\(\int e^{-2 x^2-2 x^3} (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+(250 x^4-200 x^6-300 x^7) \log (5)+(100 x^3-100 x^5-150 x^6) \log ^2(5)+e^{x^2+x^3} (160 x^3-80 x^5-120 x^6+(120 x^2-80 x^4-120 x^5) \log (5))) \, dx\) [5081]

Optimal result

Integrand size = 131, antiderivative size = 25 \[ \int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx=\left (4 x+5 e^{-x \left (x+x^2\right )} x^2 (x+\log (5))\right )^2 \]

[Out]

(4*x+5*(ln(5)+x)*x^2/exp(x*(x^2+x)))^2

Rubi [F]

\[ \int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx=\int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx \]

[In]

Int[E^(-2*x^2 - 2*x^3)*(32*E^(2*x^2 + 2*x^3)*x + 150*x^5 - 100*x^7 - 150*x^8 + (250*x^4 - 200*x^6 - 300*x^7)*L
og[5] + (100*x^3 - 100*x^5 - 150*x^6)*Log[5]^2 + E^(x^2 + x^3)*(160*x^3 - 80*x^5 - 120*x^6 + (120*x^2 - 80*x^4
- 120*x^5)*Log[5])),x]

[Out]

16*x^2 + (50*E^(-2*x^2 - 2*x^3)*x^4*(2*x^2 + 3*x^3)*Log[5])/(2*x + 3*x^2) + (25*E^(-2*x^2 - 2*x^3)*x^3*(2*x^2
+ 3*x^3)*Log[5]^2)/(2*x + 3*x^2) + (40*E^(-x^2 - x^3)*x^2*(3*x^4 + x^2*Log[25] + x^3*(2 + Log[125])))/(2*x + 3
*x^2) + 150*Defer[Int][E^(-2*x^2 - 2*x^3)*x^5, x] - 100*Defer[Int][E^(-2*x^2 - 2*x^3)*x^7, x] - 150*Defer[Int]
[E^(-2*x^2 - 2*x^3)*x^8, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (32 e^{-2 x^2-2 x^3+2 x^2 (1+x)} x+150 e^{-2 x^2-2 x^3} x^5-100 e^{-2 x^2-2 x^3} x^7-150 e^{-2 x^2-2 x^3} x^8-50 e^{-2 x^2-2 x^3} x^4 \left (-5+4 x^2+6 x^3\right ) \log (5)-50 e^{-2 x^2-2 x^3} x^3 \left (-2+2 x^2+3 x^3\right ) \log ^2(5)+40 e^{-x^2-x^3} x^2 \left (4 x-3 x^4+3 \log (5)-x^2 \log (25)-x^3 (2+\log (125))\right )\right ) \, dx \\ & = 32 \int e^{-2 x^2-2 x^3+2 x^2 (1+x)} x \, dx+40 \int e^{-x^2-x^3} x^2 \left (4 x-3 x^4+3 \log (5)-x^2 \log (25)-x^3 (2+\log (125))\right ) \, dx-100 \int e^{-2 x^2-2 x^3} x^7 \, dx+150 \int e^{-2 x^2-2 x^3} x^5 \, dx-150 \int e^{-2 x^2-2 x^3} x^8 \, dx-(50 \log (5)) \int e^{-2 x^2-2 x^3} x^4 \left (-5+4 x^2+6 x^3\right ) \, dx-\left (50 \log ^2(5)\right ) \int e^{-2 x^2-2 x^3} x^3 \left (-2+2 x^2+3 x^3\right ) \, dx \\ & = \frac {50 e^{-2 x^2-2 x^3} x^4 \left (2 x^2+3 x^3\right ) \log (5)}{2 x+3 x^2}+\frac {25 e^{-2 x^2-2 x^3} x^3 \left (2 x^2+3 x^3\right ) \log ^2(5)}{2 x+3 x^2}+\frac {40 e^{-x^2-x^3} x^2 \left (3 x^4+x^2 \log (25)+x^3 (2+\log (125))\right )}{2 x+3 x^2}+32 \int x \, dx-100 \int e^{-2 x^2-2 x^3} x^7 \, dx+150 \int e^{-2 x^2-2 x^3} x^5 \, dx-150 \int e^{-2 x^2-2 x^3} x^8 \, dx \\ & = 16 x^2+\frac {50 e^{-2 x^2-2 x^3} x^4 \left (2 x^2+3 x^3\right ) \log (5)}{2 x+3 x^2}+\frac {25 e^{-2 x^2-2 x^3} x^3 \left (2 x^2+3 x^3\right ) \log ^2(5)}{2 x+3 x^2}+\frac {40 e^{-x^2-x^3} x^2 \left (3 x^4+x^2 \log (25)+x^3 (2+\log (125))\right )}{2 x+3 x^2}-100 \int e^{-2 x^2-2 x^3} x^7 \, dx+150 \int e^{-2 x^2-2 x^3} x^5 \, dx-150 \int e^{-2 x^2-2 x^3} x^8 \, dx \\ \end{align*}

