\(\int \frac {-20 x+20 x^3+4 x^4+e^{3+x} (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7)+(60 x+150 x^2+60 x^3+6 x^4+e^{3+x} (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6)) \log (\frac {2 x+e^{3+x} (-10-25 x-10 x^2-x^3)}{10+25 x+10 x^2+x^3})}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10})} \, dx\) [7872]

Optimal result

Integrand size = 214, antiderivative size = 29 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (-e^{3+x}+\frac {x}{5+\frac {1}{2} x (5+x)^2}\right )}{x^3} \]

[Out]

ln(x/(5+1/2*x*(5+x)^2)-exp(3+x))/x^3

Rubi [A] (verified)

Time = 5.66 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07, number of steps used = 24, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6820, 6874, 14, 2631} \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (\frac {2 x}{x^3+10 x^2+25 x+10}-e^{x+3}\right )}{x^3} \]

[In]

Int[(-20*x + 20*x^3 + 4*x^4 + E^(3 + x)*(100*x + 500*x^2 + 825*x^3 + 520*x^4 + 150*x^5 + 20*x^6 + x^7) + (60*x
+ 150*x^2 + 60*x^3 + 6*x^4 + E^(3 + x)*(-300 - 1500*x - 2475*x^2 - 1560*x^3 - 450*x^4 - 60*x^5 - 3*x^6))*Log[
(2*x + E^(3 + x)*(-10 - 25*x - 10*x^2 - x^3))/(10 + 25*x + 10*x^2 + x^3)])/(-20*x^5 - 50*x^6 - 20*x^7 - 2*x^8
+ E^(3 + x)*(100*x^4 + 500*x^5 + 825*x^6 + 520*x^7 + 150*x^8 + 20*x^9 + x^10)),x]

[Out]

Log[-E^(3 + x) + (2*x)/(10 + 25*x + 10*x^2 + x^3)]/x^3

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 2631

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 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 {20 x-20 x^3-4 x^4-e^{3+x} x \left (10+25 x+10 x^2+x^3\right )^2+3 \left (10+25 x+10 x^2+x^3\right ) \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right ) \log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4 \left (10+25 x+10 x^2+x^3\right ) \left (2 x-e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx \\ & = \int \left (\frac {2 \left (-10+10 x+35 x^2+12 x^3+x^4\right )}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {x-3 \log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4}\right ) \, dx \\ & = 2 \int \frac {-10+10 x+35 x^2+12 x^3+x^4}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\int \frac {x-3 \log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4} \, dx \\ & = 2 \int \left (-\frac {1}{x^3 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {7}{2 x^2 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}-\frac {17}{4 x \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {337+160 x+17 x^2}{4 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx+\int \left (\frac {1}{x^3}-\frac {3 \log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4}\right ) \, dx \\ & = -\frac {1}{2 x^2}+\frac {1}{2} \int \frac {337+160 x+17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-2 \int \frac {1}{x^3 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-3 \int \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4} \, dx+7 \int \frac {1}{x^2 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx \\ & = -\frac {1}{2 x^2}+\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}+\frac {1}{2} \int \frac {-337-160 x-17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (2 x-e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-\int \frac {-4 \left (-5+5 x^2+x^3\right )-e^{3+x} \left (10+25 x+10 x^2+x^3\right )^2}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (2 x-e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx \\ & = -\frac {1}{2 x^2}+\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}+\frac {1}{2} \int \left (\frac {337}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {160 x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-\int \left (\frac {1}{x^3}+\frac {2 \left (-10+10 x+35 x^2+12 x^3+x^4\right )}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx \\ & = \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-2 \int \frac {-10+10 x+35 x^2+12 x^3+x^4}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx \\ & = \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-2 \int \left (-\frac {1}{x^3 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {7}{2 x^2 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}-\frac {17}{4 x \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {337+160 x+17 x^2}{4 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx \\ & = \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-\frac {1}{2} \int \frac {337+160 x+17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+2 \int \frac {1}{x^3 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-7 \int \frac {1}{x^2 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+\frac {17}{2} \int \frac {1}{x \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx \\ & = \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-\frac {1}{2} \int \frac {-337-160 x-17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (2 x-e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx \\ & = \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-\frac {1}{2} \int \left (\frac {337}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {160 x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx \\ & = \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3} \]

[In]

