\(\int \frac {x+e^x (-3 x+x^2)[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx\) [4228]

Optimal result

Integrand size = 53, antiderivative size = 20 \[ \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx=e^x+e^{(4+\log (28 x))^2}+\log (3-x) \]

[Out]

exp(x)+ln(-x+3)+exp((4+ln(28*x))^2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {1607, 6820, 2225, 2326} \[ \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx=377801998336 x^8 e^{\log ^2(28 x)+16}+e^x+\log (3-x) \]

[In]

Int[(x + E^x*(-3*x + x^2) + 377801998336*E^(16 + Log[28*x]^2)*x^8*(-24 + 8*x + (-6 + 2*x)*Log[28*x]))/(-3*x +
x^2),x]

[Out]

E^x + 377801998336*E^(16 + Log[28*x]^2)*x^8 + Log[3 - x]

Rule 1607

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

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{(-3+x) x} \, dx \\ & = \int \left (e^x+\frac {1}{-3+x}[phone] e^{16+\log ^2(28 x)} x^7 (4+\log (28 x))\right ) \, dx \\ & = \log (3-x)[phone] \int e^{16+\log ^2(28 x)} x^7 (4+\log (28 x)) \, dx+\int e^x \, dx \\ & = e^x+377801998336 e^{16+\log ^2(28 x)} x^8+\log (3-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx=e^x+377801998336 e^{16+\log ^2(28 x)} x^8+\log (-3+x) \]

[In]

Integrate[(x + E^x*(-3*x + x^2) + 377801998336*E^(16 + Log[28*x]^2)*x^8*(-24 + 8*x + (-6 + 2*x)*Log[28*x]))/(-
3*x + x^2),x]

[Out]

E^x + 377801998336*E^(16 + Log[28*x]^2)*x^8 + Log[-3 + x]

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

method	result	size
risch	\(\ln \left (-3+x \right )[phone] x^{8} {\mathrm e}^{\ln \left (28 x \right )^{2}+16}+{\mathrm e}^{x}\)	\(22\)
default	\(\ln \left (-3+x \right )+{\mathrm e}^{\ln \left (28 x \right )^{2}+8 \ln \left (28 x \right )+16}+{\mathrm e}^{x}\)	\(23\)
parallelrisch	\(\ln \left (-3+x \right )+{\mathrm e}^{\ln \left (28 x \right )^{2}+8 \ln \left (28 x \right )+16}+{\mathrm e}^{x}\)	\(23\)
parts	\(\ln \left (-3+x \right )+{\mathrm e}^{\ln \left (28 x \right )^{2}+8 \ln \left (28 x \right )+16}+{\mathrm e}^{x}\)	\(23\)

[In]

int((((2*x-6)*ln(28*x)+8*x-24)*exp(ln(28*x)^2+8*ln(28*x)+16)+(x^2-3*x)*exp(x)+x)/(x^2-3*x),x,method=_RETURNVER
BOSE)

[Out]

ln(-3+x)[phone]*x^8*exp(ln(28*x)^2+16)+exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx=e^{\left (\log \left (28 \, x\right )^{2} + 8 \, \log \left (28 \, x\right ) + 16\right )} + e^{x} + \log \left (x - 3\right ) \]

[In]

integrate((((2*x-6)*log(28*x)+8*x-24)*exp(log(28*x)^2+8*log(28*x)+16)+(x^2-3*x)*exp(x)+x)/(x^2-3*x),x, algorit
hm="fricas")

[Out]

e^(log(28*x)^2 + 8*log(28*x) + 16) + e^x + log(x - 3)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx=377801998336 x^{8} e^{\log {\left (28 x \right )}^{2} + 16} + e^{x} + \log {\left (x - 3 \right )} \]

[In]

integrate((((2*x-6)*ln(28*x)+8*x-24)*exp(ln(28*x)**2+8*ln(28*x)+16)+(x**2-3*x)*exp(x)+x)/(x**2-3*x),x)

[Out]

377801998336*x**8*exp(log(28*x)**2 + 16) + exp(x) + log(x - 3)

Maxima [F]

\[ \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx=\int { \frac {2 \, {\left ({\left (x - 3\right )} \log \left (28 \, x\right ) + 4 \, x - 12\right )} e^{\left (\log \left (28 \, x\right )^{2} + 8 \, \log \left (28 \, x\right ) + 16\right )} + {\left (x^{2} - 3 \, x\right )} e^{x} + x}{x^{2} - 3 \, x} \,d x } \]

[In]

integrate((((2*x-6)*log(28*x)+8*x-24)*exp(log(28*x)^2+8*log(28*x)+16)+(x^2-3*x)*exp(x)+x)/(x^2-3*x),x, algorit
hm="maxima")

[Out]

5764801*2^(4*log(7) + 16)*x^8*e^(log(7)^2 + 4*log(2)^2 + 2*log(7)*log(x) + 4*log(2)*log(x) + log(x)^2 + 16) +
3*e^3*exp_integral_e(1, -x + 3) + integrate(x*e^x/(x - 3), x) + log(x - 3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx=377801998336 \, x^{8} e^{\left (\log \left (28 \, x\right )^{2} + 16\right )} + e^{x} + \log \left (x - 3\right ) \]

[In]

integrate((((2*x-6)*log(28*x)+8*x-24)*exp(log(28*x)^2+8*log(28*x)+16)+(x^2-3*x)*exp(x)+x)/(x^2-3*x),x, algorit
hm="giac")

[Out]

377801998336*x^8*e^(log(28*x)^2 + 16) + e^x + log(x - 3)

Mupad [B] (verification not implemented)

Time = 11.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50 \[ \int \frac {x+e^x \left (-3 x+x^2\right )[phone] e^{16+\log ^2(28 x)} x^8 (-24+8 x+(-6+2 x) \log (28 x))}{-3 x+x^2} \, dx=\ln \left (x-3\right )+{\mathrm {e}}^x+377801998336\,x^{2\,\ln \left (28\right )}\,x^8\,{\mathrm {e}}^{{\ln \left (28\right )}^2}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{{\ln \left (x\right )}^2} \]

[In]

int(-(x + exp(8*log(28*x) + log(28*x)^2 + 16)*(8*x + log(28*x)*(2*x - 6) - 24) - exp(x)*(3*x - x^2))/(3*x - x^
2),x)

[Out]

log(x - 3) + exp(x) + 377801998336*x^(2*log(28))*x^8*exp(log(28)^2)*exp(16)*exp(log(x)^2)