Mathematica [F]

\[ \int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx=\int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx \]

[In]

Integrate[E^(-2*x^2 - 2*x^3)*(32*E^(2*x^2 + 2*x^3)*x + 150*x^5 - 100*x^7 - 150*x^8 + (250*x^4 - 200*x^6 - 300*
x^7)*Log[5] + (100*x^3 - 100*x^5 - 150*x^6)*Log[5]^2 + E^(x^2 + x^3)*(160*x^3 - 80*x^5 - 120*x^6 + (120*x^2 -
80*x^4 - 120*x^5)*Log[5])),x]

[Out]

Integrate[E^(-2*x^2 - 2*x^3)*(32*E^(2*x^2 + 2*x^3)*x + 150*x^5 - 100*x^7 - 150*x^8 + (250*x^4 - 200*x^6 - 300*
x^7)*Log[5] + (100*x^3 - 100*x^5 - 150*x^6)*Log[5]^2 + E^(x^2 + x^3)*(160*x^3 - 80*x^5 - 120*x^6 + (120*x^2 -
80*x^4 - 120*x^5)*Log[5])), x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(25)=50\).

Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48

method	result	size
risch	\(16 x^{2}+\left (40 x^{3} \ln \left (5\right )+40 x^{4}\right ) {\mathrm e}^{-x^{2} \left (1+x \right )}+\left (25 x^{4} \ln \left (5\right )^{2}+50 x^{5} \ln \left (5\right )+25 x^{6}\right ) {\mathrm e}^{-2 x^{2} \left (1+x \right )}\)	\(62\)
parts	\(16 x^{2}+\left (40 x^{3} \ln \left (5\right )+40 x^{4}\right ) {\mathrm e}^{-x^{3}-x^{2}}+\left (25 x^{4} \ln \left (5\right )^{2}+50 x^{5} \ln \left (5\right )+25 x^{6}\right ) {\mathrm e}^{-2 x^{3}-2 x^{2}}\)	\(64\)
norman	\(\left (25 x^{6}+16 x^{2} {\mathrm e}^{2 x^{3}+2 x^{2}}+25 x^{4} \ln \left (5\right )^{2}+50 x^{5} \ln \left (5\right )+40 \,{\mathrm e}^{x^{3}+x^{2}} x^{4}+40 \ln \left (5\right ) {\mathrm e}^{x^{3}+x^{2}} x^{3}\right ) {\mathrm e}^{-2 x^{3}-2 x^{2}}\)	\(77\)
parallelrisch	\(\left (25 x^{4} \ln \left (5\right )^{2}+50 x^{5} \ln \left (5\right )+25 x^{6}+40 \,{\mathrm e}^{x^{2} \left (1+x \right )} \ln \left (5\right ) x^{3}+40 \,{\mathrm e}^{x^{2} \left (1+x \right )} x^{4}+16 x^{2} {\mathrm e}^{2 x^{2} \left (1+x \right )}\right ) {\mathrm e}^{-2 x^{2} \left (1+x \right )}\)	\(77\)

