Integrand size = 119, antiderivative size = 31 \[ \int \frac {e^{2 x} \left (-150+400 x^2+310 x^3-574 x^4-740 x^5-16 x^6+310 x^7+100 x^8\right )+e^{2 x} \left (-300+50 x+400 x^2+110 x^3-360 x^4-200 x^5\right ) \log (3+2 x)+e^{2 x} \left (-150+50 x+100 x^2\right ) \log ^2(3+2 x)}{3 x^3+2 x^4} \, dx=e^{2 x} \left (x+\frac {5 \left (1-x \left (-1+x^2\right )+\log (3+2 x)\right )}{x}\right )^2 \]
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{2 x} \left (-150+400 x^2+310 x^3-574 x^4-740 x^5-16 x^6+310 x^7+100 x^8\right )+e^{2 x} \left (-300+50 x+400 x^2+110 x^3-360 x^4-200 x^5\right ) \log (3+2 x)+e^{2 x} \left (-150+50 x+100 x^2\right ) \log ^2(3+2 x)}{3 x^3+2 x^4} \, dx=\frac {e^{2 x} \left (5+5 x+x^2-5 x^3+5 \log (3+2 x)\right )^2}{x^2} \]
Integrate[(E^(2*x)*(-150 + 400*x^2 + 310*x^3 - 574*x^4 - 740*x^5 - 16*x^6 + 310*x^7 + 100*x^8) + E^(2*x)*(-300 + 50*x + 400*x^2 + 110*x^3 - 360*x^4 - 200*x^5)*Log[3 + 2*x] + E^(2*x)*(-150 + 50*x + 100*x^2)*Log[3 + 2*x]^2)/ (3*x^3 + 2*x^4),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{2 x} \left (100 x^2+50 x-150\right ) \log ^2(2 x+3)+e^{2 x} \left (-200 x^5-360 x^4+110 x^3+400 x^2+50 x-300\right ) \log (2 x+3)+e^{2 x} \left (100 x^8+310 x^7-16 x^6-740 x^5-574 x^4+310 x^3+400 x^2-150\right )}{2 x^4+3 x^3} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{2 x} \left (100 x^2+50 x-150\right ) \log ^2(2 x+3)+e^{2 x} \left (-200 x^5-360 x^4+110 x^3+400 x^2+50 x-300\right ) \log (2 x+3)+e^{2 x} \left (100 x^8+310 x^7-16 x^6-740 x^5-574 x^4+310 x^3+400 x^2-150\right )}{x^3 (2 x+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 e^{2 x} \left (50 x^8+155 x^7-8 x^6-370 x^5-100 x^5 \log (2 x+3)-287 x^4-180 x^4 \log (2 x+3)+155 x^3+55 x^3 \log (2 x+3)+200 x^2+50 x^2 \log ^2(2 x+3)+200 x^2 \log (2 x+3)+25 x \log ^2(2 x+3)-75 \log ^2(2 x+3)+25 x \log (2 x+3)-150 \log (2 x+3)-75\right )}{27 x}-\frac {16 e^{2 x} \left (50 x^8+155 x^7-8 x^6-370 x^5-100 x^5 \log (2 x+3)-287 x^4-180 x^4 \log (2 x+3)+155 x^3+55 x^3 \log (2 x+3)+200 x^2+50 x^2 \log ^2(2 x+3)+200 x^2 \log (2 x+3)+25 x \log ^2(2 x+3)-75 \log ^2(2 x+3)+25 x \log (2 x+3)-150 \log (2 x+3)-75\right )}{27 (2 x+3)}-\frac {4 e^{2 x} \left (50 x^8+155 x^7-8 x^6-370 x^5-100 x^5 \log (2 x+3)-287 x^4-180 x^4 \log (2 x+3)+155 x^3+55 x^3 \log (2 x+3)+200 x^2+50 x^2 \log ^2(2 x+3)+200 x^2 \log (2 x+3)+25 x \log ^2(2 x+3)-75 \log ^2(2 x+3)+25 x \log (2 x+3)-150 \log (2 x+3)-75\right )}{9 x^2}+\frac {2 e^{2 x} \left (50 x^8+155 x^7-8 x^6-370 x^5-100 x^5 \log (2 