\(\int \frac {e^x (-48-42 x^2-45 x^3+6 x^4)+e^x (-6-3 x^2-6 x^3) \log (2+x^2+2 x^3)}{2+x^2+2 x^3} \, dx\) [7930]

Optimal result

Integrand size = 62, antiderivative size = 21 \[ \int \frac {e^x \left (-48-42 x^2-45 x^3+6 x^4\right )+e^x \left (-6-3 x^2-6 x^3\right ) \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx=3 e^x \left (-9+x-\log \left (2+x^2+2 x^3\right )\right ) \]

[Out]

3*exp(x)*(x-ln(2*x^3+x^2+2)-9)

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33, number of steps used = 33, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6874, 2225, 2207, 2634, 12} \[ \int \frac {e^x \left (-48-42 x^2-45 x^3+6 x^4\right )+e^x \left (-6-3 x^2-6 x^3\right ) \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx=-3 e^x \log \left (2 x^3+x^2+2\right )+3 e^x x-27 e^x \]

[In]

Int[(E^x*(-48 - 42*x^2 - 45*x^3 + 6*x^4) + E^x*(-6 - 3*x^2 - 6*x^3)*Log[2 + x^2 + 2*x^3])/(2 + x^2 + 2*x^3),x]

[Out]

-27*E^x + 3*E^x*x - 3*E^x*Log[2 + x^2 + 2*x^3]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

