3.100.79 \(\int \frac {5+20 x+19 x^2+6 x^3+e (-1-4 x-2 x^2)}{-5 x-8 x^2-3 x^3+e (x+x^2)} \, dx\)

Optimal. Leaf size=29 \[ 4-2 \left (3+e^2+x\right )+\log \left (\frac {5-\frac {e+2 x}{1+x}}{x}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 23, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 1, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2074} \begin {gather*} -2 x-\log (x)-\log (x+1)+\log (3 x-e+5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + 20*x + 19*x^2 + 6*x^3 + E*(-1 - 4*x - 2*x^2))/(-5*x - 8*x^2 - 3*x^3 + E*(x + x^2)),x]

[Out]

-2*x - Log[x] - Log[1 + x] + Log[5 - E + 3*x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.79 \begin {gather*} -2 x-\log (x)-\log (1+x)+\log (5-e+3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 20*x + 19*x^2 + 6*x^3 + E*(-1 - 4*x - 2*x^2))/(-5*x - 8*x^2 - 3*x^3 + E*(x + x^2)),x]

[Out]

-2*x - Log[x] - Log[1 + x] + Log[5 - E + 3*x]

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fricas [A]  time = 0.71, size = 22, normalized size = 0.76 \begin {gather*} -2 \, x - \log \left (x^{2} + x\right ) + \log \left (3 \, x - e + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-4*x-1)*exp(1)+6*x^3+19*x^2+20*x+5)/((x^2+x)*exp(1)-3*x^3-8*x^2-5*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 27, normalized size = 0.93 \begin {gather*} -2 \, x + \log \left ({\left | 3 \, x - e + 5 \right |}\right ) - \log \left ({\left | x + 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-4*x-1)*exp(1)+6*x^3+19*x^2+20*x+5)/((x^2+x)*exp(1)-3*x^3-8*x^2-5*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 23, normalized size = 0.79

method	result	size
norman	\(-2 x -\ln \relax (x )-\ln \left (x +1\right )+\ln \left (-3 x +{\mathrm e}-5\right )\)	\(23\)
risch	\(-2 x +\ln \left (3 x -{\mathrm e}+5\right )-\ln \left (x^{2}+x \right )\)	\(23\)
default	\(-2 x -\ln \relax (x )+\ln \left (3 x -{\mathrm e}+5\right )-\ln \left (x +1\right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2-4*x-1)*exp(1)+6*x^3+19*x^2+20*x+5)/((x^2+x)*exp(1)-3*x^3-8*x^2-5*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.37, size = 24, normalized size = 0.83 \begin {gather*} -2 \, x + \log \left (3 \, x - e + 5\right ) - \log \left (x + 1\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-4*x-1)*exp(1)+6*x^3+19*x^2+20*x+5)/((x^2+x)*exp(1)-3*x^3-8*x^2-5*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 8.83, size = 20, normalized size = 0.69 \begin {gather*} \ln \left (x-\frac {\mathrm {e}}{3}+\frac {5}{3}\right )-2\,x-\ln \left (x\,\left (x+1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*x - exp(1)*(4*x + 2*x^2 + 1) + 19*x^2 + 6*x^3 + 5)/(5*x - exp(1)*(x + x^2) + 8*x^2 + 3*x^3),x)

[Out]

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

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sympy [A]  time = 0.48, size = 20, normalized size = 0.69 \begin {gather*} - 2 x - \log {\left (x^{2} + x \right )} + \log {\left (x - \frac {e}{3} + \frac {5}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2-4*x-1)*exp(1)+6*x**3+19*x**2+20*x+5)/((x**2+x)*exp(1)-3*x**3-8*x**2-5*x),x)

[Out]

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

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