3.9.50 \(\int \frac {-3+2401 e^8}{e^4 (2401 e^8+4802 e^4 x+2401 x^2)} \, dx\)

Optimal. Leaf size=17 \[ \frac {\frac {3}{2401 e^4}+x}{e^4+x} \]

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 27, 32} \begin {gather*} \frac {3-2401 e^8}{2401 e^4 \left (x+e^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 2401*E^8)/(E^4*(2401*E^8 + 4802*E^4*x + 2401*x^2)),x]

[Out]

(3 - 2401*E^8)/(2401*E^4*(E^4 + x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\left (3-2401 e^8\right ) \int \frac {1}{2401 e^8+4802 e^4 x+2401 x^2} \, dx}{e^4}\\ &=-\frac {\left (3-2401 e^8\right ) \int \frac {1}{2401 \left (e^4+x\right )^2} \, dx}{e^4}\\ &=-\frac {\left (3-2401 e^8\right ) \int \frac {1}{\left (e^4+x\right )^2} \, dx}{2401 e^4}\\ &=\frac {3-2401 e^8}{2401 e^4 \left (e^4+x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.24 \begin {gather*} -\frac {-3+2401 e^8}{2401 e^4 \left (e^4+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 2401*E^8)/(E^4*(2401*E^8 + 4802*E^4*x + 2401*x^2)),x]

[Out]

-1/2401*(-3 + 2401*E^8)/(E^4*(E^4 + x))

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fricas [A]  time = 1.02, size = 17, normalized size = 1.00 \begin {gather*} -\frac {2401 \, e^{8} - 3}{2401 \, {\left (x e^{4} + e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2401*exp(2)^2*exp(4)-3)/(2401*exp(2)^4+4802*x*exp(2)^2+2401*x^2)/exp(4),x, algorithm="fricas")

[Out]

-1/2401*(2401*e^8 - 3)/(x*e^4 + e^8)

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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((2401*exp(2)^2*exp(4)-3)/(2401*exp(2)^4+4802*x*exp(2)^2+2401*x^2)/exp(4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (2401*exp(8)-3)/exp(4)/2401*1/2/sqrt(exp
(4)^2-exp(8))*ln(sqrt((2*sageVARx+2*exp(4))^2+(-2*sqrt(-exp(4)^2+exp(8)))^2)/sqrt((2*sageVARx+2*exp(4))^2+(2*s
qrt(-exp(4)^2+exp(8))

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maple [A]  time = 0.22, size = 24, normalized size = 1.41

method	result	size
risch	\(-\frac {{\mathrm e}^{-4} {\mathrm e}^{8}}{x +{\mathrm e}^{4}}+\frac {3 \,{\mathrm e}^{-4}}{2401 \left (x +{\mathrm e}^{4}\right )}\)	\(24\)
gosper	\(-\frac {\left (2401 \left ({\mathrm e}^{4}\right )^{2}-3\right ) {\mathrm e}^{-4}}{2401 \left (x +{\mathrm e}^{4}\right )}\)	\(25\)
norman	\(-\frac {\left (2401 \left ({\mathrm e}^{4}\right )^{2}-3\right ) {\mathrm e}^{-4}}{2401 \left (x +{\mathrm e}^{4}\right )}\)	\(25\)
meijerg	\(-\frac {3 \,{\mathrm e}^{-12} x}{2401 \left (1+x \,{\mathrm e}^{-4}\right )}+\frac {{\mathrm e}^{-4} x}{1+x \,{\mathrm e}^{-4}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2401*exp(2)^2*exp(4)-3)/(2401*exp(2)^4+4802*x*exp(2)^2+2401*x^2)/exp(4),x,method=_RETURNVERBOSE)

[Out]

-exp(-4)/(x+exp(4))*exp(8)+3/2401*exp(-4)/(x+exp(4))

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maxima [A]  time = 0.44, size = 16, normalized size = 0.94 \begin {gather*} -\frac {{\left (2401 \, e^{8} - 3\right )} e^{\left (-4\right )}}{2401 \, {\left (x + e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2401*exp(2)^2*exp(4)-3)/(2401*exp(2)^4+4802*x*exp(2)^2+2401*x^2)/exp(4),x, algorithm="maxima")

[Out]

-1/2401*(2401*e^8 - 3)*e^(-4)/(x + e^4)

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mupad [B]  time = 0.12, size = 15, normalized size = 0.88 \begin {gather*} -\frac {{\mathrm {e}}^8-\frac {3}{2401}}{{\mathrm {e}}^8+x\,{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4)*(2401*exp(8) - 3))/(2401*exp(8) + 4802*x*exp(4) + 2401*x^2),x)

[Out]

-(exp(8) - 3/2401)/(exp(8) + x*exp(4))

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sympy [A]  time = 0.12, size = 19, normalized size = 1.12 \begin {gather*} - \frac {-3 + 2401 e^{8}}{2401 x e^{4} + 2401 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2401*exp(2)**2*exp(4)-3)/(2401*exp(2)**4+4802*x*exp(2)**2+2401*x**2)/exp(4),x)

[Out]

-(-3 + 2401*exp(8))/(2401*x*exp(4) + 2401*exp(8))

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