 \log (x)-140 x \log (x)-44 x^2 \log (x)-4 x^3 \log (x)\right )}{(5+x)^2}\right ) \, dx \\ & = -\frac {75}{5+x}-\frac {5}{4} \int \frac {e^x \left (100+430 x+243 x^2+48 x^3+3 x^4-100 \log (x)-140 x \log (x)-44 x^2 \log (x)-4 x^3 \log (x)\right )}{(5+x)^2} \, dx-3 \int \frac {x^2 \log (x)}{(5+x)^2} \, dx+\frac {9}{2} \int \frac {x^3}{(5+x)^2} \, dx-30 \int \frac {x \log (x)}{(5+x)^2} \, dx+51 \int \frac {x^2}{(5+x)^2} \, dx-75 \int \frac {\log (x)}{(5+x)^2} \, dx+\frac {345}{2} \int \frac {x}{(5+x)^2} \, dx \\ & = -\frac {75}{5+x}+\frac {15 x \log (x)}{5+x}+\frac {3 x^2 \log (x)}{5+x}-\frac {5}{4} \int \frac {e^x \left (100+430 x+243 x^2+48 x^3+3 x^4-4 (1+x) (5+x)^2 \log (x)\right )}{(5+x)^2} \, dx-3 \int \frac {x (1+2 \log (x))}{5+x} \, dx+\frac {9}{2} \int \left (-10+x-\frac {125}{(5+x)^2}+\frac {75}{5+x}\right ) \, dx+15 \int \frac {1}{5+x} \, dx-30 \int \frac {1+\log (x)}{5+x} \, dx+51 \int \left (1+\frac {25}{(5+x)^2}-\frac {10}{5+x}\right ) \, dx+\frac {345}{2} \int \left (-\frac {5}{(5+x)^2}+\frac {1}{5+x}\right ) \, dx \\ & = 6 x+\frac {9 x^2}{4}+\frac {75}{5+x}+\frac {15 x \log (x)}{5+x}+\frac {3 x^2 \log (x)}{5+x}-30 \log \left (1+\frac {x}{5}\right ) (1+\log (x))+15 \log (5+x)-\frac {5}{4} \int \left (\frac {100 e^x}{(5+x)^2}+\frac {430 e^x x}{(5+x)^2}+\frac {243 e^x x^2}{(5+x)^2}+\frac {48 e^x x^3}{(5+x)^2}+\frac {3 e^x x^4}{(5+x)^2}-4 e^x (1+x) \log (x)\right ) \, dx-3 \int \left (1+2 \log (x)-\frac {5 (1+2 \log (x))}{5+x}\right ) \, dx+30 \int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx \\ & = 3 x+\frac {9 x^2}{4}+\frac {75}{5+x}+\frac {15 x \log (x)}{5+x}+\frac {3 x^2 \log (x)}{5+x}-30 \log \left (1+\frac {x}{5}\right ) (1+\log (x))+15 \log (5+x)-30 \operatorname {PolyLog}\left (2,-\frac {x}{5}\right )-\frac {15}{4} \int \frac {e^x x^4}{(5+x)^2} \, dx+5 \int e^x (1+x) \log (x) \, dx-6 \int \log (x) \, dx+15 \int \frac {1+2 \log (x)}{5+x} \, dx-60 \int \frac {e^x x^3}{(5+x)^2} \, dx-125 \int \frac {e^x}{(5+x)^2} \, dx-\frac {1215}{4} \int \frac {e^x x^2}{(5+x)^2} \, dx-\frac {1075}{2} \int \frac {e^x x}{(5+x)^2} \, dx \\ & = 9 x+\frac {9 x^2}{4}+\frac {75}{5+x}+\frac {125 e^x}{5+x}-5 e^x \log (x)-6 x \log (x)+5 e^x (1+x) \log (x)+\frac {15 x \log (x)}{5+x}+\frac {3 x^2 \log (x)}{5+x}-30 \log \left (1+\frac {x}{5}\right ) (1+\log (x))+15 \log \left (1+\frac {x}{5}\right ) (1+2 \log (x))+15 \log (5+x)-30 \operatorname {PolyLog}\left (2,-\frac {x}{5}\right )-\frac {15}{4} \int \left (75 e^x-10 e^x x+e^x x^2+\frac {625 e^x}{(5+x)^2}-\frac {500 e^x}{5+x}\right ) \, dx-5 \int e^x \, dx-30 \int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx-60 \int \left (-10 e^x+e^x x-\frac {125 e^x}{(5+x)^2}+\frac {75 e^x}{5+x}\right ) \, dx-125 \int \frac {e^x}{5+x} \, dx-\frac {1215}{4} \int \left (e^x+\frac {25 e^x}{(5+x)^2}-\frac {10 e^x}{5+x}\right ) \, dx-\frac {1075}{2} \int \left (-\frac {5 e^x}{(5+x)^2}+\frac {e^x}{5+x}\right ) \, dx \\ & = -5 e^x+9 x+\frac {9 x^2}{4}+\frac {75}{5+x}+\frac {125 e^x}{5+x}-\frac {125 \operatorname {ExpIntegralEi}(5+x)}{e^5}-5 e^x \log (x)-6 x \log (x)+5 e^x (1+x) \log (x)+\frac {15 x \log (x)}{5+x}+\frac {3 x^2 \log (x)}{5+x}-30 \log \left (1+\frac {x}{5}\right ) (1+\log (x))+15 \log \left (1+\frac {x}{5}\right ) (1+2 \log (x))+15 \log (5+x)-\frac {15}{4} \int e^x x^2 \, dx+\frac {75}{2} \int e^x x \, dx-60 \int e^x x \, dx-\frac {1125 \int e^x \, dx}{4}-\frac {1215 \int e^x \, dx}{4}-\frac {1075}{2} \int \frac {e^x}{5+x} \, dx+600 \int e^x \, dx+1875 \int \frac {e^x}{5+x} \, dx-\frac {9375}{4} \int \frac {e^x}{(5+x)^2} \, dx+\frac {5375}{2} \int \frac {e^x}{(5+x)^2} \, dx+\frac {6075}{2} \int \frac {e^x}{5+x} \, dx-4500 \int \frac {e^x}{5+x} \, dx+7500 \int \frac {e^x}{(5+x)^2} \, dx-\frac {30375}{4} \int \frac {e^x}{(5+x)^2} \, dx \\ & = 10 e^x+9 x-\frac {45 e^x x}{2}+\frac {9 x^2}{4}-\frac {15 e^x x^2}{4}+\frac {75}{5+x}-\frac {125 e^x}{5+x}-\frac {250 \operatorname {ExpIntegralEi}(5+x)}{e^5}-5 e^x \log (x)-6 x \log (x)+5 e^x (1+x) \log (x)+\frac {15 x \log (x)}{5+x}+\frac {3 x^2 \log (x)}{5+x}-30 \log \left (1+\frac {x}{5}\right ) (1+\log (x))+15 \log \left (1+\frac {x}{5}\right ) (1+2 \log (x))+15 \log (5+x)+\frac {15}{2} \int e^x x \, dx-\frac {75 \int e^x \, dx}{2}+60 \int e^x \, dx-\frac {9375}{4} \int \frac {e^x}{5+x} \, dx+\frac {5375}{2} \int \frac {e^x}{5+x} \, dx+7500 \int \frac {e^x}{5+x} \, dx-\frac {30375}{4} \int \frac {e^x}{5+x} \, dx \\ & = \frac {65 e^x}{2}+9 x-15 e^x x+\frac {9 x^2}{4}-\frac {15 e^x x^2}{4}+\frac {75}{5+x}-\frac {125 e^x}{5+x}-5 e^x \log (x)-6 x \log (x)+5 e^x (1+x) \log (x)+\frac {15 x \log (x)}{5+x}+\frac {3 x^2 \log (x)}{5+x}-30 \log \left (1+\frac {x}{5}\right ) (1+\log (x))+15 \log \left (1+\frac {x}{5}\right ) (1+2 \log (x))+15 \log (5+x)-\frac {15 \int e^x \, dx}{2} \\ & = 25 e^x+9 x-15 e^x x+\frac {9 x^2}{4}-\frac {15 e^x x^2}{4}+\frac {75}{5+x}-\frac {125 e^x}{5+x}-5 e^x \log (x)-6 x \log (x)+5 e^x (1+x) \log (x)+\frac {15 x \log (x)}{5+x}+\frac {3 x^2 \log (x)}{5+x}-30 \log \left (1+\frac {x}{5}\right ) (1+\log (x))+15 \log \left (1+\frac {x}{5}\right ) (1+2 \log (x))+15 \log (5+x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(26)=52\).

