3.15.10 \(\int \frac {-4-6 x-70 x^2+e^x x^2+23 x^3+3 x^4-x^5}{-4 x-37 x^2+15 x^3+2 x^4-x^5+e^x (4 x^2+x^3)} \, dx\)

Optimal. Leaf size=28 \[ 2+x+\log \left (\frac {4+x}{e^x-(3-x)^2-\frac {1}{x}}\right ) \]

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Rubi [F]  time = 1.83, 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 {-4-6 x-70 x^2+e^x x^2+23 x^3+3 x^4-x^5}{-4 x-37 x^2+15 x^3+2 x^4-x^5+e^x \left (4 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 - 6*x - 70*x^2 + E^x*x^2 + 23*x^3 + 3*x^4 - x^5)/(-4*x - 37*x^2 + 15*x^3 + 2*x^4 - x^5 + E^x*(4*x^2 +
x^3)),x]

[Out]

Log[4 + x] + Defer[Int][(1 + 9*x - E^x*x - 6*x^2 + x^3)^(-1), x] + Defer[Int][1/(x*(1 + 9*x - E^x*x - 6*x^2 +
x^3)), x] + 15*Defer[Int][x/(1 + 9*x - E^x*x - 6*x^2 + x^3), x] - 8*Defer[Int][x^2/(1 + 9*x - E^x*x - 6*x^2 +
x^3), x] + Defer[Int][x^3/(1 + 9*x - E^x*x - 6*x^2 + x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+6 x+70 x^2-e^x x^2-23 x^3-3 x^4+x^5}{x (4+x) \left (1+9 x-e^x x-6 x^2+x^3\right )} \, dx\\ &=\int \left (\frac {1}{4+x}+\frac {1+x+15 x^2-8 x^3+x^4}{x \left (1+9 x-e^x x-6 x^2+x^3\right )}\right ) \, dx\\ &=\log (4+x)+\int \frac {1+x+15 x^2-8 x^3+x^4}{x \left (1+9 x-e^x x-6 x^2+x^3\right )} \, dx\\ &=\log (4+x)+\int \left (\frac {1}{1+9 x-e^x x-6 x^2+x^3}+\frac {1}{x \left (1+9 x-e^x x-6 x^2+x^3\right )}+\frac {15 x}{1+9 x-e^x x-6 x^2+x^3}-\frac {8 x^2}{1+9 x-e^x x-6 x^2+x^3}+\frac {x^3}{1+9 x-e^x x-6 x^2+x^3}\right ) \, dx\\ &=\log (4+x)-8 \int \frac {x^2}{1+9 x-e^x x-6 x^2+x^3} \, dx+15 \int \frac {x}{1+9 x-e^x x-6 x^2+x^3} \, dx+\int \frac {1}{1+9 x-e^x x-6 x^2+x^3} \, dx+\int \frac {1}{x \left (1+9 x-e^x x-6 x^2+x^3\right )} \, dx+\int \frac {x^3}{1+9 x-e^x x-6 x^2+x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 30, normalized size = 1.07 \begin {gather*} x+\log (x)+\log (4+x)-\log \left (1+9 x-e^x x-6 x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 6*x - 70*x^2 + E^x*x^2 + 23*x^3 + 3*x^4 - x^5)/(-4*x - 37*x^2 + 15*x^3 + 2*x^4 - x^5 + E^x*(4*
x^2 + x^3)),x]

[Out]

x + Log[x] + Log[4 + x] - Log[1 + 9*x - E^x*x - 6*x^2 + x^3]

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fricas [A]  time = 0.71, size = 32, normalized size = 1.14 \begin {gather*} x + \log \left (x + 4\right ) - \log \left (-\frac {x^{3} - 6 \, x^{2} - x e^{x} + 9 \, x + 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x^2-x^5+3*x^4+23*x^3-70*x^2-6*x-4)/((x^3+4*x^2)*exp(x)-x^5+2*x^4+15*x^3-37*x^2-4*x),x, algor
ithm="fricas")

[Out]

x + log(x + 4) - log(-(x^3 - 6*x^2 - x*e^x + 9*x + 1)/x)

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giac [A]  time = 0.28, size = 30, normalized size = 1.07 \begin {gather*} x - \log \left (-x^{3} + 6 \, x^{2} + x e^{x} - 9 \, x - 1\right ) + \log \left (x + 4\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x^2-x^5+3*x^4+23*x^3-70*x^2-6*x-4)/((x^3+4*x^2)*exp(x)-x^5+2*x^4+15*x^3-37*x^2-4*x),x, algor
ithm="giac")

[Out]

x - log(-x^3 + 6*x^2 + x*e^x - 9*x - 1) + log(x + 4) + log(x)

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maple [A]  time = 0.05, size = 30, normalized size = 1.07

method	result	size
norman	\(x -\ln \left (x^{3}-6 x^{2}-{\mathrm e}^{x} x +9 x +1\right )+\ln \relax (x )+\ln \left (4+x \right )\)	\(30\)
risch	\(\ln \left (4+x \right )+x -\ln \left ({\mathrm e}^{x}-\frac {x^{3}-6 x^{2}+9 x +1}{x}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*x^2-x^5+3*x^4+23*x^3-70*x^2-6*x-4)/((x^3+4*x^2)*exp(x)-x^5+2*x^4+15*x^3-37*x^2-4*x),x,method=_RETU
RNVERBOSE)

[Out]

x-ln(x^3-6*x^2-exp(x)*x+9*x+1)+ln(x)+ln(4+x)

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maxima [A]  time = 0.53, size = 32, normalized size = 1.14 \begin {gather*} x + \log \left (x + 4\right ) - \log \left (-\frac {x^{3} - 6 \, x^{2} - x e^{x} + 9 \, x + 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x^2-x^5+3*x^4+23*x^3-70*x^2-6*x-4)/((x^3+4*x^2)*exp(x)-x^5+2*x^4+15*x^3-37*x^2-4*x),x, algor
ithm="maxima")

[Out]

x + log(x + 4) - log(-(x^3 - 6*x^2 - x*e^x + 9*x + 1)/x)

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mupad [B]  time = 1.03, size = 29, normalized size = 1.04 \begin {gather*} x+\ln \left (x+4\right )-\ln \left (9\,x-x\,{\mathrm {e}}^x-6\,x^2+x^3+1\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - x^2*exp(x) + 70*x^2 - 23*x^3 - 3*x^4 + x^5 + 4)/(4*x - exp(x)*(4*x^2 + x^3) + 37*x^2 - 15*x^3 - 2*x
^4 + x^5),x)

[Out]

x + log(x + 4) - log(9*x - x*exp(x) - 6*x^2 + x^3 + 1) + log(x)

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sympy [A]  time = 0.23, size = 26, normalized size = 0.93 \begin {gather*} x + \log {\left (x + 4 \right )} - \log {\left (e^{x} + \frac {- x^{3} + 6 x^{2} - 9 x - 1}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x**2-x**5+3*x**4+23*x**3-70*x**2-6*x-4)/((x**3+4*x**2)*exp(x)-x**5+2*x**4+15*x**3-37*x**2-4*
x),x)

[Out]

x + log(x + 4) - log(exp(x) + (-x**3 + 6*x**2 - 9*x - 1)/x)

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