x+3)-287 x^4-180 x^4 \log (2 x+3)+155 x^3+55 x^3 \log (2 x+3)+200 x^2+50 x^2 \log ^2(2 x+3)+200 x^2 \log (2 x+3)+25 x \log ^2(2 x+3)-75 \log ^2(2 x+3)+25 x \log (2 x+3)-150 \log (2 x+3)-75\right )}{3 x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{2 x} \left (50 x^8+155 x^7-8 x^6-370 x^5-287 x^4+155 x^3+200 x^2+25 \left (2 x^2+x-3\right ) \log ^2(2 x+3)-5 \left (20 x^5+36 x^4-11 x^3-40 x^2-5 x+30\right ) \log (2 x+3)-75\right )}{x^3 (2 x+3)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {e^{2 x} \left (-50 x^8-155 x^7+8 x^6+370 x^5+287 x^4-155 x^3-200 x^2+25 \left (-2 x^2-x+3\right ) \log ^2(2 x+3)+5 \left (20 x^5+36 x^4-11 x^3-40 x^2-5 x+30\right ) \log (2 x+3)+75\right )}{x^3 (2 x+3)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {e^{2 x} \left (-50 x^8-155 x^7+8 x^6+370 x^5+287 x^4-155 x^3-200 x^2+25 \left (-2 x^2-x+3\right ) \log ^2(2 x+3)+5 \left (20 x^5+36 x^4-11 x^3-40 x^2-5 x+30\right ) \log (2 x+3)+75\right )}{x^3 (2 x+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {50 e^{2 x} x^5}{2 x+3}-\frac {155 e^{2 x} x^4}{2 x+3}+\frac {8 e^{2 x} x^3}{2 x+3}+\frac {370 e^{2 x} x^2}{2 x+3}+\frac {287 e^{2 x} x}{2 x+3}-\frac {155 e^{2 x}}{2 x+3}-\frac {200 e^{2 x}}{(2 x+3) x}-\frac {25 e^{2 x} (x-1) \log ^2(2 x+3)}{x^3}+\frac {5 e^{2 x} \left (20 x^5+36 x^4-11 x^3-40 x^2-5 x+30\right ) \log (2 x+3)}{(2 x+3) x^3}+\frac {75 e^{2 x}}{(2 x+3) x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {400}{9} \int \frac {\operatorname {ExpIntegralEi}(2 x)}{2 x+3}dx+\frac {200 \int \frac {\operatorname {ExpIntegralEi}(2 x+3)}{2 x+3}dx}{9 e^3}+25 \int \frac {e^{2 x} \log ^2(2 x+3)}{x^3}dx-25 \int \frac {e^{2 x} \log ^2(2 x+3)}{x^2}dx-\frac {100 \operatorname {ExpIntegralEi}(2 x)}{9}+\frac {100 \operatorname {ExpIntegralEi}(2 x+3)}{9 e^3}-\frac {200}{9} \operatorname {ExpIntegralEi}(2 x) \log (2 x+3)-\frac {100 \operatorname {ExpIntegralEi}(2 x+3) \log (2 x+3)}{9 e^3}-\frac {25}{2} e^{2 x} x^4+5 e^{2 x} x^3+\frac {49}{2} e^{2 x} x^2-\frac {25 e^{2 x}}{2 x^2}-\frac {25 e^{2 x} \log (2 x+3)}{x^2}+20 e^{2 x} x-\frac {35 e^{2 x}}{2}-\frac {25 e^{2 x}}{x}+25 e^{2 x} x \log (2 x+3)-5 e^{2 x} \log (2 x+3)-\frac {25 e^{2 x} \log (2 x+3)}{3 x}\right )\) |
Int[(E^(2*x)*(-150 + 400*x^2 + 310*x^3 - 574*x^4 - 740*x^5 - 16*x^6 + 310* x^7 + 100*x^8) + E^(2*x)*(-300 + 50*x + 400*x^2 + 110*x^3 - 360*x^4 - 200* x^5)*Log[3 + 2*x] + E^(2*x)*(-150 + 50*x + 100*x^2)*Log[3 + 2*x]^2)/(3*x^3 + 2*x^4),x]
3.14.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(30)=60\).
