\(\int \frac {x}{2+13 x+15 x^2} \, dx\) [2239]

Optimal result

Integrand size = 14, antiderivative size = 21 \[ \int \frac {x}{2+13 x+15 x^2} \, dx=\frac {2}{21} \log (2+3 x)-\frac {1}{35} \log (1+5 x) \]

[Out]

2/21*ln(2+3*x)-1/35*ln(1+5*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {646, 31} \[ \int \frac {x}{2+13 x+15 x^2} \, dx=\frac {2}{21} \log (3 x+2)-\frac {1}{35} \log (5 x+1) \]

[In]

Int[x/(2 + 13*x + 15*x^2),x]

[Out]

(2*Log[2 + 3*x])/21 - Log[1 + 5*x]/35

Rule 31

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

Rule 646

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

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {3}{7} \int \frac {1}{3+15 x} \, dx\right )+\frac {10}{7} \int \frac {1}{10+15 x} \, dx \\ & = \frac {2}{21} \log (2+3 x)-\frac {1}{35} \log (1+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {x}{2+13 x+15 x^2} \, dx=\frac {2}{21} \log (2+3 x)-\frac {1}{35} \log (1+5 x) \]

[In]

Integrate[x/(2 + 13*x + 15*x^2),x]

[Out]

(2*Log[2 + 3*x])/21 - Log[1 + 5*x]/35

Maple [A] (verified)

Time = 20.75 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67

method	result	size
parallelrisch	\(\frac {2 \ln \left (\frac {2}{3}+x \right )}{21}-\frac {\ln \left (x +\frac {1}{5}\right )}{35}\)	\(14\)
default	\(\frac {2 \ln \left (2+3 x \right )}{21}-\frac {\ln \left (1+5 x \right )}{35}\)	\(18\)
norman	\(\frac {2 \ln \left (2+3 x \right )}{21}-\frac {\ln \left (1+5 x \right )}{35}\)	\(18\)
risch	\(\frac {2 \ln \left (2+3 x \right )}{21}-\frac {\ln \left (1+5 x \right )}{35}\)	\(18\)

[In]

int(x/(15*x^2+13*x+2),x,method=_RETURNVERBOSE)

[Out]

2/21*ln(2/3+x)-1/35*ln(x+1/5)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x}{2+13 x+15 x^2} \, dx=-\frac {1}{35} \, \log \left (5 \, x + 1\right ) + \frac {2}{21} \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/35*log(5*x + 1) + 2/21*log(3*x + 2)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x}{2+13 x+15 x^2} \, dx=- \frac {\log {\left (x + \frac {1}{5} \right )}}{35} + \frac {2 \log {\left (x + \frac {2}{3} \right )}}{21} \]

[In]

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

[Out]

-log(x + 1/5)/35 + 2*log(x + 2/3)/21

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x}{2+13 x+15 x^2} \, dx=-\frac {1}{35} \, \log \left (5 \, x + 1\right ) + \frac {2}{21} \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/35*log(5*x + 1) + 2/21*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x}{2+13 x+15 x^2} \, dx=-\frac {1}{35} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) + \frac {2}{21} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

-1/35*log(abs(5*x + 1)) + 2/21*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {x}{2+13 x+15 x^2} \, dx=\frac {2\,\ln \left (x+\frac {2}{3}\right )}{21}-\frac {\ln \left (x+\frac {1}{5}\right )}{35} \]

[In]

int(x/(13*x + 15*x^2 + 2),x)

[Out]

(2*log(x + 2/3))/21 - log(x + 1/5)/35