3.100.56 \(\int \frac {e^5 (-4-x)-40 x+19 x^2+2 x^3+e^5 \log (2)}{-e^5 x-5 x^2+2 x^3+e^5 \log (2)} \, dx\)

Optimal. Leaf size=28 \[ x+2 \log \left (\left (x-\frac {x (x+x (-6+2 x))}{e^5}-\log (2)\right )^2\right ) \]

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Rubi [A]  time = 0.24, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6741, 6742, 1587} \begin {gather*} 4 \log \left (-2 x^3+5 x^2+e^5 x-e^5 \log (2)\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^5*(-4 - x) - 40*x + 19*x^2 + 2*x^3 + E^5*Log[2])/(-(E^5*x) - 5*x^2 + 2*x^3 + E^5*Log[2]),x]

[Out]

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (40+e^5\right ) x-19 x^2-2 x^3+e^5 (4-\log (2))}{e^5 x+5 x^2-2 x^3-e^5 \log (2)} \, dx\\ &=\int \left (1+\frac {4 \left (e^5+10 x-6 x^2\right )}{e^5 x+5 x^2-2 x^3-e^5 \log (2)}\right ) \, dx\\ &=x+4 \int \frac {e^5+10 x-6 x^2}{e^5 x+5 x^2-2 x^3-e^5 \log (2)} \, dx\\ &=x+4 \log \left (e^5 x+5 x^2-2 x^3-e^5 \log (2)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 28, normalized size = 1.00 \begin {gather*} x+4 \log \left (e^5 x+5 x^2-2 x^3-e^5 \log (2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(-4 - x) - 40*x + 19*x^2 + 2*x^3 + E^5*Log[2])/(-(E^5*x) - 5*x^2 + 2*x^3 + E^5*Log[2]),x]

[Out]

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

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fricas [A]  time = 0.51, size = 26, normalized size = 0.93 \begin {gather*} x + 4 \, \log \left (2 \, x^{3} - 5 \, x^{2} - x e^{5} + e^{5} \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5)*log(2)+(-x-4)*exp(5)+2*x^3+19*x^2-40*x)/(exp(5)*log(2)-x*exp(5)+2*x^3-5*x^2),x, algorithm="f
ricas")

[Out]

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

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giac [A]  time = 0.14, size = 27, normalized size = 0.96 \begin {gather*} x + 4 \, \log \left ({\left | 2 \, x^{3} - 5 \, x^{2} - x e^{5} + e^{5} \log \relax (2) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5)*log(2)+(-x-4)*exp(5)+2*x^3+19*x^2-40*x)/(exp(5)*log(2)-x*exp(5)+2*x^3-5*x^2),x, algorithm="g
iac")

[Out]

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

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maple [A]  time = 0.06, size = 27, normalized size = 0.96

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5)*ln(2)+(-x-4)*exp(5)+2*x^3+19*x^2-40*x)/(exp(5)*ln(2)-x*exp(5)+2*x^3-5*x^2),x,method=_RETURNVERBOSE
)

[Out]

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

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maxima [A]  time = 0.36, size = 26, normalized size = 0.93 \begin {gather*} x + 4 \, \log \left (2 \, x^{3} - 5 \, x^{2} - x e^{5} + e^{5} \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5)*log(2)+(-x-4)*exp(5)+2*x^3+19*x^2-40*x)/(exp(5)*log(2)-x*exp(5)+2*x^3-5*x^2),x, algorithm="m
axima")

[Out]

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

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mupad [B]  time = 6.89, size = 26, normalized size = 0.93 \begin {gather*} x+4\,\ln \left (2\,x^3-5\,x^2-{\mathrm {e}}^5\,x+{\mathrm {e}}^5\,\ln \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5)*log(2) - 40*x - exp(5)*(x + 4) + 19*x^2 + 2*x^3)/(exp(5)*log(2) - x*exp(5) - 5*x^2 + 2*x^3),x)

[Out]

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

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sympy [A]  time = 0.44, size = 26, normalized size = 0.93 \begin {gather*} x + 4 \log {\left (2 x^{3} - 5 x^{2} - x e^{5} + e^{5} \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5)*ln(2)+(-x-4)*exp(5)+2*x**3+19*x**2-40*x)/(exp(5)*ln(2)-x*exp(5)+2*x**3-5*x**2),x)

[Out]

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

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