3.29.36 \(\int \frac {1+e^2+e (2-2 x)-2 x+x^2+e^{3 x} (-3 x^2-3 e^2 x^2+6 x^3-3 x^4+e (-6 x^2+6 x^3))+x^2 \log (5)}{1+2 x-3 x^2-4 x^3+4 x^4+e^2 (1+4 x+4 x^2)+e (2+6 x-8 x^3)+e^{6 x} (x^2+e^2 x^2-2 x^3+x^4+e (2 x^2-2 x^3))+(-2 x-2 x^2+4 x^3+e (-2 x-4 x^2)) \log (5)+x^2 \log ^2(5)+e^{3 x} (2 x-6 x^3+4 x^4+e^2 (2 x+4 x^2)+e (4 x+4 x^2-8 x^3)+(-2 x^2-2 e x^2+2 x^3) \log (5))} \, dx\)

Optimal. Leaf size=31 \[ \frac {x}{1+2 x-x \left (-e^{3 x}+\frac {\log (5)}{1+e-x}\right )} \]

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Rubi [F]  time = 3.90, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+e^2+e (2-2 x)-2 x+x^2+e^{3 x} \left (-3 x^2-3 e^2 x^2+6 x^3-3 x^4+e \left (-6 x^2+6 x^3\right )\right )+x^2 \log (5)}{1+2 x-3 x^2-4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (2+6 x-8 x^3\right )+e^{6 x} \left (x^2+e^2 x^2-2 x^3+x^4+e \left (2 x^2-2 x^3\right )\right )+\left (-2 x-2 x^2+4 x^3+e \left (-2 x-4 x^2\right )\right ) \log (5)+x^2 \log ^2(5)+e^{3 x} \left (2 x-6 x^3+4 x^4+e^2 \left (2 x+4 x^2\right )+e \left (4 x+4 x^2-8 x^3\right )+\left (-2 x^2-2 e x^2+2 x^3\right ) \log (5)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + E^2 + E*(2 - 2*x) - 2*x + x^2 + E^(3*x)*(-3*x^2 - 3*E^2*x^2 + 6*x^3 - 3*x^4 + E*(-6*x^2 + 6*x^3)) + x
^2*Log[5])/(1 + 2*x - 3*x^2 - 4*x^3 + 4*x^4 + E^2*(1 + 4*x + 4*x^2) + E*(2 + 6*x - 8*x^3) + E^(6*x)*(x^2 + E^2
*x^2 - 2*x^3 + x^4 + E*(2*x^2 - 2*x^3)) + (-2*x - 2*x^2 + 4*x^3 + E*(-2*x - 4*x^2))*Log[5] + x^2*Log[5]^2 + E^
(3*x)*(2*x - 6*x^3 + 4*x^4 + E^2*(2*x + 4*x^2) + E*(4*x + 4*x^2 - 8*x^3) + (-2*x^2 - 2*E*x^2 + 2*x^3)*Log[5]))
,x]

[Out]

