3.101.85 \(\int \frac {-5 x^2+x^3+e^x (3+8 x)}{-x^3+e^x (x+2 x^2)} \, dx\)

Optimal. Leaf size=20 \[ \log \left (4 x^3 \left (-1-2 x+e^{-x} x^2\right )\right ) \]

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Rubi [F]  time = 0.76, 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 {-5 x^2+x^3+e^x (3+8 x)}{-x^3+e^x \left (x+2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.23, size = 24, normalized size = 1.20 \begin {gather*} -x+3 \log (x)+\log \left (e^x+2 e^x x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x^2 + x^3 + E^x*(3 + 8*x))/(-x^3 + E^x*(x + 2*x^2)),x]

[Out]

-x + 3*Log[x] + Log[E^x + 2*E^x*x - x^2]

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fricas [A]  time = 0.60, size = 37, normalized size = 1.85 \begin {gather*} -x + \log \left (2 \, x + 1\right ) + 3 \, \log \relax (x) + \log \left (-\frac {x^{2} - {\left (2 \, x + 1\right )} e^{x}}{2 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+3)*exp(x)+x^3-5*x^2)/((2*x^2+x)*exp(x)-x^3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 22, normalized size = 1.10 \begin {gather*} -x + \log \left (-x^{2} + 2 \, x e^{x} + e^{x}\right ) + 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+3)*exp(x)+x^3-5*x^2)/((2*x^2+x)*exp(x)-x^3),x, algorithm="giac")

[Out]

-x + log(-x^2 + 2*x*e^x + e^x) + 3*log(x)

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maple [A]  time = 0.03, size = 23, normalized size = 1.15

method	result	size
norman	\(-x +3 \ln \relax (x )+\ln \left (x^{2}-2 \,{\mathrm e}^{x} x -{\mathrm e}^{x}\right )\)	\(23\)
risch	\(3 \ln \relax (x )+\ln \left (2 x +1\right )-x +\ln \left ({\mathrm e}^{x}-\frac {x^{2}}{2 x +1}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x+3*ln(x)+ln(x^2-2*exp(x)*x-exp(x))

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maxima [A]  time = 0.37, size = 37, normalized size = 1.85 \begin {gather*} -x + \log \left (2 \, x + 1\right ) + 3 \, \log \relax (x) + \log \left (-\frac {x^{2} - {\left (2 \, x + 1\right )} e^{x}}{2 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+3)*exp(x)+x^3-5*x^2)/((2*x^2+x)*exp(x)-x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.28, size = 22, normalized size = 1.10 \begin {gather*} \ln \left (x^2-2\,x\,{\mathrm {e}}^x-{\mathrm {e}}^x\right )-x+3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(8*x + 3) - 5*x^2 + x^3)/(exp(x)*(x + 2*x^2) - x^3),x)

[Out]

log(x^2 - 2*x*exp(x) - exp(x)) - x + 3*log(x)

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sympy [A]  time = 0.30, size = 26, normalized size = 1.30 \begin {gather*} - x + 3 \log {\relax (x )} + \log {\left (x + \frac {1}{2} \right )} + \log {\left (- \frac {x^{2}}{2 x + 1} + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+3)*exp(x)+x**3-5*x**2)/((2*x**2+x)*exp(x)-x**3),x)

[Out]

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

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