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 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {48 e^x}{2+x^2+2 x^3}-\frac {42 e^x x^2}{2+x^2+2 x^3}-\frac {45 e^x x^3}{2+x^2+2 x^3}+\frac {6 e^x x^4}{2+x^2+2 x^3}-\frac {6 e^x \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3}-\frac {3 e^x x^2 \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3}-\frac {6 e^x x^3 \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3}\right ) \, dx \\ & = -\left (3 \int \frac {e^x x^2 \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx\right )+6 \int \frac {e^x x^4}{2+x^2+2 x^3} \, dx-6 \int \frac {e^x \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx-6 \int \frac {e^x x^3 \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx-42 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx-45 \int \frac {e^x x^3}{2+x^2+2 x^3} \, dx-48 \int \frac {e^x}{2+x^2+2 x^3} \, dx \\ & = -3 e^x \log \left (2+x^2+2 x^3\right )+3 \int \frac {2 x (1+3 x) \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+6 \int \left (-\frac {e^x}{4}+\frac {e^x x}{2}+\frac {e^x \left (2-4 x+x^2\right )}{4 \left (2+x^2+2 x^3\right )}\right ) \, dx+6 \int \frac {2 x (1+3 x) \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+6 \int \frac {x (1+3 x) \left (e^x-2 \int \frac {e^x}{2+x^2+2 x^3} \, dx-\int \frac {e^x x^2}{2+x^2+2 x^3} \, dx\right )}{2+x^2+2 x^3} \, dx-42 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx-45 \int \left (\frac {e^x}{2}-\frac {e^x \left (2+x^2\right )}{2 \left (2+x^2+2 x^3\right )}\right ) \, dx-48 \int \frac {e^x}{2+x^2+2 x^3} \, dx \\ & = -3 e^x \log \left (2+x^2+2 x^3\right )-\frac {3 \int e^x \, dx}{2}+\frac {3}{2} \int \frac {e^x \left (2-4 x+x^2\right )}{2+x^2+2 x^3} \, dx+3 \int e^x x \, dx+6 \int \frac {x (1+3 x) \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+6 \int \left (\frac {e^x x (1+3 x)}{2+x^2+2 x^3}-\frac {x (1+3 x) \left (2 \int \frac {e^x}{2+x^2+2 x^3} \, dx+\int \frac {e^x x^2}{2+x^2+2 x^3} \, dx\right )}{2+x^2+2 x^3}\right ) \, dx+12 \int \frac {x (1+3 x) \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-\frac {45 \int e^x \, dx}{2}+\frac {45}{2} \int \frac {e^x \left (2+x^2\right )}{2+x^2+2 x^3} \, dx-42 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx-48 \int \frac {e^x}{2+x^2+2 x^3} \, dx \\ & = -24 e^x+3 e^x x-3 e^x \log \left (2+x^2+2 x^3\right )+\frac {3}{2} \int \left (\frac {2 e^x}{2+x^2+2 x^3}-\frac {4 e^x x}{2+x^2+2 x^3}+\frac {e^x x^2}{2+x^2+2 x^3}\right ) \, dx-3 \int e^x \, dx+6 \int \frac {e^x x (1+3 x)}{2+x^2+2 x^3} \, dx-6 \int \frac {x (1+3 x) \left (2 \int \frac {e^x}{2+x^2+2 x^3} \, dx+\int \frac {e^x x^2}{2+x^2+2 x^3} \, dx\right )}{2+x^2+2 x^3} \, dx+6 \int \left (\frac {x \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}+\frac {3 x^2 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}\right ) \, dx+12 \int \left (\frac {x \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}+\frac {3 x^2 \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}\right ) \, dx+\frac {45}{2} \int \left (\frac {2 e^x}{2+x^2+2 x^3}+\frac {e^x x^2}{2+x^2+2 x^3}\right ) \, dx-42 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx-48 \int \frac {e^x}{2+x^2+2 x^3} \, dx \\ & = -27 e^x+3 e^x x-3 e^x \log \left (2+x^2+2 x^3\right )+\frac {3}{2} \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+3 \int \frac {e^x}{2+x^2+2 x^3} \, dx-6 \int \frac {e^x x}{2+x^2+2 x^3} \, dx+6 \int \left (\frac {e^x x}{2+x^2+2 x^3}+\frac {3 e^x x^2}{2+x^2+2 x^3}\right ) \, dx+6 \int \frac {x \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-6 \int \left (\frac {2 x (1+3 x) \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}+\frac {x (1+3 x) \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}\right ) \, dx+12 \int \frac {x \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+18 \int \frac {x^2 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+\frac {45}{2} \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+36 \int \frac {x^2 \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-42 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+45 \int \frac {e^x}{2+x^2+2 x^3} \, dx-48 \int \frac {e^x}{2+x^2+2 x^3} \, dx \\ & = -27 e^x+3 e^x x-3 e^x \log \left (2+x^2+2 x^3\right )+\frac {3}{2} \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+3 \int \frac {e^x}{2+x^2+2 x^3} \, dx+6 \int \frac {x \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-6 \int \frac {x (1+3 x) \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+12 \int \frac {x \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-12 \int \frac {x (1+3 x) \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+18 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+18 \int \frac {x^2 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+\frac {45}{2} \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+36 \int \frac {x^2 \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-42 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+45 \int \frac {e^x}{2+x^2+2 x^3} \, dx-48 \int \frac {e^x}{2+x^2+2 x^3} \, dx \\ & = -27 e^x+3 e^x x-3 e^x \log \left (2+x^2+2 x^3\right )+\frac {3}{2} \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+3 \int \frac {e^x}{2+x^2+2 x^3} \, dx+6 \int \frac {x \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-6 \int \left (\frac {x \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}+\frac {3 x^2 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}\right ) \, dx+12 \int \frac {x \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-12 \int \left (\frac {x \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}+\frac {3 x^2 \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3}\right ) \, dx+18 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+18 \int \frac {x^2 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx+\frac {45}{2} \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+36 \int \frac {x^2 \int \frac {e^x}{2+x^2+2 x^3} \, dx}{2+x^2+2 x^3} \, dx-42 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+45 \int \frac {e^x}{2+x^2+2 x^3} \, dx-48 \int \frac {e^x}{2+x^2+2 x^3} \, dx \\ & = -27 e^x+3 e^x x-3 e^x \log \left (2+x^2+2 x^3\right )+\frac {3}{2} \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+3 \int \frac {e^x}{2+x^2+2 x^3} \, dx+18 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+\frac {45}{2} \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx-42 \int \frac {e^x x^2}{2+x^2+2 x^3} \, dx+45 \int \frac {e^x}{2+x^2+2 x^3} \, dx-48 \int \frac {e^x}{2+x^2+2 x^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-48-42 x^2-45 x^3+6 x^4\right )+e^x \left (-6-3 x^2-6 x^3\right ) \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx=3 e^x \left (-9+x-\log \left (2+x^2+2 x^3\right )\right ) \]

[In]

Integrate[(E^x*(-48 - 42*x^2 - 45*x^3 + 6*x^4) + E^x*(-6 - 3*x^2 - 6*x^3)*Log[2 + x^2 + 2*x^3])/(2 + x^2 + 2*x
^3),x]

[Out]