(1 + E)^2*Defer[Int][(1 + E + E^(3*x)*(1 + E)*x - 2*x^2 - E^(3*x)*x^2 + x*(1 + 2*E - Log[5]))^(-2), x] + (1 +
E)*(1 + 3*E)*Defer[Int][x/(1 + E + E^(3*x)*(1 + E)*x - 2*x^2 - E^(3*x)*x^2 + x*(1 + 2*E - Log[5]))^2, x] + (1
+ 6*E^2 + 3*E*(2 - Log[5]) - 2*Log[5])*Defer[Int][x^2/(1 + E + E^(3*x)*(1 + E)*x - 2*x^2 - E^(3*x)*x^2 + x*(1
+ 2*E - Log[5]))^2, x] - 3*(3 + 4*E - Log[5])*Defer[Int][x^3/(1 + E + E^(3*x)*(1 + E)*x - 2*x^2 - E^(3*x)*x^2
+ x*(1 + 2*E - Log[5]))^2, x] + 6*Defer[Int][x^4/(1 + E + E^(3*x)*(1 + E)*x - 2*x^2 - E^(3*x)*x^2 + x*(1 + 2*E
- Log[5]))^2, x] - 3*(1 + E)*Defer[Int][x/(1 + E + E^(3*x)*(1 + E)*x - 2*x^2 - E^(3*x)*x^2 + x*(1 + 2*E - Log
[5])), x] + 3*Defer[Int][x^2/(1 + E + E^(3*x)*(1 + E)*x - 2*x^2 - E^(3*x)*x^2 + x*(1 + 2*E - Log[5])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+e^2+e (2-2 x)-2 x+e^{3 x} \left (-3 x^2-3 e^2 x^2+6 x^3-3 x^4+e \left (-6 x^2+6 x^3\right )\right )+x^2 (1+\log (5))}{1+2 x-3 x^2-4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (2+6 x-8 x^3\right )+e^{6 x} \left (x^2+e^2 x^2-2 x^3+x^4+e \left (2 x^2-2 x^3\right )\right )+\left (-2 x-2 x^2+4 x^3+e \left (-2 x-4 x^2\right )\right ) \log (5)+x^2 \log ^2(5)+e^{3 x} \left (2 x-6 x^3+4 x^4+e^2 \left (2 x+4 x^2\right )+e \left (4 x+4 x^2-8 x^3\right )+\left (-2 x^2-2 e x^2+2 x^3\right ) \log (5)\right )} \, dx\\ &=\int \frac {1+e^2+e (2-2 x)-2 x+e^{3 x} \left (-3 x^2-3 e^2 x^2+6 x^3-3 x^4+e \left (-6 x^2+6 x^3\right )\right )+x^2 (1+\log (5))}{1+2 x-4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (2+6 x-8 x^3\right )+e^{6 x} \left (x^2+e^2 x^2-2 x^3+x^4+e \left (2 x^2-2 x^3\right )\right )+\left (-2 x-2 x^2+4 x^3+e \left (-2 x-4 x^2\right )\right ) \log (5)+e^{3 x} \left (2 x-6 x^3+4 x^4+e^2 \left (2 x+4 x^2\right )+e \left (4 x+4 x^2-8 x^3\right )+\left (-2 x^2-2 e x^2+2 x^3\right ) \log (5)\right )+x^2 \left (-3+\log ^2(5)\right )} \, dx\\ &=\int \frac {1+e^2-2 e (-1+x)-2 x-3 e^{2+3 x} x^2+6 e^{1+3 x} (-1+x) x^2-3 e^{3 x} (-1+x)^2 x^2+x^2 (1+\log (5))}{\left (1+e+e^{1+3 x} x-e^{3 x} (-1+x) x-2 x^2+x (1+2 e-\log (5))\right )^2} \, dx\\ &=\int \left (\frac {3 x (-1-e+x)}{1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))}+\frac {(1+e)^2+(1+e) (1+3 e) x+6 x^4+x^2 \left (1+6 e^2+3 e (2-\log (5))-2 \log (5)\right )-3 x^3 (3+4 e-\log (5))}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2}\right ) \, dx\\ &=3 \int \frac {x (-1-e+x)}{1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))} \, dx+\int \frac {(1+e)^2+(1+e) (1+3 e) x+6 x^4+x^2 \left (1+6 e^2+3 e (2-\log (5))-2 \log (5)\right )-3 x^3 (3+4 e-\log (5))}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2} \, dx\\ &=3 \int \left (\frac {(-1-e) x}{1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))}+\frac {x^2}{1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))}\right ) \, dx+\int \left (\frac {(1+e)^2}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2}+\frac {(1+e) (1+3 e) x}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2}+\frac {6 x^4}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2}+\frac {x^2 \left (1+6 e^2+3 e (2-\log (5))-2 \log (5)\right )}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2}+\frac {3 x^3 (-3-4 e+\log (5))}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2}\right ) \, dx\\ &=3 \int \frac {x^2}{1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))} \, dx+6 \int \frac {x^4}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2} \, dx-(3 (1+e)) \int \frac {x}{1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))} \, dx+(1+e)^2 \int \frac {1}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2} \, dx+((1+e) (1+3 e)) \int \frac {x}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2} \, dx+\left (1+6 e^2+3 e (2-\log (5))-2 \log (5)\right ) \int \frac {x^2}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2} \, dx-(3 (3+4 e-\log (5))) \int \frac {x^3}{\left (1+e+e^{3 x} (1+e) x-2 x^2-e^{3 x} x^2+x (1+2 e-\log (5))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 48, normalized size = 1.55 \begin {gather*} \frac {(1+e-x) x}{1+e+x+2 e x+e^{1+3 x} x-e^{3 x} (-1+x) x-2 x^2-x \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^2 + E*(2 - 2*x) - 2*x + x^2 + E^(3*x)*(-3*x^2 - 3*E^2*x^2 + 6*x^3 - 3*x^4 + E*(-6*x^2 + 6*x^3
)) + x^2*Log[5])/(1 + 2*x - 3*x^2 - 4*x^3 + 4*x^4 + E^2*(1 + 4*x + 4*x^2) + E*(2 + 6*x - 8*x^3) + E^(6*x)*(x^2
+ E^2*x^2 - 2*x^3 + x^4 + E*(2*x^2 - 2*x^3)) + (-2*x - 2*x^2 + 4*x^3 + E*(-2*x - 4*x^2))*Log[5] + x^2*Log[5]^
2 + E^(3*x)*(2*x - 6*x^3 + 4*x^4 + E^2*(2*x + 4*x^2) + E*(4*x + 4*x^2 - 8*x^3) + (-2*x^2 - 2*E*x^2 + 2*x^3)*Lo
g[5])),x]