Integrate[(-20*x + 20*x^3 + 4*x^4 + E^(3 + x)*(100*x + 500*x^2 + 825*x^3 + 520*x^4 + 150*x^5 + 20*x^6 + x^7) +
(60*x + 150*x^2 + 60*x^3 + 6*x^4 + E^(3 + x)*(-300 - 1500*x - 2475*x^2 - 1560*x^3 - 450*x^4 - 60*x^5 - 3*x^6)
)*Log[(2*x + E^(3 + x)*(-10 - 25*x - 10*x^2 - x^3))/(10 + 25*x + 10*x^2 + x^3)])/(-20*x^5 - 50*x^6 - 20*x^7 -
2*x^8 + E^(3 + x)*(100*x^4 + 500*x^5 + 825*x^6 + 520*x^7 + 150*x^8 + 20*x^9 + x^10)),x]

[Out]

Log[-E^(3 + x) + (2*x)/(10 + 25*x + 10*x^2 + x^3)]/x^3

Maple [A] (verified)

Time = 35.57 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {\left (-x^{3}-10 x^{2}-25 x -10\right ) {\mathrm e}^{3+x}+2 x}{x^{3}+10 x^{2}+25 x +10}\right )}{x^{3}}\)	\(46\)
risch	\(\frac {\ln \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}}-\frac {2 i \pi {\operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )\right ) \operatorname {csgn}\left (\frac {i}{x^{3}+10 x^{2}+25 x +10}\right ) \operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )-i \pi \,\operatorname {csgn}\left (i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{x^{3}+10 x^{2}+25 x +10}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )}^{2}-i \pi {\operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )}^{3}-2 i \pi +2 \ln \left (x^{3}+10 x^{2}+25 x +10\right )}{2 x^{3}}\)	\(474\)

[In]

int((((-3*x^6-60*x^5-450*x^4-1560*x^3-2475*x^2-1500*x-300)*exp(3+x)+6*x^4+60*x^3+150*x^2+60*x)*ln(((-x^3-10*x^
2-25*x-10)*exp(3+x)+2*x)/(x^3+10*x^2+25*x+10))+(x^7+20*x^6+150*x^5+520*x^4+825*x^3+500*x^2+100*x)*exp(3+x)+4*x
^4+20*x^3-20*x)/((x^10+20*x^9+150*x^8+520*x^7+825*x^6+500*x^5+100*x^4)*exp(3+x)-2*x^8-20*x^7-50*x^6-20*x^5),x,
method=_RETURNVERBOSE)

[Out]

ln(((-x^3-10*x^2-25*x-10)*exp(3+x)+2*x)/(x^3+10*x^2+25*x+10))/x^3

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (-\frac {{\left (x^{3} + 10 \, x^{2} + 25 \, x + 10\right )} e^{\left (x + 3\right )} - 2 \, x}{x^{3} + 10 \, x^{2} + 25 \, x + 10}\right )}{x^{3}} \]

[In]

integrate((((-3*x^6-60*x^5-450*x^4-1560*x^3-2475*x^2-1500*x-300)*exp(3+x)+6*x^4+60*x^3+150*x^2+60*x)*log(((-x^
3-10*x^2-25*x-10)*exp(3+x)+2*x)/(x^3+10*x^2+25*x+10))+(x^7+20*x^6+150*x^5+520*x^4+825*x^3+500*x^2+100*x)*exp(3
+x)+4*x^4+20*x^3-20*x)/((x^10+20*x^9+150*x^8+520*x^7+825*x^6+500*x^5+100*x^4)*exp(3+x)-2*x^8-20*x^7-50*x^6-20*
x^5),x, algorithm="fricas")

[Out]

log(-((x^3 + 10*x^2 + 25*x + 10)*e^(x + 3) - 2*x)/(x^3 + 10*x^2 + 25*x + 10))/x^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).

Time = 0.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log {\left (\frac {2 x + \left (- x^{3} - 10 x^{2} - 25 x - 10\right ) e^{x + 3}}{x^{3} + 10 x^{2} + 25 x + 10} \right )}}{x^{3}} \]

[In]

integrate((((-3*x**6-60*x**5-450*x**4-1560*x**3-2475*x**2-1500*x-300)*exp(3+x)+6*x**4+60*x**3+150*x**2+60*x)*l
n(((-x**3-10*x**2-25*x-10)*exp(3+x)+2*x)/(x**3+10*x**2+25*x+10))+(x**7+20*x**6+150*x**5+520*x**4+825*x**3+500*
x**2+100*x)*exp(3+x)+4*x**4+20*x**3-20*x)/((x**10+20*x**9+150*x**8+520*x**7+825*x**6+500*x**5+100*x**4)*exp(3+
x)-2*x**8-20*x**7-50*x**6-20*x**5),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).

