3.83.90 \(\int \frac {-3 e^{1+\frac {3 e+75 x}{5 x}}-5 x^2}{5 x^2} \, dx\)

Optimal. Leaf size=19 \[ 3+e^{\frac {15 \left (\frac {e}{25}+x\right )}{x}}-x \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 14, 2209} \begin {gather*} e^{\frac {3 e}{5 x}+15}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*E^(1 + (3*E + 75*x)/(5*x)) - 5*x^2)/(5*x^2),x]

[Out]

E^(15 + (3*E)/(5*x)) - 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {-3 e^{1+\frac {3 e+75 x}{5 x}}-5 x^2}{x^2} \, dx\\ &=\frac {1}{5} \int \left (-5-\frac {3 e^{16+\frac {3 e}{5 x}}}{x^2}\right ) \, dx\\ &=-x-\frac {3}{5} \int \frac {e^{16+\frac {3 e}{5 x}}}{x^2} \, dx\\ &=e^{15+\frac {3 e}{5 x}}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.84 \begin {gather*} e^{15+\frac {3 e}{5 x}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*E^(1 + (3*E + 75*x)/(5*x)) - 5*x^2)/(5*x^2),x]

[Out]

E^(15 + (3*E)/(5*x)) - x

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fricas [A]  time = 0.59, size = 25, normalized size = 1.32 \begin {gather*} -{\left (x e - e^{\left (\frac {80 \, x + 3 \, e}{5 \, x}\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-3*exp(1)*exp(1/5*(3*exp(1)+75*x)/x)-5*x^2)/x^2,x, algorithm="fricas")

[Out]

-(x*e - e^(1/5*(80*x + 3*e)/x))*e^(-1)

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giac [B]  time = 0.15, size = 66, normalized size = 3.47 \begin {gather*} \frac {{\left (\frac {{\left (80 \, x + 3 \, e\right )} e^{\left (\frac {80 \, x + 3 \, e}{5 \, x}\right )}}{x} - 3 \, e^{2} - 80 \, e^{\left (\frac {80 \, x + 3 \, e}{5 \, x}\right )}\right )} e^{\left (-1\right )}}{\frac {80 \, x + 3 \, e}{x} - 80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-3*exp(1)*exp(1/5*(3*exp(1)+75*x)/x)-5*x^2)/x^2,x, algorithm="giac")

[Out]

((80*x + 3*e)*e^(1/5*(80*x + 3*e)/x)/x - 3*e^2 - 80*e^(1/5*(80*x + 3*e)/x))*e^(-1)/((80*x + 3*e)/x - 80)

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maple [A]  time = 0.06, size = 15, normalized size = 0.79

method	result	size
derivativedivides	\(-x +{\mathrm e}^{15+\frac {3 \,{\mathrm e}}{5 x}}\)	\(15\)
default	\(-x +{\mathrm e}^{15+\frac {3 \,{\mathrm e}}{5 x}}\)	\(15\)
risch	\(-x +{\mathrm e}^{\frac {\frac {3 \,{\mathrm e}}{5}+15 x}{x}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-3*exp(1)*exp(1/5*(3*exp(1)+75*x)/x)-5*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x+exp(15+3/5*exp(1)/x)

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maxima [A]  time = 0.37, size = 14, normalized size = 0.74 \begin {gather*} -x + e^{\left (\frac {3 \, e}{5 \, x} + 15\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-3*exp(1)*exp(1/5*(3*exp(1)+75*x)/x)-5*x^2)/x^2,x, algorithm="maxima")

[Out]

-x + e^(3/5*e/x + 15)

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mupad [B]  time = 5.34, size = 14, normalized size = 0.74 \begin {gather*} {\mathrm {e}}^{\frac {3\,\mathrm {e}}{5\,x}+15}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*exp(1)*exp((15*x + (3*exp(1))/5)/x))/5 + x^2)/x^2,x)

[Out]

exp((3*exp(1))/(5*x) + 15) - x

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sympy [A]  time = 0.19, size = 14, normalized size = 0.74 \begin {gather*} - x + e^{\frac {15 x + \frac {3 e}{5}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-3*exp(1)*exp(1/5*(3*exp(1)+75*x)/x)-5*x**2)/x**2,x)

[Out]

-x + exp((15*x + 3*E/5)/x)

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