3.25.52 \(\int \frac {e^{\frac {1}{31} (54+6 x)} (-31+6 x)}{31 e^{\frac {1}{31} (54+6 x)} x-465 x^2} \, dx\)

Optimal. Leaf size=18 \[ \log \left (-16+\frac {e^{\frac {6 (9+x)}{31}}+x}{x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.73, antiderivative size = 19, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6741, 12, 6712, 31} \begin {gather*} \log \left (15-\frac {e^{\frac {6 x}{31}+\frac {54}{31}}}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((54 + 6*x)/31)*(-31 + 6*x))/(31*E^((54 + 6*x)/31)*x - 465*x^2),x]

[Out]

Log[15 - E^(54/31 + (6*x)/31)/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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 6712

Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x
] - q*v*D[w, x])]}, -Dist[c*q, Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ
[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]

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 {54}{31}+\frac {6 x}{31}} (-31+6 x)}{31 \left (e^{\frac {54}{31}+\frac {6 x}{31}}-15 x\right ) x} \, dx\\ &=\frac {1}{31} \int \frac {e^{\frac {54}{31}+\frac {6 x}{31}} (-31+6 x)}{\left (e^{\frac {54}{31}+\frac {6 x}{31}}-15 x\right ) x} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{-15+x} \, dx,x,\frac {e^{\frac {54}{31}+\frac {6 x}{31}}}{x}\right )\\ &=\log \left (15-\frac {e^{\frac {54}{31}+\frac {6 x}{31}}}{x}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 27, normalized size = 1.50 \begin {gather*} \frac {1}{31} \left (31 \log \left (e^{\frac {54}{31}+\frac {6 x}{31}}-15 x\right )-31 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((54 + 6*x)/31)*(-31 + 6*x))/(31*E^((54 + 6*x)/31)*x - 465*x^2),x]

[Out]

(31*Log[E^(54/31 + (6*x)/31) - 15*x] - 31*Log[x])/31

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 16, normalized size = 0.89 \begin {gather*} -\log \relax (x) + \log \left (-15 \, x + e^{\left (\frac {6}{31} \, x + \frac {54}{31}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x-31)*exp(6/31*x+54/31)/(31*x*exp(6/31*x+54/31)-465*x^2),x, algorithm="fricas")

[Out]

-log(x) + log(-15*x + e^(6/31*x + 54/31))

________________________________________________________________________________________

giac [A]  time = 0.32, size = 18, normalized size = 1.00 \begin {gather*} \log \left (15 \, x - e^{\left (\frac {6}{31} \, x + \frac {54}{31}\right )}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x-31)*exp(6/31*x+54/31)/(31*x*exp(6/31*x+54/31)-465*x^2),x, algorithm="giac")

[Out]

log(15*x - e^(6/31*x + 54/31)) - log(x)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 18, normalized size = 1.00

method	result	size
risch	\(-\ln \relax (x )-\frac {54}{31}+\ln \left ({\mathrm e}^{\frac {6 x}{31}+\frac {54}{31}}-15 x \right )\)	\(18\)
norman	\(-\ln \relax (x )+\ln \left (15 x -{\mathrm e}^{\frac {6 x}{31}+\frac {54}{31}}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x-31)*exp(6/31*x+54/31)/(31*x*exp(6/31*x+54/31)-465*x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(x)-54/31+ln(exp(6/31*x+54/31)-15*x)

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 24, normalized size = 1.33 \begin {gather*} \log \left (-{\left (15 \, x e^{\frac {8}{31}} - e^{\left (\frac {6}{31} \, x + 2\right )}\right )} e^{\left (-2\right )}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x-31)*exp(6/31*x+54/31)/(31*x*exp(6/31*x+54/31)-465*x^2),x, algorithm="maxima")

[Out]

log(-(15*x*e^(8/31) - e^(6/31*x + 2))*e^(-2)) - log(x)

________________________________________________________________________________________

mupad [B]  time = 1.43, size = 16, normalized size = 0.89 \begin {gather*} \ln \left (x-\frac {{\mathrm {e}}^{54/31}\,{\left ({\mathrm {e}}^x\right )}^{6/31}}{15}\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((6*x)/31 + 54/31)*(6*x - 31))/(31*x*exp((6*x)/31 + 54/31) - 465*x^2),x)

[Out]

log(x - (exp(54/31)*exp(x)^(6/31))/15) - log(x)

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 17, normalized size = 0.94 \begin {gather*} - \log {\relax (x )} + \log {\left (- 15 x + e^{\frac {6 x}{31} + \frac {54}{31}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x-31)*exp(6/31*x+54/31)/(31*x*exp(6/31*x+54/31)-465*x**2),x)

[Out]

-log(x) + log(-15*x + exp(6*x/31 + 54/31))

________________________________________________________________________________________