\(\int \frac {e^{225/x} (2 x-x^9)+e^{225/x} (-450+2 x+225 x^8+7 x^9) \log (x)}{4 x-4 x^9+x^{17}} \, dx\) [7454]

Optimal result

Integrand size = 58, antiderivative size = 22 \[ \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{4 x-4 x^9+x^{17}} \, dx=5+\frac {e^{225/x} x \log (x)}{2-x^8} \]

[Out]

5+exp(225/x)*ln(x)*x/(-x^8+2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1608, 28, 6873, 2326} \[ \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{4 x-4 x^9+x^{17}} \, dx=\frac {e^{225/x} x \left (2 \log (x)-x^8 \log (x)\right )}{\left (2-x^8\right )^2} \]

[In]

Int[(E^(225/x)*(2*x - x^9) + E^(225/x)*(-450 + 2*x + 225*x^8 + 7*x^9)*Log[x])/(4*x - 4*x^9 + x^17),x]

[Out]

(E^(225/x)*x*(2*Log[x] - x^8*Log[x]))/(2 - x^8)^2

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1608

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 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{x \left (4-4 x^8+x^{16}\right )} \, dx \\ & = \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{x \left (-2+x^8\right )^2} \, dx \\ & = \int \frac {e^{225/x} \left (2 x-x^9-450 \log (x)+2 x \log (x)+225 x^8 \log (x)+7 x^9 \log (x)\right )}{x \left (2-x^8\right )^2} \, dx \\ & = \frac {e^{225/x} x \left (2 \log (x)-x^8 \log (x)\right )}{\left (2-x^8\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{4 x-4 x^9+x^{17}} \, dx=\frac {e^{225/x} x \log (x)}{2-x^8} \]

[In]

Integrate[(E^(225/x)*(2*x - x^9) + E^(225/x)*(-450 + 2*x + 225*x^8 + 7*x^9)*Log[x])/(4*x - 4*x^9 + x^17),x]

[Out]

(E^(225/x)*x*Log[x])/(2 - x^8)

Maple [A] (verified)

Time = 11.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
risch	\(-\frac {x \,{\mathrm e}^{\frac {225}{x}} \ln \left (x \right )}{x^{8}-2}\)	\(19\)
parallelrisch	\(-\frac {x \,{\mathrm e}^{\frac {225}{x}} \ln \left (x \right )}{x^{8}-2}\)	\(19\)

[In]

int(((7*x^9+225*x^8+2*x-450)*exp(225/x)*ln(x)+(-x^9+2*x)*exp(225/x))/(x^17-4*x^9+4*x),x,method=_RETURNVERBOSE)

[Out]

-x/(x^8-2)*exp(225/x)*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{4 x-4 x^9+x^{17}} \, dx=-\frac {x e^{\frac {225}{x}} \log \left (x\right )}{x^{8} - 2} \]

[In]

integrate(((7*x^9+225*x^8+2*x-450)*exp(225/x)*log(x)+(-x^9+2*x)*exp(225/x))/(x^17-4*x^9+4*x),x, algorithm="fri
cas")

[Out]

-x*e^(225/x)*log(x)/(x^8 - 2)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{4 x-4 x^9+x^{17}} \, dx=- \frac {x e^{\frac {225}{x}} \log {\left (x \right )}}{x^{8} - 2} \]

[In]

integrate(((7*x**9+225*x**8+2*x-450)*exp(225/x)*ln(x)+(-x**9+2*x)*exp(225/x))/(x**17-4*x**9+4*x),x)

[Out]

-x*exp(225/x)*log(x)/(x**8 - 2)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{4 x-4 x^9+x^{17}} \, dx=-\frac {x e^{\frac {225}{x}} \log \left (x\right )}{x^{8} - 2} \]

[In]

integrate(((7*x^9+225*x^8+2*x-450)*exp(225/x)*log(x)+(-x^9+2*x)*exp(225/x))/(x^17-4*x^9+4*x),x, algorithm="max
ima")

[Out]

-x*e^(225/x)*log(x)/(x^8 - 2)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{4 x-4 x^9+x^{17}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((7*x^9+225*x^8+2*x-450)*exp(225/x)*log(x)+(-x^9+2*x)*exp(225/x))/(x^17-4*x^9+4*x),x, algorithm="gia
c")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Polynomial exponent overflow. Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{225/x} \left (2 x-x^9\right )+e^{225/x} \left (-450+2 x+225 x^8+7 x^9\right ) \log (x)}{4 x-4 x^9+x^{17}} \, dx=\int \frac {{\mathrm {e}}^{225/x}\,\left (2\,x-x^9\right )+{\mathrm {e}}^{225/x}\,\ln \left (x\right )\,\left (7\,x^9+225\,x^8+2\,x-450\right )}{x^{17}-4\,x^9+4\,x} \,d x \]

[In]

int((exp(225/x)*(2*x - x^9) + exp(225/x)*log(x)*(2*x + 225*x^8 + 7*x^9 - 450))/(4*x - 4*x^9 + x^17),x)

[Out]

int((exp(225/x)*(2*x - x^9) + exp(225/x)*log(x)*(2*x + 225*x^8 + 7*x^9 - 450))/(4*x - 4*x^9 + x^17), x)