[Out]

((1 + E - x)*x)/(1 + E + x + 2*E*x + E^(1 + 3*x)*x - E^(3*x)*(-1 + x)*x - 2*x^2 - x*Log[5])

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fricas [A]  time = 1.02, size = 55, normalized size = 1.77 \begin {gather*} \frac {x^{2} - x e - x}{2 \, x^{2} - {\left (2 \, x + 1\right )} e + {\left (x^{2} - x e - x\right )} e^{\left (3 \, x\right )} + x \log \relax (5) - x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2*exp(1)^2+(6*x^3-6*x^2)*exp(1)-3*x^4+6*x^3-3*x^2)*exp(3*x)+x^2*log(5)+exp(1)^2+(-2*x+2)*exp(
1)+x^2-2*x+1)/((x^2*exp(1)^2+(-2*x^3+2*x^2)*exp(1)+x^4-2*x^3+x^2)*exp(3*x)^2+((-2*x^2*exp(1)+2*x^3-2*x^2)*log(
5)+(4*x^2+2*x)*exp(1)^2+(-8*x^3+4*x^2+4*x)*exp(1)+4*x^4-6*x^3+2*x)*exp(3*x)+x^2*log(5)^2+((-4*x^2-2*x)*exp(1)+
4*x^3-2*x^2-2*x)*log(5)+(4*x^2+4*x+1)*exp(1)^2+(-8*x^3+6*x+2)*exp(1)+4*x^4-4*x^3-3*x^2+2*x+1),x, algorithm="fr
icas")

[Out]

(x^2 - x*e - x)/(2*x^2 - (2*x + 1)*e + (x^2 - x*e - x)*e^(3*x) + x*log(5) - x - 1)

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giac [B]  time = 1.13, size = 62, normalized size = 2.00 \begin {gather*} \frac {x^{2} - x e - x}{x^{2} e^{\left (3 \, x\right )} + 2 \, x^{2} - 2 \, x e - x e^{\left (3 \, x\right )} - x e^{\left (3 \, x + 1\right )} + x \log \relax (5) - x - e - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2*exp(1)^2+(6*x^3-6*x^2)*exp(1)-3*x^4+6*x^3-3*x^2)*exp(3*x)+x^2*log(5)+exp(1)^2+(-2*x+2)*exp(
1)+x^2-2*x+1)/((x^2*exp(1)^2+(-2*x^3+2*x^2)*exp(1)+x^4-2*x^3+x^2)*exp(3*x)^2+((-2*x^2*exp(1)+2*x^3-2*x^2)*log(
5)+(4*x^2+2*x)*exp(1)^2+(-8*x^3+4*x^2+4*x)*exp(1)+4*x^4-6*x^3+2*x)*exp(3*x)+x^2*log(5)^2+((-4*x^2-2*x)*exp(1)+
4*x^3-2*x^2-2*x)*log(5)+(4*x^2+4*x+1)*exp(1)^2+(-8*x^3+6*x+2)*exp(1)+4*x^4-4*x^3-3*x^2+2*x+1),x, algorithm="gi
ac")

