### 3.2240 $$\int \frac{1}{2+13 x+15 x^2} \, dx$$

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

[Out]

-Log[2 + 3*x]/7 + Log[1 + 5*x]/7

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Rubi [A]  time = 0.0058359, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {616, 31} $\frac{1}{7} \log (5 x+1)-\frac{1}{7} \log (3 x+2)$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + 13*x + 15*x^2)^(-1),x]

[Out]

-Log[2 + 3*x]/7 + Log[1 + 5*x]/7

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 31

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

Rubi steps

\begin{align*} \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 = 0.0023779, size = 21, normalized size = 1. $\frac{1}{7} \log (5 x+1)-\frac{1}{7} \log (3 x+2)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 + 13*x + 15*x^2)^(-1),x]

[Out]

-Log[2 + 3*x]/7 + Log[1 + 5*x]/7

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Maple [A]  time = 0.043, size = 18, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( 2+3\,x \right ) }{7}}+{\frac{\ln \left ( 1+5\,x \right ) }{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988588, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{7} \, \log \left (5 \, x + 1\right ) - \frac{1}{7} \, \log \left (3 \, x + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*log(5*x + 1) - 1/7*log(3*x + 2)

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Fricas [A]  time = 2.35735, size = 50, normalized size = 2.38 \begin{align*} \frac{1}{7} \, \log \left (5 \, x + 1\right ) - \frac{1}{7} \, \log \left (3 \, x + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*log(5*x + 1) - 1/7*log(3*x + 2)

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Sympy [A]  time = 0.093426, size = 15, normalized size = 0.71 \begin{align*} \frac{\log{\left (x + \frac{1}{5} \right )}}{7} - \frac{\log{\left (x + \frac{2}{3} \right )}}{7} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 1/5)/7 - log(x + 2/3)/7

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Giac [A]  time = 1.14799, size = 26, normalized size = 1.24 \begin{align*} \frac{1}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac{1}{7} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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