3.63.58 \(\int \frac {36864+e^7+384 e x+e^2 x^2}{36864+384 e x+e^2 x^2} \, dx\)

Optimal. Leaf size=23 \[ x+\frac {e^5}{-\frac {192}{e}+x^2-x (1+x)} \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {27, 683} \begin {gather*} x-\frac {e^6}{e x+192} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36864 + E^7 + 384*E*x + E^2*x^2)/(36864 + 384*E*x + E^2*x^2),x]

[Out]

x - E^6/(192 + E*x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36864+e^7+384 e x+e^2 x^2}{(192+e x)^2} \, dx\\ &=\int \left (1+\frac {e^7}{(192+e x)^2}\right ) \, dx\\ &=x-\frac {e^6}{192+e x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.61 \begin {gather*} x-\frac {e^6}{192+e x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36864 + E^7 + 384*E*x + E^2*x^2)/(36864 + 384*E*x + E^2*x^2),x]

[Out]

x - E^6/(192 + E*x)

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fricas [A]  time = 0.52, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^{2} e + 192 \, x - e^{6}}{x e + 192} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)^2*exp(5)+x^2*exp(1)^2+384*x*exp(1)+36864)/(x^2*exp(1)^2+384*x*exp(1)+36864),x, algorithm="fr
icas")

[Out]

(x^2*e + 192*x - e^6)/(x*e + 192)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)^2*exp(5)+x^2*exp(1)^2+384*x*exp(1)+36864)/(x^2*exp(1)^2+384*x*exp(1)+36864),x, algorithm="gi
ac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: sageVARx*exp(2)/exp(2)+exp(7)*1/384/sqrt
(exp(1)^2-exp(2))*ln(sqrt((2*sageVARx*exp(2)+384*exp(1))^2+(-384*sqrt(-exp(1)^2+exp(2)))^2)/sqrt((2*sageVARx*e
xp(2)+384*exp(1))^2+(

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maple [A]  time = 0.41, size = 15, normalized size = 0.65

method	result	size
risch	\(x -\frac {{\mathrm e}^{6}}{x \,{\mathrm e}+192}\)	\(15\)
gosper	\(\frac {x \left ({\mathrm e}^{2} {\mathrm e}^{5}+192 x \,{\mathrm e}+36864\right )}{192 x \,{\mathrm e}+36864}\)	\(26\)
norman	\(\frac {x^{2} {\mathrm e}+\left (\frac {{\mathrm e}^{2} {\mathrm e}^{5}}{192}+192\right ) x}{x \,{\mathrm e}+192}\)	\(29\)
meijerg	\(\frac {x}{1+\frac {x \,{\mathrm e}}{192}}+\frac {x \,{\mathrm e}^{7}}{192 x \,{\mathrm e}+36864}+192 \,{\mathrm e}^{-1} \left (\frac {x \,{\mathrm e} \left (\frac {x \,{\mathrm e}}{64}+6\right )}{576+3 x \,{\mathrm e}}-2 \ln \left (1+\frac {x \,{\mathrm e}}{192}\right )\right )+384 \,{\mathrm e}^{-1} \left (-\frac {x \,{\mathrm e}}{192 \left (1+\frac {x \,{\mathrm e}}{192}\right )}+\ln \left (1+\frac {x \,{\mathrm e}}{192}\right )\right )\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)^2*exp(5)+x^2*exp(1)^2+384*x*exp(1)+36864)/(x^2*exp(1)^2+384*x*exp(1)+36864),x,method=_RETURNVERBOS
E)

[Out]

x-exp(6)/(x*exp(1)+192)

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maxima [A]  time = 0.48, size = 14, normalized size = 0.61 \begin {gather*} x - \frac {e^{6}}{x e + 192} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)^2*exp(5)+x^2*exp(1)^2+384*x*exp(1)+36864)/(x^2*exp(1)^2+384*x*exp(1)+36864),x, algorithm="ma
xima")

[Out]

x - e^6/(x*e + 192)

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mupad [B]  time = 4.05, size = 14, normalized size = 0.61 \begin {gather*} x-\frac {{\mathrm {e}}^6}{x\,\mathrm {e}+192} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(7) + 384*x*exp(1) + x^2*exp(2) + 36864)/(384*x*exp(1) + x^2*exp(2) + 36864),x)

[Out]

x - exp(6)/(x*exp(1) + 192)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.43 \begin {gather*} x - \frac {e^{6}}{e x + 192} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)**2*exp(5)+x**2*exp(1)**2+384*x*exp(1)+36864)/(x**2*exp(1)**2+384*x*exp(1)+36864),x)

[Out]

x - exp(6)/(E*x + 192)

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