[Out]

(x^2 - x*e - x)/(x^2*e^(3*x) + 2*x^2 - 2*x*e - x*e^(3*x) - x*e^(3*x + 1) + x*log(5) - x - e - 1)

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maple [A]  time = 0.72, size = 60, normalized size = 1.94

method	result	size
risch	\(-\frac {\left ({\mathrm e}-x +1\right ) x}{-x \,{\mathrm e}^{3 x +1}+x^{2} {\mathrm e}^{3 x}+x \ln \relax (5)-2 x \,{\mathrm e}+2 x^{2}-x \,{\mathrm e}^{3 x}-{\mathrm e}-x -1}\)	\(60\)
norman	\(\frac {x^{2}+\left (-{\mathrm e}-1\right ) x}{-{\mathrm e}^{3 x} {\mathrm e} x +x^{2} {\mathrm e}^{3 x}+x \ln \relax (5)-2 x \,{\mathrm e}+2 x^{2}-x \,{\mathrm e}^{3 x}-{\mathrm e}-x -1}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^2*exp(1)^2+(6*x^3-6*x^2)*exp(1)-3*x^4+6*x^3-3*x^2)*exp(3*x)+x^2*ln(5)+exp(1)^2+(-2*x+2)*exp(1)+x^2-
2*x+1)/((x^2*exp(1)^2+(-2*x^3+2*x^2)*exp(1)+x^4-2*x^3+x^2)*exp(3*x)^2+((-2*x^2*exp(1)+2*x^3-2*x^2)*ln(5)+(4*x^
2+2*x)*exp(1)^2+(-8*x^3+4*x^2+4*x)*exp(1)+4*x^4-6*x^3+2*x)*exp(3*x)+x^2*ln(5)^2+((-4*x^2-2*x)*exp(1)+4*x^3-2*x
^2-2*x)*ln(5)+(4*x^2+4*x+1)*exp(1)^2+(-8*x^3+6*x+2)*exp(1)+4*x^4-4*x^3-3*x^2+2*x+1),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.97, size = 54, normalized size = 1.74 \begin {gather*} \frac {x^{2} - x {\left (e + 1\right )}}{2 \, x^{2} - x {\left (2 \, e - \log \relax (5) + 1\right )} + {\left (x^{2} - x {\left (e + 1\right )}\right )} e^{\left (3 \, x\right )} - e - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2*exp(1)^2+(6*x^3-6*x^2)*exp(1)-3*x^4+6*x^3-3*x^2)*exp(3*x)+x^2*log(5)+exp(1)^2+(-2*x+2)*exp(
1)+x^2-2*x+1)/((x^2*exp(1)^2+(-2*x^3+2*x^2)*exp(1)+x^4-2*x^3+x^2)*exp(3*x)^2+((-2*x^2*exp(1)+2*x^3-2*x^2)*log(
5)+(4*x^2+2*x)*exp(1)^2+(-8*x^3+4*x^2+4*x)*exp(1)+4*x^4-6*x^3+2*x)*exp(3*x)+x^2*log(5)^2+((-4*x^2-2*x)*exp(1)+
4*x^3-2*x^2-2*x)*log(5)+(4*x^2+4*x+1)*exp(1)^2+(-8*x^3+6*x+2)*exp(1)+4*x^4-4*x^3-3*x^2+2*x+1),x, algorithm="ma
xima")

[Out]

