3.94.62 \(\int \frac {x+\frac {11 e^2 x}{5}+6 e^x x}{6 x} \, dx\)

Optimal. Leaf size=18 \[ e^x+\frac {1}{6} \left (x+\frac {11 e^2 x}{5}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6, 12, 14, 2194} \begin {gather*} \frac {1}{30} \left (5+11 e^2\right ) x+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + (11*E^2*x)/5 + 6*E^x*x)/(6*x),x]

[Out]

E^x + ((5 + 11*E^2)*x)/30

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 e^x x+\left (1+\frac {11 e^2}{5}\right ) x}{6 x} \, dx\\ &=\frac {1}{6} \int \frac {6 e^x x+\left (1+\frac {11 e^2}{5}\right ) x}{x} \, dx\\ &=\frac {1}{6} \int \left (6 e^x+\frac {1}{5} \left (5+11 e^2\right )\right ) \, dx\\ &=\frac {1}{30} \left (5+11 e^2\right ) x+\int e^x \, dx\\ &=e^x+\frac {1}{30} \left (5+11 e^2\right ) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.94 \begin {gather*} e^x+\frac {x}{6}+\frac {11 e^2 x}{30} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + (11*E^2*x)/5 + 6*E^x*x)/(6*x),x]

[Out]

E^x + x/6 + (11*E^2*x)/30

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fricas [A]  time = 0.53, size = 11, normalized size = 0.61 \begin {gather*} \frac {11}{30} \, x e^{2} + \frac {1}{6} \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(-exp(log(-11/5*x)+2)+6*exp(x)*x+x)/x,x, algorithm="fricas")

[Out]

11/30*x*e^2 + 1/6*x + e^x

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giac [A]  time = 0.29, size = 12, normalized size = 0.67 \begin {gather*} \frac {1}{30} \, x {\left (11 \, e^{2} + 5\right )} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(-exp(log(-11/5*x)+2)+6*exp(x)*x+x)/x,x, algorithm="giac")

[Out]

1/30*x*(11*e^2 + 5) + e^x

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maple [A]  time = 0.04, size = 12, normalized size = 0.67

method	result	size
norman	\(\left (\frac {11 \,{\mathrm e}^{2}}{30}+\frac {1}{6}\right ) x +{\mathrm e}^{x}\)	\(12\)
risch	\(\frac {11 \,{\mathrm e}^{2} x}{30}+\frac {x}{6}+{\mathrm e}^{x}\)	\(12\)
default	\({\mathrm e}^{x}+\frac {x}{6}-\frac {{\mathrm e}^{\ln \left (-\frac {11 x}{5}\right )+2}}{6}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/6*(-exp(ln(-11/5*x)+2)+6*exp(x)*x+x)/x,x,method=_RETURNVERBOSE)

[Out]

(11/30*exp(2)+1/6)*x+exp(x)

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maxima [A]  time = 0.50, size = 11, normalized size = 0.61 \begin {gather*} \frac {11}{30} \, x e^{2} + \frac {1}{6} \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(-exp(log(-11/5*x)+2)+6*exp(x)*x+x)/x,x, algorithm="maxima")

[Out]

11/30*x*e^2 + 1/6*x + e^x

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mupad [B]  time = 7.97, size = 11, normalized size = 0.61 \begin {gather*} {\mathrm {e}}^x+x\,\left (\frac {11\,{\mathrm {e}}^2}{30}+\frac {1}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x/6 - exp(log(-(11*x)/5) + 2)/6 + x*exp(x))/x,x)

[Out]

exp(x) + x*((11*exp(2))/30 + 1/6)

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sympy [A]  time = 0.09, size = 14, normalized size = 0.78 \begin {gather*} x \left (\frac {1}{6} + \frac {11 e^{2}}{30}\right ) + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(-exp(ln(-11/5*x)+2)+6*exp(x)*x+x)/x,x)

[Out]

x*(1/6 + 11*exp(2)/30) + exp(x)

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