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

Optimal. Leaf size=28 \[ -x+\log \left (-4-x+\frac {e^5 \left (-5+\left (1-x^2\right )^2\right )}{x}\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 37, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 2, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2074, 1587} \begin {gather*} \log \left (-e^5 x^4+\left (1+2 e^5\right ) x^2+4 x+4 e^5\right )-x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - Log[x] + Log[4*E^5 + 4*x + (1 + 2*E^5)*x^2 - E^5*x^4]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, 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 (-1-\frac {1}{x}+\frac {2 \left (2+\left (1+2 e^5\right ) x-2 e^5 x^3\right )}{4 e^5+4 x+\left (1+2 e^5\right ) x^2-e^5 x^4}\right ) \, dx\\ &=-x-\log (x)+2 \int \frac {2+\left (1+2 e^5\right ) x-2 e^5 x^3}{4 e^5+4 x+\left (1+2 e^5\right ) x^2-e^5 x^4} \, dx\\ &=-x-\log (x)+\log \left (4 e^5+4 x+\left (1+2 e^5\right ) x^2-e^5 x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 38, normalized size = 1.36 \begin {gather*} -x-\log (x)+\log \left (-4 e^5-4 x-x^2-2 e^5 x^2+e^5 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - Log[x] + Log[-4*E^5 - 4*x - x^2 - 2*E^5*x^2 + E^5*x^4]

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fricas [A]  time = 0.73, size = 31, normalized size = 1.11 \begin {gather*} -x + \log \left (-x^{2} + {\left (x^{4} - 2 \, x^{2} - 4\right )} e^{5} - 4 \, x\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(-x^2 + (x^4 - 2*x^2 - 4)*e^5 - 4*x) - log(x)

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giac [A]  time = 0.40, size = 37, normalized size = 1.32 \begin {gather*} -x + \log \left ({\left | x^{4} e^{5} - 2 \, x^{2} e^{5} - x^{2} - 4 \, x - 4 \, e^{5} \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(abs(x^4*e^5 - 2*x^2*e^5 - x^2 - 4*x - 4*e^5)) - log(abs(x))

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maple [A]  time = 0.06, size = 35, normalized size = 1.25

method	result	size
risch	\(-x -\ln \relax (x )+\ln \left (-x^{4} {\mathrm e}^{5}+\left (2 \,{\mathrm e}^{5}+1\right ) x^{2}+4 x +4 \,{\mathrm e}^{5}\right )\)	\(35\)
default	\(-x +\ln \left (x^{4} {\mathrm e}^{5}-2 x^{2} {\mathrm e}^{5}-x^{2}-4 \,{\mathrm e}^{5}-4 x \right )-\ln \relax (x )\)	\(36\)
norman	\(-x +\ln \left (x^{4} {\mathrm e}^{5}-2 x^{2} {\mathrm e}^{5}-x^{2}-4 \,{\mathrm e}^{5}-4 x \right )-\ln \relax (x )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x-ln(x)+ln(-x^4*exp(5)+(2*exp(5)+1)*x^2+4*x+4*exp(5))

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maxima [A]  time = 0.43, size = 34, normalized size = 1.21 \begin {gather*} -x + \log \left (x^{4} e^{5} - x^{2} {\left (2 \, e^{5} + 1\right )} - 4 \, x - 4 \, e^{5}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(x^4*e^5 - x^2*(2*e^5 + 1) - 4*x - 4*e^5) - log(x)

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mupad [B]  time = 0.46, size = 31, normalized size = 1.11 \begin {gather*} \ln \left (x^4-x^2\,{\mathrm {e}}^{-5}-2\,x^2-4\,x\,{\mathrm {e}}^{-5}-4\right )-x-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^4 - x^2*exp(-5) - 2*x^2 - 4*x*exp(-5) - 4) - x - log(x)

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sympy [A]  time = 1.57, size = 32, normalized size = 1.14 \begin {gather*} - x - \log {\relax (x )} + \log {\left (x^{4} + \frac {x^{2} \left (- 2 e^{5} - 1\right )}{e^{5}} - \frac {4 x}{e^{5}} - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(x) + log(x**4 + x**2*(-2*exp(5) - 1)*exp(-5) - 4*x*exp(-5) - 4)

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