Time = 0.35 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=-\frac {\log \left (x^{3} + 10 \, x^{2} + 25 \, x + 10\right ) - \log \left (-{\left (x^{3} e^{3} + 10 \, x^{2} e^{3} + 25 \, x e^{3} + 10 \, e^{3}\right )} e^{x} + 2 \, x\right )}{x^{3}} \]

[In]

integrate((((-3*x^6-60*x^5-450*x^4-1560*x^3-2475*x^2-1500*x-300)*exp(3+x)+6*x^4+60*x^3+150*x^2+60*x)*log(((-x^
3-10*x^2-25*x-10)*exp(3+x)+2*x)/(x^3+10*x^2+25*x+10))+(x^7+20*x^6+150*x^5+520*x^4+825*x^3+500*x^2+100*x)*exp(3
+x)+4*x^4+20*x^3-20*x)/((x^10+20*x^9+150*x^8+520*x^7+825*x^6+500*x^5+100*x^4)*exp(3+x)-2*x^8-20*x^7-50*x^6-20*
x^5),x, algorithm="maxima")

[Out]

-(log(x^3 + 10*x^2 + 25*x + 10) - log(-(x^3*e^3 + 10*x^2*e^3 + 25*x*e^3 + 10*e^3)*e^x + 2*x))/x^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.61 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (-\frac {x^{3} e^{\left (x + 3\right )} + 10 \, x^{2} e^{\left (x + 3\right )} + 25 \, x e^{\left (x + 3\right )} - 2 \, x + 10 \, e^{\left (x + 3\right )}}{x^{3} + 10 \, x^{2} + 25 \, x + 10}\right )}{x^{3}} \]

[In]

integrate((((-3*x^6-60*x^5-450*x^4-1560*x^3-2475*x^2-1500*x-300)*exp(3+x)+6*x^4+60*x^3+150*x^2+60*x)*log(((-x^
3-10*x^2-25*x-10)*exp(3+x)+2*x)/(x^3+10*x^2+25*x+10))+(x^7+20*x^6+150*x^5+520*x^4+825*x^3+500*x^2+100*x)*exp(3
+x)+4*x^4+20*x^3-20*x)/((x^10+20*x^9+150*x^8+520*x^7+825*x^6+500*x^5+100*x^4)*exp(3+x)-2*x^8-20*x^7-50*x^6-20*
x^5),x, algorithm="giac")

[Out]

log(-(x^3*e^(x + 3) + 10*x^2*e^(x + 3) + 25*x*e^(x + 3) - 2*x + 10*e^(x + 3))/(x^3 + 10*x^2 + 25*x + 10))/x^3

Mupad [B] (verification not implemented)

Time = 12.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\ln \left (\frac {2\,x-{\mathrm {e}}^3\,{\mathrm {e}}^x\,\left (x^3+10\,x^2+25\,x+10\right )}{x^3+10\,x^2+25\,x+10}\right )}{x^3} \]

[In]

int(-(log((2*x - exp(x + 3)*(25*x + 10*x^2 + x^3 + 10))/(25*x + 10*x^2 + x^3 + 10))*(60*x + 150*x^2 + 60*x^3 +
6*x^4 - exp(x + 3)*(1500*x + 2475*x^2 + 1560*x^3 + 450*x^4 + 60*x^5 + 3*x^6 + 300)) - 20*x + exp(x + 3)*(100*
x + 500*x^2 + 825*x^3 + 520*x^4 + 150*x^5 + 20*x^6 + x^7) + 20*x^3 + 4*x^4)/(20*x^5 - exp(x + 3)*(100*x^4 + 50
0*x^5 + 825*x^6 + 520*x^7 + 150*x^8 + 20*x^9 + x^10) + 50*x^6 + 20*x^7 + 2*x^8),x)

[Out]

log((2*x - exp(3)*exp(x)*(25*x + 10*x^2 + x^3 + 10))/(25*x + 10*x^2 + x^3 + 10))/x^3