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

Optimal. Leaf size=20 \[ e^x+e^{(4+\log (28 x))^2}+\log (3-x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.54, antiderivative size = 25, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 4, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {1593, 6688, 2194, 2288} \begin {gather*} 377801998336 x^8 e^{\log ^2(28 x)+16}+e^x+\log (3-x) \end {gather*}

Antiderivative was successfully verified.

[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 1593

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 2194

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 2288

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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 23, normalized size = 1.15 \begin {gather*} e^x+377801998336 e^{16+\log ^2(28 x)} x^8+\log (-3+x) \end {gather*}

Antiderivative was successfully verified.

[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]

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 22, normalized size = 1.10 \begin {gather*} e^{\left (\log \left (28 \, x\right )^{2} + 8 \, \log \left (28 \, x\right ) + 16\right )} + e^{x} + \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 21, normalized size = 1.05 \begin {gather*} 377801998336 \, x^{8} e^{\left (\log \left (28 \, x\right )^{2} + 16\right )} + e^{x} + \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.09, size = 22, normalized size = 1.10

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

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 5764801 \cdot 2^{4 \, \log \relax (7) + 16} x^{8} e^{\left (\log \relax (7)^{2} + 4 \, \log \relax (2)^{2} + 2 \, \log \relax (7) \log \relax (x) + 4 \, \log \relax (2) \log \relax (x) + \log \relax (x)^{2} + 16\right )} + 3 \, e^{3} E_{1}\left (-x + 3\right ) + \int \frac {x e^{x}}{x - 3}\,{d x} + \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 3.09, size = 30, normalized size = 1.50 \begin {gather*} \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 \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.46, size = 22, normalized size = 1.10 \begin {gather*} 377801998336 x^{8} e^{\log {\left (28 x \right )}^{2} + 16} + e^{x} + \log {\left (x - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________