Time = 1.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{100+40 x+4 x^2} \, dx=\frac {1}{4} \left (36 x+9 x^2+\frac {300}{5+x}+e^x \left (100-60 x-15 x^2-\frac {500}{5+x}\right )+4 \left (-3+5 e^x\right ) x \log (x)\right ) \]

[In]

Integrate[(300 + 690*x + 204*x^2 + 18*x^3 + E^x*(-500 - 2150*x - 1215*x^2 - 240*x^3 - 15*x^4) + (-300 - 120*x
- 12*x^2 + E^x*(500 + 700*x + 220*x^2 + 20*x^3))*Log[x])/(100 + 40*x + 4*x^2),x]

[Out]

(36*x + 9*x^2 + 300/(5 + x) + E^x*(100 - 60*x - 15*x^2 - 500/(5 + x)) + 4*(-3 + 5*E^x)*x*Log[x])/4

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(23)=46\).

Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.12

method	result	size
risch	\(\left (5 \,{\mathrm e}^{x} x -3 x \right ) \ln \left (x \right )-\frac {15 \,{\mathrm e}^{x} x^{3}-9 x^{3}+135 \,{\mathrm e}^{x} x^{2}-81 x^{2}+200 \,{\mathrm e}^{x} x -180 x -300}{4 \left (5+x \right )}\)	\(55\)
parts	\(\frac {-\frac {15 \,{\mathrm e}^{x} x^{3}}{4}-50 \,{\mathrm e}^{x} x -\frac {135 \,{\mathrm e}^{x} x^{2}}{4}+5 x^{2} {\mathrm e}^{x} \ln \left (x \right )+25 x \,{\mathrm e}^{x} \ln \left (x \right )}{5+x}+\frac {9 x^{2}}{4}+9 x +\frac {75}{5+x}-3 x \ln \left (x \right )\)	\(64\)
default	\(\frac {-15 \,{\mathrm e}^{x} x^{3}-200 \,{\mathrm e}^{x} x -135 \,{\mathrm e}^{x} x^{2}+20 x^{2} {\mathrm e}^{x} \ln \left (x \right )+100 x \,{\mathrm e}^{x} \ln \left (x \right )}{20+4 x}+\frac {9 x^{2}}{4}+9 x +\frac {75}{5+x}-3 x \ln \left (x \right )\)	\(65\)
norman	\(\frac {\frac {81 x^{2}}{4}+\frac {9 x^{3}}{4}-15 x \ln \left (x \right )-3 x^{2} \ln \left (x \right )-50 \,{\mathrm e}^{x} x -\frac {135 \,{\mathrm e}^{x} x^{2}}{4}-\frac {15 \,{\mathrm e}^{x} x^{3}}{4}+25 x \,{\mathrm e}^{x} \ln \left (x \right )+5 x^{2} {\mathrm e}^{x} \ln \left (x \right )-150}{5+x}\)	\(66\)
parallelrisch	\(-\frac {75 \,{\mathrm e}^{x} x^{3}-100 x^{2} {\mathrm e}^{x} \ln \left (x \right )-45 x^{3}+60 x^{2} \ln \left (x \right )+675 \,{\mathrm e}^{x} x^{2}-500 x \,{\mathrm e}^{x} \ln \left (x \right )-405 x^{2}+300 x \ln \left (x \right )+1000 \,{\mathrm e}^{x} x +3000}{20 \left (5+x \right )}\)	\(67\)

[In]

int((((20*x^3+220*x^2+700*x+500)*exp(x)-12*x^2-120*x-300)*ln(x)+(-15*x^4-240*x^3-1215*x^2-2150*x-500)*exp(x)+1
8*x^3+204*x^2+690*x+300)/(4*x^2+40*x+100),x,method=_RETURNVERBOSE)

[Out]

(5*exp(x)*x-3*x)*ln(x)-1/4*(15*exp(x)*x^3-9*x^3+135*exp(x)*x^2-81*x^2+200*exp(x)*x-180*x-300)/(5+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{100+40 x+4 x^2} \, dx=\frac {9 \, x^{3} + 81 \, x^{2} - 5 \, {\left (3 \, x^{3} + 27 \, x^{2} + 40 \, x\right )} e^{x} - 4 \, {\left (3 \, x^{2} - 5 \, {\left (x^{2} + 5 \, x\right )} e^{x} + 15 \, x\right )} \log \left (x\right ) + 180 \, x + 300}{4 \, {\left (x + 5\right )}} \]

[In]

integrate((((20*x^3+220*x^2+700*x+500)*exp(x)-12*x^2-120*x-300)*log(x)+(-15*x^4-240*x^3-1215*x^2-2150*x-500)*e
xp(x)+18*x^3+204*x^2+690*x+300)/(4*x^2+40*x+100),x, algorithm="fricas")

[Out]

1/4*(9*x^3 + 81*x^2 - 5*(3*x^3 + 27*x^2 + 40*x)*e^x - 4*(3*x^2 - 5*(x^2 + 5*x)*e^x + 15*x)*log(x) + 180*x + 30
0)/(x + 5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).

Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{100+40 x+4 x^2} \, dx=\frac {9 x^{2}}{4} - 3 x \log {\left (x \right )} + 9 x + \frac {\left (- 15 x^{3} + 20 x^{2} \log {\left (x \right )} - 135 x^{2} + 100 x \log {\left (x \right )} - 200 x\right ) e^{x}}{4 x + 20} + \frac {75}{x + 5} \]

[In]

integrate((((20*x**3+220*x**2+700*x+500)*exp(x)-12*x**2-120*x-300)*ln(x)+(-15*x**4-240*x**3-1215*x**2-2150*x-5
00)*exp(x)+18*x**3+204*x**2+690*x+300)/(4*x**2+40*x+100),x)

[Out]

9*x**2/4 - 3*x*log(x) + 9*x + (-15*x**3 + 20*x**2*log(x) - 135*x**2 + 100*x*log(x) - 200*x)*exp(x)/(4*x + 20)
+ 75/(x + 5)

Maxima [F]

\[ \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{100+40 x+4 x^2} \, dx=\int { \frac {18 \, x^{3} + 204 \, x^{2} - 5 \, {\left (3 \, x^{4} + 48 \, x^{3} + 243 \, x^{2} + 430 \, x + 100\right )} e^{x} - 4 \, {\left (3 \, x^{2} - 5 \, {\left (x^{3} + 11 \, x^{2} + 35 \, x + 25\right )} e^{x} + 30 \, x + 75\right )} \log \left (x\right ) + 690 \, x + 300}{4 \, {\left (x^{2} + 10 \, x + 25\right )}} \,d x } \]

[In]

integrate((((20*x^3+220*x^2+700*x+500)*exp(x)-12*x^2-120*x-300)*log(x)+(-15*x^4-240*x^3-1215*x^2-2150*x-500)*e
xp(x)+18*x^3+204*x^2+690*x+300)/(4*x^2+40*x+100),x, algorithm="maxima")

[Out]

5*x*e^x*log(x) + 9/4*x^2 - 3*x*log(x) + 9*x + 125*e^(-5)*exp_integral_e(2, -x - 5)/(x + 5) + 75/(x + 5) - 1/4*
integrate(5*(3*x^4 + 48*x^3 + 247*x^2 + 470*x + 100)*e^x/(x^2 + 10*x + 25), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.65 \[ \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{100+40 x+4 x^2} \, dx=-\frac {15 \, x^{3} e^{x} - 20 \, x^{2} e^{x} \log \left (x\right ) - 9 \, x^{3} + 135 \, x^{2} e^{x} + 12 \, x^{2} \log \left (x\right ) - 100 \, x e^{x} \log \left (x\right ) - 81 \, x^{2} + 200 \, x e^{x} + 60 \, x \log \left (x\right ) - 180 \, x - 300}{4 \, {\left (x + 5\right )}} \]

[In]

integrate((((20*x^3+220*x^2+700*x+500)*exp(x)-12*x^2-120*x-300)*log(x)+(-15*x^4-240*x^3-1215*x^2-2150*x-500)*e
xp(x)+18*x^3+204*x^2+690*x+300)/(4*x^2+40*x+100),x, algorithm="giac")

[Out]

-1/4*(15*x^3*e^x - 20*x^2*e^x*log(x) - 9*x^3 + 135*x^2*e^x + 12*x^2*log(x) - 100*x*e^x*log(x) - 81*x^2 + 200*x
*e^x + 60*x*log(x) - 180*x - 300)/(x + 5)

Mupad [B] (verification not implemented)

Time = 16.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {300+690 x+204 x^2+18 x^3+e^x \left (-500-2150 x-1215 x^2-240 x^3-15 x^4\right )+\left (-300-120 x-12 x^2+e^x \left (500+700 x+220 x^2+20 x^3\right )\right ) \log (x)}{100+40 x+4 x^2} \, dx=9\,x-\ln \left (x\right )\,\left (3\,x-5\,x\,{\mathrm {e}}^x\right )+\frac {75}{x+5}+\frac {9\,x^2}{4}-\frac {{\mathrm {e}}^x\,\left (\frac {15\,x^3}{4}+\frac {135\,x^2}{4}+50\,x\right )}{x+5} \]

[In]

int((690*x - exp(x)*(2150*x + 1215*x^2 + 240*x^3 + 15*x^4 + 500) - log(x)*(120*x + 12*x^2 - exp(x)*(700*x + 22
0*x^2 + 20*x^3 + 500) + 300) + 204*x^2 + 18*x^3 + 300)/(40*x + 4*x^2 + 100),x)

[Out]

9*x - log(x)*(3*x - 5*x*exp(x)) + 75/(x + 5) + (9*x^2)/4 - (exp(x)*(50*x + (135*x^2)/4 + (15*x^3)/4))/(x + 5)