Time = 0.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81
| method | result | size |
| risch | \(\frac {25 \,{\mathrm e}^{2 x} \ln \left (3+2 x \right )^{2}}{x^{2}}-\frac {10 \left (5 x^{3}-x^{2}-5 x -5\right ) {\mathrm e}^{2 x} \ln \left (3+2 x \right )}{x^{2}}+\frac {\left (25 x^{6}-10 x^{5}-49 x^{4}-40 x^{3}+35 x^{2}+50 x +25\right ) {\mathrm e}^{2 x}}{x^{2}}\) | \(87\) |
| parallelrisch | \(-\frac {-300 \,{\mathrm e}^{2 x} x^{6}+120 x^{5} {\mathrm e}^{2 x}+588 \,{\mathrm e}^{2 x} x^{4}+600 \,{\mathrm e}^{2 x} \ln \left (3+2 x \right ) x^{3}+480 \,{\mathrm e}^{2 x} x^{3}-120 \,{\mathrm e}^{2 x} \ln \left (3+2 x \right ) x^{2}-420 \,{\mathrm e}^{2 x} x^{2}-600 \,{\mathrm e}^{2 x} \ln \left (3+2 x \right ) x -300 \,{\mathrm e}^{2 x} \ln \left (3+2 x \right )^{2}-600 x \,{\mathrm e}^{2 x}-600 \,{\mathrm e}^{2 x} \ln \left (3+2 x \right )-300 \,{\mathrm e}^{2 x}}{12 x^{2}}\) | \(134\) |
int(((100*x^2+50*x-150)*exp(x)^2*ln(3+2*x)^2+(-200*x^5-360*x^4+110*x^3+400 *x^2+50*x-300)*exp(x)^2*ln(3+2*x)+(100*x^8+310*x^7-16*x^6-740*x^5-574*x^4+ 310*x^3+400*x^2-150)*exp(x)^2)/(2*x^4+3*x^3),x,method=_RETURNVERBOSE)
25/x^2*exp(2*x)*ln(3+2*x)^2-10*(5*x^3-x^2-5*x-5)/x^2*exp(2*x)*ln(3+2*x)+(2 5*x^6-10*x^5-49*x^4-40*x^3+35*x^2+50*x+25)/x^2*exp(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (31) = 62\).
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.68 \[ \int \frac {e^{2 x} \left (-150+400 x^2+310 x^3-574 x^4-740 x^5-16 x^6+310 x^7+100 x^8\right )+e^{2 x} \left (-300+50 x+400 x^2+110 x^3-360 x^4-200 x^5\right ) \log (3+2 x)+e^{2 x} \left (-150+50 x+100 x^2\right ) \log ^2(3+2 x)}{3 x^3+2 x^4} \, dx=-\frac {10 \, {\left (5 \, x^{3} - x^{2} - 5 \, x - 5\right )} e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right ) - 25 \, e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right )^{2} - {\left (25 \, x^{6} - 10 \, x^{5} - 49 \, x^{4} - 40 \, x^{3} + 35 \, x^{2} + 50 \, x + 25\right )} e^{\left (2 \, x\right )}}{x^{2}} \]
integrate(((100*x^2+50*x-150)*exp(x)^2*log(3+2*x)^2+(-200*x^5-360*x^4+110* x^3+400*x^2+50*x-300)*exp(x)^2*log(3+2*x)+(100*x^8+310*x^7-16*x^6-740*x^5- 574*x^4+310*x^3+400*x^2-150)*exp(x)^2)/(2*x^4+3*x^3),x, algorithm=\
-(10*(5*x^3 - x^2 - 5*x - 5)*e^(2*x)*log(2*x + 3) - 25*e^(2*x)*log(2*x + 3 )^2 - (25*x^6 - 10*x^5 - 49*x^4 - 40*x^3 + 35*x^2 + 50*x + 25)*e^(2*x))/x^ 2
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (27) = 54\).
Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.90 \[ \int \frac {e^{2 x} \left (-150+400 x^2+310 x^3-574 x^4-740 x^5-16 x^6+310 x^7+100 x^8\right )+e^{2 x} \left (-300+50 x+400 x^2+110 x^3-360 x^4-200 x^5\right ) \log (3+2 x)+e^{2 x} \left (-150+50 x+100 x^2\right ) \log ^2(3+2 x)}{3 x^3+2 x^4} \, dx=\frac {\left (25 x^{6} - 10 x^{5} - 49 x^{4} - 50 x^{3} \log {\left (2 x + 3 \right )} - 40 x^{3} + 10 x^{2} \log {\left (2 x + 3 \right )} + 35 x^{2} + 50 x \log {\left (2 x + 3 \right )} + 50 x + 25 \log {\left (2 x + 3 \right )}^{2} + 50 \log {\left (2 x + 3 \right )} + 25\right ) e^{2 x}}{x^{2}} \]
integrate(((100*x**2+50*x-150)*exp(x)**2*ln(3+2*x)**2+(-200*x**5-360*x**4+ 110*x**3+400*x**2+50*x-300)*exp(x)**2*ln(3+2*x)+(100*x**8+310*x**7-16*x**6 -740*x**5-574*x**4+310*x**3+400*x**2-150)*exp(x)**2)/(2*x**4+3*x**3),x)
(25*x**6 - 10*x**5 - 49*x**4 - 50*x**3*log(2*x + 3) - 40*x**3 + 10*x**2*lo g(2*x + 3) + 35*x**2 + 50*x*log(2*x + 3) + 50*x + 25*log(2*x + 3)**2 + 50* log(2*x + 3) + 25)*exp(2*x)/x**2
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (31) = 62\).
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.68 \[ \int \frac {e^{2 x} \left (-150+400 x^2+310 x^3-574 x^4-740 x^5-16 x^6+310 x^7+100 x^8\right )+e^{2 x} \left (-300+50 x+400 x^2+110 x^3-360 x^4-200 x^5\right ) \log (3+2 x)+e^{2 x} \left (-150+50 x+100 x^2\right ) \log ^2(3+2 x)}{3 x^3+2 x^4} \, dx=-\frac {10 \, {\left (5 \, x^{3} - x^{2} - 5 \, x - 5\right )} e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right ) - 25 \, e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right )^{2} - {\left (25 \, x^{6} - 10 \, x^{5} - 49 \, x^{4} - 40 \, x^{3} + 35 \, x^{2} + 50 \, x + 25\right )} e^{\left (2 \, x\right )}}{x^{2}} \]
integrate(((100*x^2+50*x-150)*exp(x)^2*log(3+2*x)^2+(-200*x^5-360*x^4+110* x^3+400*x^2+50*x-300)*exp(x)^2*log(3+2*x)+(100*x^8+310*x^7-16*x^6-740*x^5- 574*x^4+310*x^3+400*x^2-150)*exp(x)^2)/(2*x^4+3*x^3),x, algorithm=\
-(10*(5*x^3 - x^2 - 5*x - 5)*e^(2*x)*log(2*x + 3) - 25*e^(2*x)*log(2*x + 3 )^2 - (25*x^6 - 10*x^5 - 49*x^4 - 40*x^3 + 35*x^2 + 50*x + 25)*e^(2*x))/x^ 2
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (31) = 62\).