3*E^x*(-9 + x - Log[2 + x^2 + 2*x^3])

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14

method	result	size
risch	\(-3 \,{\mathrm e}^{x} \ln \left (2 x^{3}+x^{2}+2\right )+3 \left (x -9\right ) {\mathrm e}^{x}\)	\(24\)
norman	\(3 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{x} \ln \left (2 x^{3}+x^{2}+2\right )-27 \,{\mathrm e}^{x}\)	\(26\)
parallelrisch	\(3 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{x} \ln \left (2 x^{3}+x^{2}+2\right )-27 \,{\mathrm e}^{x}\)	\(26\)

[In]

int(((-6*x^3-3*x^2-6)*exp(x)*ln(2*x^3+x^2+2)+(6*x^4-45*x^3-42*x^2-48)*exp(x))/(2*x^3+x^2+2),x,method=_RETURNVE
RBOSE)

[Out]

-3*exp(x)*ln(2*x^3+x^2+2)+3*(x-9)*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {e^x \left (-48-42 x^2-45 x^3+6 x^4\right )+e^x \left (-6-3 x^2-6 x^3\right ) \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx=3 \, {\left (x - 9\right )} e^{x} - 3 \, e^{x} \log \left (2 \, x^{3} + x^{2} + 2\right ) \]

[In]

integrate(((-6*x^3-3*x^2-6)*exp(x)*log(2*x^3+x^2+2)+(6*x^4-45*x^3-42*x^2-48)*exp(x))/(2*x^3+x^2+2),x, algorith
m="fricas")

[Out]

3*(x - 9)*e^x - 3*e^x*log(2*x^3 + x^2 + 2)

Sympy [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^x \left (-48-42 x^2-45 x^3+6 x^4\right )+e^x \left (-6-3 x^2-6 x^3\right ) \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx=\left (3 x - 3 \log {\left (2 x^{3} + x^{2} + 2 \right )} - 27\right ) e^{x} \]

[In]

integrate(((-6*x**3-3*x**2-6)*exp(x)*ln(2*x**3+x**2+2)+(6*x**4-45*x**3-42*x**2-48)*exp(x))/(2*x**3+x**2+2),x)

[Out]

(3*x - 3*log(2*x**3 + x**2 + 2) - 27)*exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {e^x \left (-48-42 x^2-45 x^3+6 x^4\right )+e^x \left (-6-3 x^2-6 x^3\right ) \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx=3 \, {\left (x - 9\right )} e^{x} - 3 \, e^{x} \log \left (2 \, x^{3} + x^{2} + 2\right ) \]

[In]

integrate(((-6*x^3-3*x^2-6)*exp(x)*log(2*x^3+x^2+2)+(6*x^4-45*x^3-42*x^2-48)*exp(x))/(2*x^3+x^2+2),x, algorith
m="maxima")

[Out]

3*(x - 9)*e^x - 3*e^x*log(2*x^3 + x^2 + 2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^x \left (-48-42 x^2-45 x^3+6 x^4\right )+e^x \left (-6-3 x^2-6 x^3\right ) \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx=3 \, x e^{x} - 3 \, e^{x} \log \left (2 \, x^{3} + x^{2} + 2\right ) - 27 \, e^{x} \]

[In]

integrate(((-6*x^3-3*x^2-6)*exp(x)*log(2*x^3+x^2+2)+(6*x^4-45*x^3-42*x^2-48)*exp(x))/(2*x^3+x^2+2),x, algorith
m="giac")

[Out]

3*x*e^x - 3*e^x*log(2*x^3 + x^2 + 2) - 27*e^x

Mupad [B] (verification not implemented)

Time = 14.91 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^x \left (-48-42 x^2-45 x^3+6 x^4\right )+e^x \left (-6-3 x^2-6 x^3\right ) \log \left (2+x^2+2 x^3\right )}{2+x^2+2 x^3} \, dx=-3\,{\mathrm {e}}^x\,\left (\ln \left (2\,x^3+x^2+2\right )-x+9\right ) \]

[In]

int(-(exp(x)*(42*x^2 + 45*x^3 - 6*x^4 + 48) + exp(x)*log(x^2 + 2*x^3 + 2)*(3*x^2 + 6*x^3 + 6))/(x^2 + 2*x^3 +
2),x)

[Out]

-3*exp(x)*(log(x^2 + 2*x^3 + 2) - x + 9)