### 3.2236 $$\int \frac{x^4}{2+13 x+15 x^2} \, dx$$

Optimal. Leaf size=40 $\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}$

[Out]

(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375

________________________________________________________________________________________

Rubi [A]  time = 0.0172645, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.188, Rules used = {701, 632, 31} $\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^4/(2 + 13*x + 15*x^2),x]

[Out]

(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375

Rule 701

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, 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] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

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{x^4}{2+13 x+15 x^2} \, dx &=\int \left (\frac{139}{3375}-\frac{13 x}{225}+\frac{x^2}{15}-\frac{278+1417 x}{3375 \left (2+13 x+15 x^2\right )}\right ) \, dx\\ &=\frac{139 x}{3375}-\frac{13 x^2}{450}+\frac{x^3}{45}-\frac{\int \frac{278+1417 x}{2+13 x+15 x^2} \, dx}{3375}\\ &=\frac{139 x}{3375}-\frac{13 x^2}{450}+\frac{x^3}{45}+\frac{3}{875} \int \frac{1}{3+15 x} \, dx-\frac{80}{189} \int \frac{1}{10+15 x} \, dx\\ &=\frac{139 x}{3375}-\frac{13 x^2}{450}+\frac{x^3}{45}-\frac{16}{567} \log (2+3 x)+\frac{\log (1+5 x)}{4375}\\ \end{align*}

Mathematica [A]  time = 0.0052034, size = 40, normalized size = 1. $\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^4/(2 + 13*x + 15*x^2),x]

[Out]

(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 31, normalized size = 0.8 \begin{align*}{\frac{139\,x}{3375}}-{\frac{13\,{x}^{2}}{450}}+{\frac{{x}^{3}}{45}}-{\frac{16\,\ln \left ( 2+3\,x \right ) }{567}}+{\frac{\ln \left ( 1+5\,x \right ) }{4375}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(15*x^2+13*x+2),x)

[Out]

139/3375*x-13/450*x^2+1/45*x^3-16/567*ln(2+3*x)+1/4375*ln(1+5*x)

________________________________________________________________________________________

Maxima [A]  time = 0.97224, size = 41, normalized size = 1.02 \begin{align*} \frac{1}{45} \, x^{3} - \frac{13}{450} \, x^{2} + \frac{139}{3375} \, x + \frac{1}{4375} \, \log \left (5 \, x + 1\right ) - \frac{16}{567} \, \log \left (3 \, x + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(15*x^2+13*x+2),x, algorithm="maxima")

[Out]

1/45*x^3 - 13/450*x^2 + 139/3375*x + 1/4375*log(5*x + 1) - 16/567*log(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 2.27832, size = 108, normalized size = 2.7 \begin{align*} \frac{1}{45} \, x^{3} - \frac{13}{450} \, x^{2} + \frac{139}{3375} \, x + \frac{1}{4375} \, \log \left (5 \, x + 1\right ) - \frac{16}{567} \, \log \left (3 \, x + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(15*x^2+13*x+2),x, algorithm="fricas")

[Out]

1/45*x^3 - 13/450*x^2 + 139/3375*x + 1/4375*log(5*x + 1) - 16/567*log(3*x + 2)

________________________________________________________________________________________

Sympy [A]  time = 0.105138, size = 34, normalized size = 0.85 \begin{align*} \frac{x^{3}}{45} - \frac{13 x^{2}}{450} + \frac{139 x}{3375} + \frac{\log{\left (x + \frac{1}{5} \right )}}{4375} - \frac{16 \log{\left (x + \frac{2}{3} \right )}}{567} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(15*x**2+13*x+2),x)

[Out]

x**3/45 - 13*x**2/450 + 139*x/3375 + log(x + 1/5)/4375 - 16*log(x + 2/3)/567

________________________________________________________________________________________

Giac [A]  time = 1.09413, size = 43, normalized size = 1.08 \begin{align*} \frac{1}{45} \, x^{3} - \frac{13}{450} \, x^{2} + \frac{139}{3375} \, x + \frac{1}{4375} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac{16}{567} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(15*x^2+13*x+2),x, algorithm="giac")

[Out]

1/45*x^3 - 13/450*x^2 + 139/3375*x + 1/4375*log(abs(5*x + 1)) - 16/567*log(abs(3*x + 2))