Time = 0.32 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.26 \[ \int \frac {e^{2 x} \left (-150+400 x^2+310 x^3-574 x^4-740 x^5-16 x^6+310 x^7+100 x^8\right )+e^{2 x} \left (-300+50 x+400 x^2+110 x^3-360 x^4-200 x^5\right ) \log (3+2 x)+e^{2 x} \left (-150+50 x+100 x^2\right ) \log ^2(3+2 x)}{3 x^3+2 x^4} \, dx=\frac {25 \, x^{6} e^{\left (2 \, x\right )} - 10 \, x^{5} e^{\left (2 \, x\right )} - 49 \, x^{4} e^{\left (2 \, x\right )} - 50 \, x^{3} e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right ) - 40 \, x^{3} e^{\left (2 \, x\right )} + 10 \, x^{2} e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right ) + 35 \, x^{2} e^{\left (2 \, x\right )} + 50 \, x e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right ) + 25 \, e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right )^{2} + 50 \, x e^{\left (2 \, x\right )} + 50 \, e^{\left (2 \, x\right )} \log \left (2 \, x + 3\right ) + 25 \, e^{\left (2 \, x\right )}}{x^{2}} \]
integrate(((100*x^2+50*x-150)*exp(x)^2*log(3+2*x)^2+(-200*x^5-360*x^4+110* x^3+400*x^2+50*x-300)*exp(x)^2*log(3+2*x)+(100*x^8+310*x^7-16*x^6-740*x^5- 574*x^4+310*x^3+400*x^2-150)*exp(x)^2)/(2*x^4+3*x^3),x, algorithm=\
(25*x^6*e^(2*x) - 10*x^5*e^(2*x) - 49*x^4*e^(2*x) - 50*x^3*e^(2*x)*log(2*x + 3) - 40*x^3*e^(2*x) + 10*x^2*e^(2*x)*log(2*x + 3) + 35*x^2*e^(2*x) + 50 *x*e^(2*x)*log(2*x + 3) + 25*e^(2*x)*log(2*x + 3)^2 + 50*x*e^(2*x) + 50*e^ (2*x)*log(2*x + 3) + 25*e^(2*x))/x^2
Timed out. \[ \int \frac {e^{2 x} \left (-150+400 x^2+310 x^3-574 x^4-740 x^5-16 x^6+310 x^7+100 x^8\right )+e^{2 x} \left (-300+50 x+400 x^2+110 x^3-360 x^4-200 x^5\right ) \log (3+2 x)+e^{2 x} \left (-150+50 x+100 x^2\right ) \log ^2(3+2 x)}{3 x^3+2 x^4} \, dx=\int \frac {{\mathrm {e}}^{2\,x}\,\left (100\,x^2+50\,x-150\right )\,{\ln \left (2\,x+3\right )}^2+{\mathrm {e}}^{2\,x}\,\left (-200\,x^5-360\,x^4+110\,x^3+400\,x^2+50\,x-300\right )\,\ln \left (2\,x+3\right )+{\mathrm {e}}^{2\,x}\,\left (100\,x^8+310\,x^7-16\,x^6-740\,x^5-574\,x^4+310\,x^3+400\,x^2-150\right )}{2\,x^4+3\,x^3} \,d x \]
int((exp(2*x)*(400*x^2 + 310*x^3 - 574*x^4 - 740*x^5 - 16*x^6 + 310*x^7 + 100*x^8 - 150) + exp(2*x)*log(2*x + 3)^2*(50*x + 100*x^2 - 150) + exp(2*x) *log(2*x + 3)*(50*x + 400*x^2 + 110*x^3 - 360*x^4 - 200*x^5 - 300))/(3*x^3 + 2*x^4),x)