3.35.83 \(\int \frac {-16 x+24 x^2+e^5 (-2+4 x)}{-36-4 x^2+4 x^3+e^5 (-x+x^2)} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\left (9+\frac {1}{4} \left (e^5+4 x\right ) \left (x-x^2\right )\right )^2\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 1, number of rules used = 1, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {1587} \begin {gather*} 2 \log \left (-4 x^3+4 x^2+e^5 \left (x-x^2\right )+36\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16*x + 24*x^2 + E^5*(-2 + 4*x))/(-36 - 4*x^2 + 4*x^3 + E^5*(-x + x^2)),x]

[Out]

2*Log[36 + 4*x^2 - 4*x^3 + E^5*(x - x^2)]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 \log \left (36+4 x^2-4 x^3+e^5 \left (x-x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.04 \begin {gather*} 2 \log \left (e^5 (-1+x) x+4 \left (-9-x^2+x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x + 24*x^2 + E^5*(-2 + 4*x))/(-36 - 4*x^2 + 4*x^3 + E^5*(-x + x^2)),x]

[Out]

2*Log[E^5*(-1 + x)*x + 4*(-9 - x^2 + x^3)]

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fricas [A]  time = 0.76, size = 25, normalized size = 1.09 \begin {gather*} 2 \, \log \left (4 \, x^{3} - 4 \, x^{2} + {\left (x^{2} - x\right )} e^{5} - 36\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-2)*exp(5)+24*x^2-16*x)/((x^2-x)*exp(5)+4*x^3-4*x^2-36),x, algorithm="fricas")

[Out]

2*log(4*x^3 - 4*x^2 + (x^2 - x)*e^5 - 36)

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giac [A]  time = 0.23, size = 26, normalized size = 1.13 \begin {gather*} 2 \, \log \left ({\left | 4 \, x^{3} - 4 \, x^{2} + {\left (x^{2} - x\right )} e^{5} - 36 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-2)*exp(5)+24*x^2-16*x)/((x^2-x)*exp(5)+4*x^3-4*x^2-36),x, algorithm="giac")

[Out]

2*log(abs(4*x^3 - 4*x^2 + (x^2 - x)*e^5 - 36))

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maple [A]  time = 0.04, size = 24, normalized size = 1.04

method	result	size
risch	\(2 \ln \left (-36+4 x^{3}+\left ({\mathrm e}^{5}-4\right ) x^{2}-x \,{\mathrm e}^{5}\right )\)	\(24\)
default	\(2 \ln \left (x^{2} {\mathrm e}^{5}+4 x^{3}-x \,{\mathrm e}^{5}-4 x^{2}-36\right )\)	\(27\)
norman	\(2 \ln \left (x^{2} {\mathrm e}^{5}+4 x^{3}-x \,{\mathrm e}^{5}-4 x^{2}-36\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x-2)*exp(5)+24*x^2-16*x)/((x^2-x)*exp(5)+4*x^3-4*x^2-36),x,method=_RETURNVERBOSE)

[Out]

2*ln(-36+4*x^3+(exp(5)-4)*x^2-x*exp(5))

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maxima [A]  time = 0.56, size = 25, normalized size = 1.09 \begin {gather*} 2 \, \log \left (4 \, x^{3} - 4 \, x^{2} + {\left (x^{2} - x\right )} e^{5} - 36\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-2)*exp(5)+24*x^2-16*x)/((x^2-x)*exp(5)+4*x^3-4*x^2-36),x, algorithm="maxima")

[Out]

2*log(4*x^3 - 4*x^2 + (x^2 - x)*e^5 - 36)

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mupad [B]  time = 2.14, size = 22, normalized size = 0.96 \begin {gather*} 2\,\ln \left (x^3-\frac {x\,{\mathrm {e}}^5}{4}+\frac {x^2\,\left ({\mathrm {e}}^5-4\right )}{4}-9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(24*x^2 - 16*x + exp(5)*(4*x - 2))/(exp(5)*(x - x^2) + 4*x^2 - 4*x^3 + 36),x)

[Out]

2*log(x^3 - (x*exp(5))/4 + (x^2*(exp(5) - 4))/4 - 9)

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sympy [A]  time = 0.36, size = 22, normalized size = 0.96 \begin {gather*} 2 \log {\left (4 x^{3} + x^{2} \left (-4 + e^{5}\right ) - x e^{5} - 36 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-2)*exp(5)+24*x**2-16*x)/((x**2-x)*exp(5)+4*x**3-4*x**2-36),x)

[Out]

2*log(4*x**3 + x**2*(-4 + exp(5)) - x*exp(5) - 36)

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