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

Optimal. Leaf size=26 \[ 9-\frac {2}{x^2}+\log \left (4 e^{2 e} \left (e^{-5+2 x}-x\right )\right ) \]

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Rubi [F]  time = 0.30, 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 {-4 x-x^3+e^{-5+2 x} \left (4+2 x^3\right )}{e^{-5+2 x} x^3-x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 19, normalized size = 0.73 \begin {gather*} -\frac {2}{x^2}+\log \left (e^{2 x}-e^5 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/x^2 + Log[E^(2*x) - E^5*x]

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fricas [A]  time = 0.79, size = 21, normalized size = 0.81 \begin {gather*} \frac {x^{2} \log \left (-x + e^{\left (2 \, x - 5\right )}\right ) - 2}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*log(-x + e^(2*x - 5)) - 2)/x^2

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giac [A]  time = 0.16, size = 21, normalized size = 0.81 \begin {gather*} \frac {x^{2} \log \left (-x e^{5} + e^{\left (2 \, x\right )}\right ) - 2}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*log(-x*e^5 + e^(2*x)) - 2)/x^2

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maple [A]  time = 0.17, size = 18, normalized size = 0.69

method	result	size
norman	\(-\frac {2}{x^{2}}+\ln \left (x -{\mathrm e}^{2 x -5}\right )\)	\(18\)
risch	\(-\frac {2}{x^{2}}+5+\ln \left ({\mathrm e}^{2 x -5}-x \right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.38, size = 17, normalized size = 0.65 \begin {gather*} -\frac {2}{x^{2}} + \log \left (-x e^{5} + e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/x^2 + log(-x*e^5 + e^(2*x))

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mupad [B]  time = 0.14, size = 17, normalized size = 0.65 \begin {gather*} \ln \left (x-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-5}\right )-\frac {2}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - exp(2*x)*exp(-5)) - 2/x^2

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sympy [A]  time = 0.11, size = 14, normalized size = 0.54 \begin {gather*} \log {\left (- x + e^{2 x - 5} \right )} - \frac {2}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-x + exp(2*x - 5)) - 2/x**2

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