3.386 \(\int \frac{1}{x^5 (13+\frac{2}{x}+15 x)} \, dx\)

Optimal. Leaf size=48 \[ \frac{13}{8 x^2}-\frac{1}{6 x^3}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]

[Out]

-1/(6*x^3) + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 + (625*Log[1 + 5*x])/7

________________________________________________________________________________________

Rubi [A]  time = 0.040875, antiderivative size = 48, normalized size of antiderivative = 1., 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, 709, 800} \[ \frac{13}{8 x^2}-\frac{1}{6 x^3}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(13 + 2/x + 15*x)),x]

[Out]

-1/(6*x^3) + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 + (625*Log[1 + 5*x])/7

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]

Rule 709

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (13+\frac{2}{x}+15 x\right )} \, dx &=\int \frac{1}{x^4 \left (2+13 x+15 x^2\right )} \, dx\\ &=-\frac{1}{6 x^3}+\frac{1}{2} \int \frac{-13-15 x}{x^3 \left (2+13 x+15 x^2\right )} \, dx\\ &=-\frac{1}{6 x^3}+\frac{1}{2} \int \left (-\frac{13}{2 x^3}+\frac{139}{4 x^2}-\frac{1417}{8 x}-\frac{243}{56 (2+3 x)}+\frac{6250}{7 (1+5 x)}\right ) \, dx\\ &=-\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (2+3 x)+\frac{625}{7} \log (1+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0046693, size = 48, normalized size = 1. \[ \frac{13}{8 x^2}-\frac{1}{6 x^3}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(13 + 2/x + 15*x)),x]

[Out]

-1/(6*x^3) + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 + (625*Log[1 + 5*x])/7

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 37, normalized size = 0.8 \begin{align*} -{\frac{1}{6\,{x}^{3}}}+{\frac{13}{8\,{x}^{2}}}-{\frac{139}{8\,x}}-{\frac{1417\,\ln \left ( x \right ) }{16}}-{\frac{81\,\ln \left ( 2+3\,x \right ) }{112}}+{\frac{625\,\ln \left ( 1+5\,x \right ) }{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(13+2/x+15*x),x)

[Out]

-1/6/x^3+13/8/x^2-139/8/x-1417/16*ln(x)-81/112*ln(2+3*x)+625/7*ln(1+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.00942, size = 49, normalized size = 1.02 \begin{align*} -\frac{417 \, x^{2} - 39 \, x + 4}{24 \, x^{3}} + \frac{625}{7} \, \log \left (5 \, x + 1\right ) - \frac{81}{112} \, \log \left (3 \, x + 2\right ) - \frac{1417}{16} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(13+2/x+15*x),x, algorithm="maxima")

[Out]

-1/24*(417*x^2 - 39*x + 4)/x^3 + 625/7*log(5*x + 1) - 81/112*log(3*x + 2) - 1417/16*log(x)

________________________________________________________________________________________

Fricas [A]  time = 1.30015, size = 138, normalized size = 2.88 \begin{align*} \frac{30000 \, x^{3} \log \left (5 \, x + 1\right ) - 243 \, x^{3} \log \left (3 \, x + 2\right ) - 29757 \, x^{3} \log \left (x\right ) - 5838 \, x^{2} + 546 \, x - 56}{336 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(13+2/x+15*x),x, algorithm="fricas")

[Out]

1/336*(30000*x^3*log(5*x + 1) - 243*x^3*log(3*x + 2) - 29757*x^3*log(x) - 5838*x^2 + 546*x - 56)/x^3

________________________________________________________________________________________

Sympy [A]  time = 0.16058, size = 41, normalized size = 0.85 \begin{align*} - \frac{1417 \log{\left (x \right )}}{16} + \frac{625 \log{\left (x + \frac{1}{5} \right )}}{7} - \frac{81 \log{\left (x + \frac{2}{3} \right )}}{112} - \frac{417 x^{2} - 39 x + 4}{24 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(13+2/x+15*x),x)

[Out]

-1417*log(x)/16 + 625*log(x + 1/5)/7 - 81*log(x + 2/3)/112 - (417*x**2 - 39*x + 4)/(24*x**3)

________________________________________________________________________________________

Giac [A]  time = 1.24626, size = 53, normalized size = 1.1 \begin{align*} -\frac{417 \, x^{2} - 39 \, x + 4}{24 \, x^{3}} + \frac{625}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac{81}{112} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{1417}{16} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(13+2/x+15*x),x, algorithm="giac")

[Out]

-1/24*(417*x^2 - 39*x + 4)/x^3 + 625/7*log(abs(5*x + 1)) - 81/112*log(abs(3*x + 2)) - 1417/16*log(abs(x))