[In]

int((32*x*exp(x^3+x^2)^2+((-120*x^5-80*x^4+120*x^2)*ln(5)-120*x^6-80*x^5+160*x^3)*exp(x^3+x^2)+(-150*x^6-100*x
^5+100*x^3)*ln(5)^2+(-300*x^7-200*x^6+250*x^4)*ln(5)-150*x^8-100*x^7+150*x^5)/exp(x^3+x^2)^2,x,method=_RETURNV
ERBOSE)

[Out]

16*x^2+(40*x^3*ln(5)+40*x^4)*exp(-x^2*(1+x))+(25*x^4*ln(5)^2+50*x^5*ln(5)+25*x^6)*exp(-2*x^2*(1+x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (24) = 48\).

Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.88 \[ \int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx={\left (25 \, x^{6} + 50 \, x^{5} \log \left (5\right ) + 25 \, x^{4} \log \left (5\right )^{2} + 16 \, x^{2} e^{\left (2 \, x^{3} + 2 \, x^{2}\right )} + 40 \, {\left (x^{4} + x^{3} \log \left (5\right )\right )} e^{\left (x^{3} + x^{2}\right )}\right )} e^{\left (-2 \, x^{3} - 2 \, x^{2}\right )} \]

[In]

integrate((32*x*exp(x^3+x^2)^2+((-120*x^5-80*x^4+120*x^2)*log(5)-120*x^6-80*x^5+160*x^3)*exp(x^3+x^2)+(-150*x^
6-100*x^5+100*x^3)*log(5)^2+(-300*x^7-200*x^6+250*x^4)*log(5)-150*x^8-100*x^7+150*x^5)/exp(x^3+x^2)^2,x, algor
ithm="fricas")

[Out]

(25*x^6 + 50*x^5*log(5) + 25*x^4*log(5)^2 + 16*x^2*e^(2*x^3 + 2*x^2) + 40*(x^4 + x^3*log(5))*e^(x^3 + x^2))*e^
(-2*x^3 - 2*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).

Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx=16 x^{2} + \left (40 x^{4} + 40 x^{3} \log {\left (5 \right )}\right ) e^{- x^{3} - x^{2}} + \left (25 x^{6} + 50 x^{5} \log {\left (5 \right )} + 25 x^{4} \log {\left (5 \right )}^{2}\right ) e^{- 2 x^{3} - 2 x^{2}} \]

[In]

integrate((32*x*exp(x**3+x**2)**2+((-120*x**5-80*x**4+120*x**2)*ln(5)-120*x**6-80*x**5+160*x**3)*exp(x**3+x**2
)+(-150*x**6-100*x**5+100*x**3)*ln(5)**2+(-300*x**7-200*x**6+250*x**4)*ln(5)-150*x**8-100*x**7+150*x**5)/exp(x
**3+x**2)**2,x)

[Out]

16*x**2 + (40*x**4 + 40*x**3*log(5))*exp(-x**3 - x**2) + (25*x**6 + 50*x**5*log(5) + 25*x**4*log(5)**2)*exp(-2
*x**3 - 2*x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (24) = 48\).

Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx=16 \, x^{2} + 5 \, {\left (5 \, {\left (x^{6} + 2 \, x^{5} \log \left (5\right ) + x^{4} \log \left (5\right )^{2}\right )} e^{\left (-2 \, x^{3}\right )} + 8 \, {\left (x^{4} + x^{3} \log \left (5\right )\right )} e^{\left (-x^{3} + x^{2}\right )}\right )} e^{\left (-2 \, x^{2}\right )} \]

[In]

integrate((32*x*exp(x^3+x^2)^2+((-120*x^5-80*x^4+120*x^2)*log(5)-120*x^6-80*x^5+160*x^3)*exp(x^3+x^2)+(-150*x^
6-100*x^5+100*x^3)*log(5)^2+(-300*x^7-200*x^6+250*x^4)*log(5)-150*x^8-100*x^7+150*x^5)/exp(x^3+x^2)^2,x, algor
ithm="maxima")

[Out]

16*x^2 + 5*(5*(x^6 + 2*x^5*log(5) + x^4*log(5)^2)*e^(-2*x^3) + 8*(x^4 + x^3*log(5))*e^(-x^3 + x^2))*e^(-2*x^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (24) = 48\).

Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.96 \[ \int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx=25 \, x^{6} e^{\left (-2 \, x^{3} - 2 \, x^{2}\right )} + 50 \, x^{5} e^{\left (-2 \, x^{3} - 2 \, x^{2}\right )} \log \left (5\right ) + 25 \, x^{4} e^{\left (-2 \, x^{3} - 2 \, x^{2}\right )} \log \left (5\right )^{2} + 40 \, x^{4} e^{\left (-x^{3} - x^{2}\right )} + 40 \, x^{3} e^{\left (-x^{3} - x^{2}\right )} \log \left (5\right ) + 16 \, x^{2} \]

[In]

integrate((32*x*exp(x^3+x^2)^2+((-120*x^5-80*x^4+120*x^2)*log(5)-120*x^6-80*x^5+160*x^3)*exp(x^3+x^2)+(-150*x^
6-100*x^5+100*x^3)*log(5)^2+(-300*x^7-200*x^6+250*x^4)*log(5)-150*x^8-100*x^7+150*x^5)/exp(x^3+x^2)^2,x, algor
ithm="giac")

[Out]

25*x^6*e^(-2*x^3 - 2*x^2) + 50*x^5*e^(-2*x^3 - 2*x^2)*log(5) + 25*x^4*e^(-2*x^3 - 2*x^2)*log(5)^2 + 40*x^4*e^(
-x^3 - x^2) + 40*x^3*e^(-x^3 - x^2)*log(5) + 16*x^2

Mupad [F(-1)]

Timed out. \[ \int e^{-2 x^2-2 x^3} \left (32 e^{2 x^2+2 x^3} x+150 x^5-100 x^7-150 x^8+\left (250 x^4-200 x^6-300 x^7\right ) \log (5)+\left (100 x^3-100 x^5-150 x^6\right ) \log ^2(5)+e^{x^2+x^3} \left (160 x^3-80 x^5-120 x^6+\left (120 x^2-80 x^4-120 x^5\right ) \log (5)\right )\right ) \, dx=\int -{\mathrm {e}}^{-2\,x^3-2\,x^2}\,\left ({\ln \left (5\right )}^2\,\left (150\,x^6+100\,x^5-100\,x^3\right )-32\,x\,{\mathrm {e}}^{2\,x^3+2\,x^2}+{\mathrm {e}}^{x^3+x^2}\,\left (\ln \left (5\right )\,\left (120\,x^5+80\,x^4-120\,x^2\right )-160\,x^3+80\,x^5+120\,x^6\right )+\ln \left (5\right )\,\left (300\,x^7+200\,x^6-250\,x^4\right )-150\,x^5+100\,x^7+150\,x^8\right ) \,d x \]

[In]

int(-exp(- 2*x^2 - 2*x^3)*(log(5)^2*(100*x^5 - 100*x^3 + 150*x^6) - 32*x*exp(2*x^2 + 2*x^3) + exp(x^2 + x^3)*(
log(5)*(80*x^4 - 120*x^2 + 120*x^5) - 160*x^3 + 80*x^5 + 120*x^6) + log(5)*(200*x^6 - 250*x^4 + 300*x^7) - 150
*x^5 + 100*x^7 + 150*x^8),x)

[Out]

int(-exp(- 2*x^2 - 2*x^3)*(log(5)^2*(100*x^5 - 100*x^3 + 150*x^6) - 32*x*exp(2*x^2 + 2*x^3) + exp(x^2 + x^3)*(
log(5)*(80*x^4 - 120*x^2 + 120*x^5) - 160*x^3 + 80*x^5 + 120*x^6) + log(5)*(200*x^6 - 250*x^4 + 300*x^7) - 150
*x^5 + 100*x^7 + 150*x^8), x)