3.30.96 \(\int \frac {e^{\frac {2 (7+x)}{x}} (-896-64 x)+192 x}{81 x-54 e^{\frac {2 (7+x)}{x}} x+9 e^{\frac {4 (7+x)}{x}} x} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{4} \left (5-\frac {256 x}{9 \left (-3+e^{\frac {2 (7+x)}{x}}\right )}\right ) \]

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Rubi [A]  time = 0.29, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 3, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6741, 12, 6687} \begin {gather*} \frac {64 x}{9 \left (3-e^{\frac {14}{x}+2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((2*(7 + x))/x)*(-896 - 64*x) + 192*x)/(81*x - 54*E^((2*(7 + x))/x)*x + 9*E^((4*(7 + x))/x)*x),x]

[Out]

(64*x)/(9*(3 - E^(2 + 14/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 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {2 (7+x)}{x}} (-896-64 x)+192 x}{9 \left (3-e^{2+\frac {14}{x}}\right )^2 x} \, dx\\ &=\frac {1}{9} \int \frac {e^{\frac {2 (7+x)}{x}} (-896-64 x)+192 x}{\left (3-e^{2+\frac {14}{x}}\right )^2 x} \, dx\\ &=\frac {64 x}{9 \left (3-e^{2+\frac {14}{x}}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 18, normalized size = 0.72 \begin {gather*} -\frac {64 x}{9 \left (-3+e^{2+\frac {14}{x}}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(7 + x))/x)*(-896 - 64*x) + 192*x)/(81*x - 54*E^((2*(7 + x))/x)*x + 9*E^((4*(7 + x))/x)*x),x]

[Out]

(-64*x)/(9*(-3 + E^(2 + 14/x)))

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fricas [A]  time = 0.66, size = 16, normalized size = 0.64 \begin {gather*} -\frac {64 \, x}{9 \, {\left (e^{\left (\frac {2 \, {\left (x + 7\right )}}{x}\right )} - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x-896)*exp((x+7)/x)^2+192*x)/(9*x*exp((x+7)/x)^4-54*x*exp((x+7)/x)^2+81*x),x, algorithm="frica
s")

[Out]

-64/9*x/(e^(2*(x + 7)/x) - 3)

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giac [A]  time = 0.35, size = 22, normalized size = 0.88 \begin {gather*} -\frac {64}{9 \, {\left (\frac {e^{\left (\frac {14}{x} + 2\right )}}{x} - \frac {3}{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x-896)*exp((x+7)/x)^2+192*x)/(9*x*exp((x+7)/x)^4-54*x*exp((x+7)/x)^2+81*x),x, algorithm="giac"
)

[Out]

-64/9/(e^(14/x + 2)/x - 3/x)

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maple [A]  time = 0.23, size = 17, normalized size = 0.68

method	result	size
risch	\(-\frac {64 x}{9 \left ({\mathrm e}^{\frac {14+2 x}{x}}-3\right )}\)	\(17\)
norman	\(-\frac {64 x}{9 \left ({\mathrm e}^{\frac {14+2 x}{x}}-3\right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-64*x-896)*exp((x+7)/x)^2+192*x)/(9*x*exp((x+7)/x)^4-54*x*exp((x+7)/x)^2+81*x),x,method=_RETURNVERBOSE)

[Out]

-64/9*x/(exp(2*(x+7)/x)-3)

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maxima [A]  time = 0.42, size = 15, normalized size = 0.60 \begin {gather*} -\frac {64 \, x}{9 \, {\left (e^{\left (\frac {14}{x} + 2\right )} - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x-896)*exp((x+7)/x)^2+192*x)/(9*x*exp((x+7)/x)^4-54*x*exp((x+7)/x)^2+81*x),x, algorithm="maxim
a")

[Out]

-64/9*x/(e^(14/x + 2) - 3)

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mupad [B]  time = 1.84, size = 17, normalized size = 0.68 \begin {gather*} -\frac {64\,x}{9\,\left ({\mathrm {e}}^{\frac {14}{x}+2}-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((192*x - exp((2*(x + 7))/x)*(64*x + 896))/(81*x - 54*x*exp((2*(x + 7))/x) + 9*x*exp((4*(x + 7))/x)),x)

[Out]

-(64*x)/(9*(exp(14/x + 2) - 3))

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sympy [A]  time = 0.12, size = 15, normalized size = 0.60 \begin {gather*} - \frac {64 x}{9 e^{\frac {2 \left (x + 7\right )}{x}} - 27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x-896)*exp((x+7)/x)**2+192*x)/(9*x*exp((x+7)/x)**4-54*x*exp((x+7)/x)**2+81*x),x)

[Out]

-64*x/(9*exp(2*(x + 7)/x) - 27)

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