3.382 \(\int \frac {1}{x (13+\frac {2}{x}+15 x)} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{7} \log (5 x+1)-\frac {1}{7} \log (3 x+2) \]

[Out]

-1/7*ln(2+3*x)+1/7*ln(1+5*x)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1386, 616, 31} \[ \frac {1}{7} \log (5 x+1)-\frac {1}{7} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(13 + 2/x + 15*x)),x]

[Out]

-Log[2 + 3*x]/7 + Log[1 + 5*x]/7

Rule 31

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

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 1386

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n_.) + (b_.)*(x_)^(mn_))^(p_.), x_Symbol] :> Int[x^(m - n*p)*(b + a*x^n + c
*x^(2*n))^p, x] /; FreeQ[{a, b, c, m, n}, x] && EqQ[mn, -n] && IntegerQ[p] && PosQ[n]

Rubi steps

\begin {align*} \int \frac {1}{x \left (13+\frac {2}{x}+15 x\right )} \, dx &=\int \frac {1}{2+13 x+15 x^2} \, dx\\ &=\frac {15}{7} \int \frac {1}{3+15 x} \, dx-\frac {15}{7} \int \frac {1}{10+15 x} \, dx\\ &=-\frac {1}{7} \log (2+3 x)+\frac {1}{7} \log (1+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {1}{7} \log (5 x+1)-\frac {1}{7} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(13 + 2/x + 15*x)),x]

[Out]

-1/7*Log[2 + 3*x] + Log[1 + 5*x]/7

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 17, normalized size = 0.81 \[ \frac {1}{7} \, \log \left (5 \, x + 1\right ) - \frac {1}{7} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(13+2/x+15*x),x, algorithm="fricas")

[Out]

1/7*log(5*x + 1) - 1/7*log(3*x + 2)

________________________________________________________________________________________

giac [A]  time = 0.27, size = 19, normalized size = 0.90 \[ \frac {1}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac {1}{7} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(13+2/x+15*x),x, algorithm="giac")

[Out]

1/7*log(abs(5*x + 1)) - 1/7*log(abs(3*x + 2))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 18, normalized size = 0.86 \[ \frac {\ln \left (5 x +1\right )}{7}-\frac {\ln \left (3 x +2\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(13+2/x+15*x),x)

[Out]

-1/7*ln(3*x+2)+1/7*ln(5*x+1)

________________________________________________________________________________________

maxima [A]  time = 0.93, size = 17, normalized size = 0.81 \[ \frac {1}{7} \, \log \left (5 \, x + 1\right ) - \frac {1}{7} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(13+2/x+15*x),x, algorithm="maxima")

[Out]

1/7*log(5*x + 1) - 1/7*log(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 8, normalized size = 0.38 \[ -\frac {2\,\mathrm {atanh}\left (\frac {30\,x}{7}+\frac {13}{7}\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(15*x + 2/x + 13)),x)

[Out]

-(2*atanh((30*x)/7 + 13/7))/7

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 15, normalized size = 0.71 \[ \frac {\log {\left (x + \frac {1}{5} \right )}}{7} - \frac {\log {\left (x + \frac {2}{3} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(13+2/x+15*x),x)

[Out]

log(x + 1/5)/7 - log(x + 2/3)/7

________________________________________________________________________________________