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

Optimal. Leaf size=38 \[ \frac {x^2-\frac {\log (5)}{x}}{-x-e x-x^2+e^{2 x^2} x^2} \]

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Rubi [F]  time = 2.99, 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 {-x^3-e x^3+(-2-2 e-3 x) \log (5)+e^{2 x^2} \left (-4 x^6+\left (3 x+4 x^3\right ) \log (5)\right )}{x^3+e^2 x^3+2 x^4+x^5+e^{4 x^2} x^5+e \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (-2 x^4-2 e x^4-2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-((1 + E - 4*Log[5])*Defer[Int][(1 + E + x - E^(2*x^2)*x)^(-2), x]) + (1 + E)*Log[5]*Defer[Int][1/(x^3*(1 + E
+ x - E^(2*x^2)*x)^2), x] + 4*(1 + E)*Log[5]*Defer[Int][1/(x*(1 + E + x - E^(2*x^2)*x)^2), x] - 4*(1 + E)*Defe
r[Int][x^2/(1 + E + x - E^(2*x^2)*x)^2, x] - 4*Defer[Int][x^3/(-1 - E - x + E^(2*x^2)*x)^2, x] + Log[125]*Defe
r[Int][1/(x^3*(-1 - E - x + E^(2*x^2)*x)), x] + 4*Log[5]*Defer[Int][1/(x*(-1 - E - x + E^(2*x^2)*x)), x] - 4*D
efer[Int][x^2/(-1 - E - x + E^(2*x^2)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^3-e x^3+(-2-2 e-3 x) \log (5)+e^{2 x^2} \left (-4 x^6+\left (3 x+4 x^3\right ) \log (5)\right )}{\left (1+e^2\right ) x^3+2 x^4+x^5+e^{4 x^2} x^5+e \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (-2 x^4-2 e x^4-2 x^5\right )} \, dx\\ &=\int \frac {(-1-e) x^3+(-2-2 e-3 x) \log (5)+e^{2 x^2} \left (-4 x^6+\left (3 x+4 x^3\right ) \log (5)\right )}{\left (1+e^2\right ) x^3+2 x^4+x^5+e^{4 x^2} x^5+e \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (-2 x^4-2 e x^4-2 x^5\right )} \, dx\\ &=\int \frac {-4 e^{2 x^2} x^6-2 (1+e) \log (5)+3 \left (-1+e^{2 x^2}\right ) x \log (5)-x^3 \left (1+e-4 e^{2 x^2} \log (5)\right )}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2} \, dx\\ &=\int \left (\frac {\left (1+e+4 (1+e) x^2+4 x^3\right ) \left (-x^3+\log (5)\right )}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2}-\frac {4 x^5-4 x^2 \log (5)-\log (125)}{x^3 \left (-1-e-x+e^{2 x^2} x\right )}\right ) \, dx\\ &=\int \frac {\left (1+e+4 (1+e) x^2+4 x^3\right ) \left (-x^3+\log (5)\right )}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2} \, dx-\int \frac {4 x^5-4 x^2 \log (5)-\log (125)}{x^3 \left (-1-e-x+e^{2 x^2} x\right )} \, dx\\ &=\int \left (-\frac {4 (1+e) x^2}{\left (1+e+x-e^{2 x^2} x\right )^2}-\frac {4 x^3}{\left (-1-e-x+e^{2 x^2} x\right )^2}-\frac {1+e-4 \log (5)}{\left (1+e+x-e^{2 x^2} x\right )^2}+\frac {(1+e) \log (5)}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2}+\frac {4 (1+e) \log (5)}{x \left (1+e+x-e^{2 x^2} x\right )^2}\right ) \, dx-\int \left (\frac {4 x^2}{-1-e-x+e^{2 x^2} x}-\frac {4 \log (5)}{x \left (-1-e-x+e^{2 x^2} x\right )}-\frac {\log (125)}{x^3 \left (-1-e-x+e^{2 x^2} x\right )}\right ) \, dx\\ &=-\left (4 \int \frac {x^3}{\left (-1-e-x+e^{2 x^2} x\right )^2} \, dx\right )-4 \int \frac {x^2}{-1-e-x+e^{2 x^2} x} \, dx-(4 (1+e)) \int \frac {x^2}{\left (1+e+x-e^{2 x^2} x\right )^2} \, dx+(4 \log (5)) \int \frac {1}{x \left (-1-e-x+e^{2 x^2} x\right )} \, dx+((1+e) \log (5)) \int \frac {1}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2} \, dx+(4 (1+e) \log (5)) \int \frac {1}{x \left (1+e+x-e^{2 x^2} x\right )^2} \, dx+(-1-e+4 \log (5)) \int \frac {1}{\left (1+e+x-e^{2 x^2} x\right )^2} \, dx+\log (125) \int \frac {1}{x^3 \left (-1-e-x+e^{2 x^2} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.72, size = 32, normalized size = 0.84 \begin {gather*} -\frac {-x^3+\log (5)}{x^2 \left (-1-e-x+e^{2 x^2} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 39, normalized size = 1.03 \begin {gather*} \frac {x^{3} - \log \relax (5)}{x^{3} e^{\left (2 \, x^{2}\right )} - x^{3} - x^{2} e - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.24, size = 29, normalized size = 0.76

method	result	size
risch	\(\frac {-x^{3}+\ln \relax (5)}{x^{2} \left (-x \,{\mathrm e}^{2 x^{2}}+{\mathrm e}+x +1\right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.65, size = 36, normalized size = 0.95 \begin {gather*} \frac {x^{3} - \log \relax (5)}{x^{3} e^{\left (2 \, x^{2}\right )} - x^{3} - x^{2} {\left (e + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.72, size = 35, normalized size = 0.92 \begin {gather*} \frac {\ln \relax (5)-x^3}{x^2\,\mathrm {e}-x^3\,{\mathrm {e}}^{2\,x^2}+x^2+x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.19, size = 29, normalized size = 0.76 \begin {gather*} \frac {x^{3} - \log {\relax (5 )}}{x^{3} e^{2 x^{2}} - x^{3} - e x^{2} - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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