3.44.47 \(\int \frac {e^x (-4+9 x+3 x^2)}{121+330 x+291 x^2+90 x^3+9 x^4} \, dx\) [4347]

Optimal. Leaf size=23 \[ \frac {e^x}{-1-\frac {3 (-4-x) \left (x+x^2\right )}{x}} \]

[Out]

exp(x)/(-1-(x^2+x)/x*(-3*x-12))

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Rubi [A]
time = 0.13, antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6820, 2327} \begin {gather*} \frac {e^x}{3 x^2+15 x+11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(-4 + 9*x + 3*x^2))/(121 + 330*x + 291*x^2 + 90*x^3 + 9*x^4),x]

[Out]

E^x/(11 + 15*x + 3*x^2)

Rule 2327

Int[(F_)^(u_)*(v_)^(n_.)*(w_), x_Symbol] :> With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Simp[(Coefficient[w
, x, Exponent[w, x]]/Coefficient[z, x, Exponent[z, x]])*F^u*v^(n + 1), x] /; EqQ[Exponent[w, x], Exponent[z, x
]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]] /; FreeQ[{F, n}, x] && Po
lynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]

Rule 6820

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 {e^x \left (-4+9 x+3 x^2\right )}{\left (11+15 x+3 x^2\right )^2} \, dx\\ &=\frac {e^x}{11+15 x+3 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.06, size = 16, normalized size = 0.70 \begin {gather*} \frac {e^x}{11+15 x+3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-4 + 9*x + 3*x^2))/(121 + 330*x + 291*x^2 + 90*x^3 + 9*x^4),x]

[Out]

E^x/(11 + 15*x + 3*x^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(22)=44\).
time = 0.19, size = 65, normalized size = 2.83

method	result	size
gosper	\(\frac {{\mathrm e}^{x}}{3 x^{2}+15 x +11}\)	\(16\)
norman	\(\frac {{\mathrm e}^{x}}{3 x^{2}+15 x +11}\)	\(16\)
risch	\(\frac {{\mathrm e}^{x}}{3 x^{2}+15 x +11}\)	\(16\)
default	\(\frac {4 \,{\mathrm e}^{x} \left (5+2 x \right )}{31 \left (3 x^{2}+15 x +11\right )}+\frac {3 \,{\mathrm e}^{x} \left (15 x +22\right )}{31 \left (3 x^{2}+15 x +11\right )}-\frac {{\mathrm e}^{x} \left (53 x +55\right )}{31 \left (3 x^{2}+15 x +11\right )}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+9*x-4)*exp(x)/(9*x^4+90*x^3+291*x^2+330*x+121),x,method=_RETURNVERBOSE)

[Out]

4/31*exp(x)*(5+2*x)/(3*x^2+15*x+11)+3/31*exp(x)*(15*x+22)/(3*x^2+15*x+11)-1/31*exp(x)*(53*x+55)/(3*x^2+15*x+11
)

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Maxima [A]
time = 0.33, size = 15, normalized size = 0.65 \begin {gather*} \frac {e^{x}}{3 \, x^{2} + 15 \, x + 11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+9*x-4)*exp(x)/(9*x^4+90*x^3+291*x^2+330*x+121),x, algorithm="maxima")

[Out]

e^x/(3*x^2 + 15*x + 11)

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Fricas [A]
time = 0.37, size = 15, normalized size = 0.65 \begin {gather*} \frac {e^{x}}{3 \, x^{2} + 15 \, x + 11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+9*x-4)*exp(x)/(9*x^4+90*x^3+291*x^2+330*x+121),x, algorithm="fricas")

[Out]

e^x/(3*x^2 + 15*x + 11)

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Sympy [A]
time = 0.04, size = 12, normalized size = 0.52 \begin {gather*} \frac {e^{x}}{3 x^{2} + 15 x + 11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+9*x-4)*exp(x)/(9*x**4+90*x**3+291*x**2+330*x+121),x)

[Out]

exp(x)/(3*x**2 + 15*x + 11)

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Giac [A]
time = 0.41, size = 15, normalized size = 0.65 \begin {gather*} \frac {e^{x}}{3 \, x^{2} + 15 \, x + 11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+9*x-4)*exp(x)/(9*x^4+90*x^3+291*x^2+330*x+121),x, algorithm="giac")

[Out]

e^x/(3*x^2 + 15*x + 11)

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Mupad [B]
time = 0.28, size = 15, normalized size = 0.65 \begin {gather*} \frac {{\mathrm {e}}^x}{3\,\left (x^2+5\,x+\frac {11}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(9*x + 3*x^2 - 4))/(330*x + 291*x^2 + 90*x^3 + 9*x^4 + 121),x)

[Out]

exp(x)/(3*(5*x + x^2 + 11/3))

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