(x^2 - x*(e + 1))/(2*x^2 - x*(2*e - log(5) + 1) + (x^2 - x*(e + 1))*e^(3*x) - e - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^2-2\,x+x^2\,\ln \relax (5)-{\mathrm {e}}^{3\,x}\,\left (\mathrm {e}\,\left (6\,x^2-6\,x^3\right )+3\,x^2\,{\mathrm {e}}^2+3\,x^2-6\,x^3+3\,x^4\right )+x^2-\mathrm {e}\,\left (2\,x-2\right )+1}{2\,x+x^2\,{\ln \relax (5)}^2+{\mathrm {e}}^{6\,x}\,\left (\mathrm {e}\,\left (2\,x^2-2\,x^3\right )+x^2\,{\mathrm {e}}^2+x^2-2\,x^3+x^4\right )+{\mathrm {e}}^{3\,x}\,\left (2\,x+{\mathrm {e}}^2\,\left (4\,x^2+2\,x\right )-\ln \relax (5)\,\left (2\,x^2\,\mathrm {e}+2\,x^2-2\,x^3\right )+\mathrm {e}\,\left (-8\,x^3+4\,x^2+4\,x\right )-6\,x^3+4\,x^4\right )+{\mathrm {e}}^2\,\left (4\,x^2+4\,x+1\right )+\mathrm {e}\,\left (-8\,x^3+6\,x+2\right )-\ln \relax (5)\,\left (2\,x+\mathrm {e}\,\left (4\,x^2+2\,x\right )+2\,x^2-4\,x^3\right )-3\,x^2-4\,x^3+4\,x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2) - 2*x + x^2*log(5) - exp(3*x)*(exp(1)*(6*x^2 - 6*x^3) + 3*x^2*exp(2) + 3*x^2 - 6*x^3 + 3*x^4) + x^
2 - exp(1)*(2*x - 2) + 1)/(2*x + x^2*log(5)^2 + exp(6*x)*(exp(1)*(2*x^2 - 2*x^3) + x^2*exp(2) + x^2 - 2*x^3 +
x^4) + exp(3*x)*(2*x + exp(2)*(2*x + 4*x^2) - log(5)*(2*x^2*exp(1) + 2*x^2 - 2*x^3) + exp(1)*(4*x + 4*x^2 - 8*
x^3) - 6*x^3 + 4*x^4) + exp(2)*(4*x + 4*x^2 + 1) + exp(1)*(6*x - 8*x^3 + 2) - log(5)*(2*x + exp(1)*(2*x + 4*x^
2) + 2*x^2 - 4*x^3) - 3*x^2 - 4*x^3 + 4*x^4 + 1),x)

[Out]

int((exp(2) - 2*x + x^2*log(5) - exp(3*x)*(exp(1)*(6*x^2 - 6*x^3) + 3*x^2*exp(2) + 3*x^2 - 6*x^3 + 3*x^4) + x^
2 - exp(1)*(2*x - 2) + 1)/(2*x + x^2*log(5)^2 + exp(6*x)*(exp(1)*(2*x^2 - 2*x^3) + x^2*exp(2) + x^2 - 2*x^3 +
x^4) + exp(3*x)*(2*x + exp(2)*(2*x + 4*x^2) - log(5)*(2*x^2*exp(1) + 2*x^2 - 2*x^3) + exp(1)*(4*x + 4*x^2 - 8*
x^3) - 6*x^3 + 4*x^4) + exp(2)*(4*x + 4*x^2 + 1) + exp(1)*(6*x - 8*x^3 + 2) - log(5)*(2*x + exp(1)*(2*x + 4*x^
2) + 2*x^2 - 4*x^3) - 3*x^2 - 4*x^3 + 4*x^4 + 1), x)

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sympy [B]  time = 0.77, size = 48, normalized size = 1.55 \begin {gather*} \frac {x^{2} - e x - x}{2 x^{2} - 2 e x - x + x \log {\relax (5 )} + \left (x^{2} - e x - x\right ) e^{3 x} - e - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**2*exp(1)**2+(6*x**3-6*x**2)*exp(1)-3*x**4+6*x**3-3*x**2)*exp(3*x)+x**2*ln(5)+exp(1)**2+(-2*x
+2)*exp(1)+x**2-2*x+1)/((x**2*exp(1)**2+(-2*x**3+2*x**2)*exp(1)+x**4-2*x**3+x**2)*exp(3*x)**2+((-2*x**2*exp(1)
+2*x**3-2*x**2)*ln(5)+(4*x**2+2*x)*exp(1)**2+(-8*x**3+4*x**2+4*x)*exp(1)+4*x**4-6*x**3+2*x)*exp(3*x)+x**2*ln(5
)**2+((-4*x**2-2*x)*exp(1)+4*x**3-2*x**2-2*x)*ln(5)+(4*x**2+4*x+1)*exp(1)**2+(-8*x**3+6*x+2)*exp(1)+4*x**4-4*x
**3-3*x**2+2*x+1),x)

[Out]

(x**2 - E*x - x)/(2*x**2 - 2*E*x - x + x*log(5) + (x**2 - E*x - x)*exp(3*x) - E - 1)

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