3.35.2 \(\int \frac {-32+12 x+2 e^x x^3-8 x^4}{8 x-6 x^2+9 x^3+e^x x^3-2 x^5} \, dx\)

Optimal. Leaf size=30 \[ \log \left (\left (5+\frac {1}{2} \left (-1+e^x\right )+\frac {-3+\frac {4}{x}}{x}-x^2\right )^2\right ) \]

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Rubi [F]  time = 1.24, 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 {-32+12 x+2 e^x x^3-8 x^4}{8 x-6 x^2+9 x^3+e^x x^3-2 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-32 + 12*x + 2*E^x*x^3 - 8*x^4)/(8*x - 6*x^2 + 9*x^3 + E^x*x^3 - 2*x^5),x]

[Out]

2*x + 4*Defer[Int][(-8 + 6*x - 9*x^2 - E^x*x^2 + 2*x^4)^(-1), x] + 32*Defer[Int][1/(x*(-8 + 6*x - 9*x^2 - E^x*
x^2 + 2*x^4)), x] - 12*Defer[Int][x/(-8 + 6*x - 9*x^2 - E^x*x^2 + 2*x^4), x] + 18*Defer[Int][x^2/(-8 + 6*x - 9
*x^2 - E^x*x^2 + 2*x^4), x] + 8*Defer[Int][x^3/(-8 + 6*x - 9*x^2 - E^x*x^2 + 2*x^4), x] - 4*Defer[Int][x^4/(-8
+ 6*x - 9*x^2 - E^x*x^2 + 2*x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-16+6 x+e^x x^3-4 x^4\right )}{8 x-6 x^2+9 x^3+e^x x^3-2 x^5} \, dx\\ &=2 \int \frac {-16+6 x+e^x x^3-4 x^4}{8 x-6 x^2+9 x^3+e^x x^3-2 x^5} \, dx\\ &=2 \int \left (1-\frac {-16-2 x+6 x^2-9 x^3-4 x^4+2 x^5}{x \left (-8+6 x-9 x^2-e^x x^2+2 x^4\right )}\right ) \, dx\\ &=2 x-2 \int \frac {-16-2 x+6 x^2-9 x^3-4 x^4+2 x^5}{x \left (-8+6 x-9 x^2-e^x x^2+2 x^4\right )} \, dx\\ &=2 x-2 \int \left (-\frac {2}{-8+6 x-9 x^2-e^x x^2+2 x^4}-\frac {16}{x \left (-8+6 x-9 x^2-e^x x^2+2 x^4\right )}+\frac {6 x}{-8+6 x-9 x^2-e^x x^2+2 x^4}-\frac {9 x^2}{-8+6 x-9 x^2-e^x x^2+2 x^4}-\frac {4 x^3}{-8+6 x-9 x^2-e^x x^2+2 x^4}+\frac {2 x^4}{-8+6 x-9 x^2-e^x x^2+2 x^4}\right ) \, dx\\ &=2 x+4 \int \frac {1}{-8+6 x-9 x^2-e^x x^2+2 x^4} \, dx-4 \int \frac {x^4}{-8+6 x-9 x^2-e^x x^2+2 x^4} \, dx+8 \int \frac {x^3}{-8+6 x-9 x^2-e^x x^2+2 x^4} \, dx-12 \int \frac {x}{-8+6 x-9 x^2-e^x x^2+2 x^4} \, dx+18 \int \frac {x^2}{-8+6 x-9 x^2-e^x x^2+2 x^4} \, dx+32 \int \frac {1}{x \left (-8+6 x-9 x^2-e^x x^2+2 x^4\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 30, normalized size = 1.00 \begin {gather*} -4 \log (x)+2 \log \left (8-6 x+9 x^2+e^x x^2-2 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32 + 12*x + 2*E^x*x^3 - 8*x^4)/(8*x - 6*x^2 + 9*x^3 + E^x*x^3 - 2*x^5),x]

[Out]

-4*Log[x] + 2*Log[8 - 6*x + 9*x^2 + E^x*x^2 - 2*x^4]

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fricas [A]  time = 0.54, size = 30, normalized size = 1.00 \begin {gather*} 2 \, \log \left (-\frac {2 \, x^{4} - x^{2} e^{x} - 9 \, x^{2} + 6 \, x - 8}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)*x^3-8*x^4+12*x-32)/(exp(x)*x^3-2*x^5+9*x^3-6*x^2+8*x),x, algorithm="fricas")

[Out]

2*log(-(2*x^4 - x^2*e^x - 9*x^2 + 6*x - 8)/x^2)

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giac [A]  time = 0.13, size = 29, normalized size = 0.97 \begin {gather*} 2 \, \log \left (-2 \, x^{4} + x^{2} e^{x} + 9 \, x^{2} - 6 \, x + 8\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)*x^3-8*x^4+12*x-32)/(exp(x)*x^3-2*x^5+9*x^3-6*x^2+8*x),x, algorithm="giac")

[Out]

2*log(-2*x^4 + x^2*e^x + 9*x^2 - 6*x + 8) - 4*log(x)

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maple [A]  time = 0.03, size = 27, normalized size = 0.90

method	result	size
risch	\(2 \ln \left ({\mathrm e}^{x}-\frac {2 x^{4}-9 x^{2}+6 x -8}{x^{2}}\right )\)	\(27\)
norman	\(-4 \ln \relax (x )+2 \ln \left (2 x^{4}-{\mathrm e}^{x} x^{2}-9 x^{2}+6 x -8\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x)*x^3-8*x^4+12*x-32)/(exp(x)*x^3-2*x^5+9*x^3-6*x^2+8*x),x,method=_RETURNVERBOSE)

[Out]

2*ln(exp(x)-(2*x^4-9*x^2+6*x-8)/x^2)

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maxima [A]  time = 0.52, size = 30, normalized size = 1.00 \begin {gather*} 2 \, \log \left (-\frac {2 \, x^{4} - x^{2} e^{x} - 9 \, x^{2} + 6 \, x - 8}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)*x^3-8*x^4+12*x-32)/(exp(x)*x^3-2*x^5+9*x^3-6*x^2+8*x),x, algorithm="maxima")

[Out]

2*log(-(2*x^4 - x^2*e^x - 9*x^2 + 6*x - 8)/x^2)

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mupad [B]  time = 0.13, size = 28, normalized size = 0.93 \begin {gather*} 2\,\ln \left (3\,x-\frac {x^2\,{\mathrm {e}}^x}{2}-\frac {9\,x^2}{2}+x^4-4\right )-4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x + 2*x^3*exp(x) - 8*x^4 - 32)/(8*x + x^3*exp(x) - 6*x^2 + 9*x^3 - 2*x^5),x)

[Out]

2*log(3*x - (x^2*exp(x))/2 - (9*x^2)/2 + x^4 - 4) - 4*log(x)

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sympy [A]  time = 0.25, size = 24, normalized size = 0.80 \begin {gather*} 2 \log {\left (e^{x} + \frac {- 2 x^{4} + 9 x^{2} - 6 x + 8}{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)*x**3-8*x**4+12*x-32)/(exp(x)*x**3-2*x**5+9*x**3-6*x**2+8*x),x)

[Out]

2*log(exp(x) + (-2*x**4 + 9*x**2 - 6*x + 8)/x**2)

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