3.42.41 \(\int \frac {-2 x+e^{3 x+6 x^2+3 x^3} (1+3 x+12 x^2+9 x^3)}{e^{6 x+12 x^2+6 x^3} x^2-2 e^{3 x+6 x^2+3 x^3} x^3+x^4} \, dx\)

Optimal. Leaf size=24 \[ -3+\frac {1}{x \left (-e^{x (1+x) (3+3 x)}+x\right )} \]

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Rubi [A]  time = 0.34, antiderivative size = 21, normalized size of antiderivative = 0.88, number of steps used = 2, number of rules used = 2, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6688, 6687} \begin {gather*} -\frac {1}{\left (e^{3 x (x+1)^2}-x\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x + E^(3*x + 6*x^2 + 3*x^3)*(1 + 3*x + 12*x^2 + 9*x^3))/(E^(6*x + 12*x^2 + 6*x^3)*x^2 - 2*E^(3*x + 6*x
^2 + 3*x^3)*x^3 + x^4),x]

[Out]

-(1/((E^(3*x*(1 + x)^2) - x)*x))

Rule 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

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 {-2 x+e^{3 x (1+x)^2} \left (1+3 x+12 x^2+9 x^3\right )}{\left (e^{3 x (1+x)^2}-x\right )^2 x^2} \, dx\\ &=-\frac {1}{\left (e^{3 x (1+x)^2}-x\right ) x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 21, normalized size = 0.88 \begin {gather*} -\frac {1}{\left (e^{3 x (1+x)^2}-x\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + E^(3*x + 6*x^2 + 3*x^3)*(1 + 3*x + 12*x^2 + 9*x^3))/(E^(6*x + 12*x^2 + 6*x^3)*x^2 - 2*E^(3*x
+ 6*x^2 + 3*x^3)*x^3 + x^4),x]

[Out]

-(1/((E^(3*x*(1 + x)^2) - x)*x))

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fricas [A]  time = 0.79, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{x^{2} - x e^{\left (3 \, x^{3} + 6 \, x^{2} + 3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3+12*x^2+3*x+1)*exp(3*x^3+6*x^2+3*x)-2*x)/(x^2*exp(3*x^3+6*x^2+3*x)^2-2*x^3*exp(3*x^3+6*x^2+3*
x)+x^4),x, algorithm="fricas")

[Out]

1/(x^2 - x*e^(3*x^3 + 6*x^2 + 3*x))

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giac [A]  time = 0.21, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{x^{2} - x e^{\left (3 \, x^{3} + 6 \, x^{2} + 3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3+12*x^2+3*x+1)*exp(3*x^3+6*x^2+3*x)-2*x)/(x^2*exp(3*x^3+6*x^2+3*x)^2-2*x^3*exp(3*x^3+6*x^2+3*
x)+x^4),x, algorithm="giac")

[Out]

1/(x^2 - x*e^(3*x^3 + 6*x^2 + 3*x))

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maple [A]  time = 0.04, size = 20, normalized size = 0.83

method	result	size
risch	\(\frac {1}{x \left (x -{\mathrm e}^{3 x \left (x +1\right )^{2}}\right )}\)	\(20\)
norman	\(\frac {1}{x \left (x -{\mathrm e}^{3 x^{3}+6 x^{2}+3 x}\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*x^3+12*x^2+3*x+1)*exp(3*x^3+6*x^2+3*x)-2*x)/(x^2*exp(3*x^3+6*x^2+3*x)^2-2*x^3*exp(3*x^3+6*x^2+3*x)+x^4
),x,method=_RETURNVERBOSE)

[Out]

1/x/(x-exp(3*x*(x+1)^2))

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maxima [A]  time = 0.42, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{x^{2} - x e^{\left (3 \, x^{3} + 6 \, x^{2} + 3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3+12*x^2+3*x+1)*exp(3*x^3+6*x^2+3*x)-2*x)/(x^2*exp(3*x^3+6*x^2+3*x)^2-2*x^3*exp(3*x^3+6*x^2+3*
x)+x^4),x, algorithm="maxima")

[Out]

1/(x^2 - x*e^(3*x^3 + 6*x^2 + 3*x))

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mupad [B]  time = 3.08, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{x^2-x\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{3\,x^3}\,{\mathrm {e}}^{6\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - exp(3*x + 6*x^2 + 3*x^3)*(3*x + 12*x^2 + 9*x^3 + 1))/(x^2*exp(6*x + 12*x^2 + 6*x^3) - 2*x^3*exp(3*
x + 6*x^2 + 3*x^3) + x^4),x)

[Out]

1/(x^2 - x*exp(3*x)*exp(3*x^3)*exp(6*x^2))

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sympy [A]  time = 0.14, size = 22, normalized size = 0.92 \begin {gather*} - \frac {1}{- x^{2} + x e^{3 x^{3} + 6 x^{2} + 3 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x**3+12*x**2+3*x+1)*exp(3*x**3+6*x**2+3*x)-2*x)/(x**2*exp(3*x**3+6*x**2+3*x)**2-2*x**3*exp(3*x**
3+6*x**2+3*x)+x**4),x)

[Out]

-1/(-x**2 + x*exp(3*x**3 + 6*x**2 + 3*x))

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