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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, size = 48, normalized size = 1.00 \[ -\frac {1}{6 x^3}+\frac {13}{8 x^2}-\frac {139}{8 x}-\frac {1417 \log (x)}{16}-\frac {81}{112} \log (3 x+2)+\frac {625}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*1/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

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 44, normalized size = 0.92 \[ \frac {30000 \, x^{3} \log \left (5 \, x + 1\right ) - 243 \, x^{3} \log \left (3 \, x + 2\right ) - 29757 \, x^{3} \log \relax (x) - 5838 \, x^{2} + 546 \, x - 56}{336 \, x^{3}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.24, size = 39, normalized size = 0.81 \[ -\frac {417 \, x^{2} - 39 \, x + 4}{24 \, x^{3}} + \frac {625}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac {81}{112} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {1417}{16} \, \log \left ({\left | x \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 37, normalized size = 0.77 \[ -\frac {1417 \ln \relax (x )}{16}+\frac {625 \ln \left (5 x +1\right )}{7}-\frac {81 \ln \left (3 x +2\right )}{112}-\frac {139}{8 x}+\frac {13}{8 x^{2}}-\frac {1}{6 x^{3}} \]

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(3*x+2)+625/7*ln(5*x+1)

________________________________________________________________________________________

maxima [A]  time = 1.07, size = 36, normalized size = 0.75 \[ -\frac {417 \, x^{2} - 39 \, x + 4}{24 \, x^{3}} + \frac {625}{7} \, \log \left (5 \, x + 1\right ) - \frac {81}{112} \, \log \left (3 \, x + 2\right ) - \frac {1417}{16} \, \log \relax (x) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 32, normalized size = 0.67 \[ \frac {625\,\ln \left (x+\frac {1}{5}\right )}{7}-\frac {81\,\ln \left (x+\frac {2}{3}\right )}{112}-\frac {1417\,\ln \relax (x)}{16}-\frac {\frac {139\,x^2}{8}-\frac {13\,x}{8}+\frac {1}{6}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(15*x + 2/x + 13)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 41, normalized size = 0.85 \[ - \frac {1417 \log {\relax (x )}}{16} + \frac {625 \log {\left (x + \frac {1}{5} \right )}}{7} - \frac {81 \log {\left (x + \frac {2}{3} \right )}}{112} + \frac {- 417 x^{2} + 39 x - 4}{24 x^{3}} \]

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)

________________________________________________________________________________________