### 3.2250 $$\int \frac{1}{2 x+13 x^2+15 x^3} \, dx$$

Optimal. Leaf size=27 $\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)$

[Out]

Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7

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Rubi [A]  time = 0.0139869, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.312, Rules used = {1594, 705, 29, 632, 31} $\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[(2*x + 13*x^2 + 15*x^3)^(-1),x]

[Out]

Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 705

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 632

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]

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 x+13 x^2+15 x^3} \, dx &=\int \frac{1}{x \left (2+13 x+15 x^2\right )} \, dx\\ &=\frac{1}{2} \int \frac{1}{x} \, dx+\frac{1}{2} \int \frac{-13-15 x}{2+13 x+15 x^2} \, dx\\ &=\frac{\log (x)}{2}+\frac{45}{14} \int \frac{1}{10+15 x} \, dx-\frac{75}{7} \int \frac{1}{3+15 x} \, dx\\ &=\frac{\log (x)}{2}+\frac{3}{14} \log (2+3 x)-\frac{5}{7} \log (1+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0035077, size = 27, normalized size = 1. $\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2*x + 13*x^2 + 15*x^3)^(-1),x]

[Out]

Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7

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Maple [A]  time = 0.046, size = 22, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{2}}+{\frac{3\,\ln \left ( 2+3\,x \right ) }{14}}-{\frac{5\,\ln \left ( 1+5\,x \right ) }{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(15*x^3+13*x^2+2*x),x)

[Out]

1/2*ln(x)+3/14*ln(2+3*x)-5/7*ln(1+5*x)

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Maxima [A]  time = 0.985128, size = 28, normalized size = 1.04 \begin{align*} -\frac{5}{7} \, \log \left (5 \, x + 1\right ) + \frac{3}{14} \, \log \left (3 \, x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(15*x^3+13*x^2+2*x),x, algorithm="maxima")

[Out]

-5/7*log(5*x + 1) + 3/14*log(3*x + 2) + 1/2*log(x)

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Fricas [A]  time = 2.49202, size = 70, normalized size = 2.59 \begin{align*} -\frac{5}{7} \, \log \left (5 \, x + 1\right ) + \frac{3}{14} \, \log \left (3 \, x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(15*x^3+13*x^2+2*x),x, algorithm="fricas")

[Out]

-5/7*log(5*x + 1) + 3/14*log(3*x + 2) + 1/2*log(x)

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Sympy [A]  time = 0.128965, size = 24, normalized size = 0.89 \begin{align*} \frac{\log{\left (x \right )}}{2} - \frac{5 \log{\left (x + \frac{1}{5} \right )}}{7} + \frac{3 \log{\left (x + \frac{2}{3} \right )}}{14} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(15*x**3+13*x**2+2*x),x)

[Out]

log(x)/2 - 5*log(x + 1/5)/7 + 3*log(x + 2/3)/14

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Giac [A]  time = 1.12763, size = 32, normalized size = 1.19 \begin{align*} -\frac{5}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) + \frac{3}{14} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(15*x^3+13*x^2+2*x),x, algorithm="giac")

[Out]

-5/7*log(abs(5*x + 1)) + 3/14*log(abs(3*x + 2)) + 1/2*log(abs(x))