3.32.38 \(\int \frac {-33-18 e^5-32 x-13 x^2-2 x^3}{9+24 x+22 x^2+8 x^3+x^4+e^5 (6+12 x+6 x^2)} \, dx\)

Optimal. Leaf size=25 \[ 2+\frac {3}{1+x}-\log \left (5 \left (6 e^5+(3+x)^2\right )\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2074, 628} \begin {gather*} \frac {3}{x+1}-\log \left (x^2+6 x+3 \left (3+2 e^5\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-33 - 18*E^5 - 32*x - 13*x^2 - 2*x^3)/(9 + 24*x + 22*x^2 + 8*x^3 + x^4 + E^5*(6 + 12*x + 6*x^2)),x]

[Out]

3/(1 + x) - Log[3*(3 + 2*E^5) + 6*x + x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3}{(1+x)^2}+\frac {2 (-3-x)}{3 \left (3+2 e^5\right )+6 x+x^2}\right ) \, dx\\ &=\frac {3}{1+x}+2 \int \frac {-3-x}{3 \left (3+2 e^5\right )+6 x+x^2} \, dx\\ &=\frac {3}{1+x}-\log \left (3 \left (3+2 e^5\right )+6 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 0.88 \begin {gather*} \frac {3}{1+x}-\log \left (6 e^5+(3+x)^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-33 - 18*E^5 - 32*x - 13*x^2 - 2*x^3)/(9 + 24*x + 22*x^2 + 8*x^3 + x^4 + E^5*(6 + 12*x + 6*x^2)),x]

[Out]

3/(1 + x) - Log[6*E^5 + (3 + x)^2]

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fricas [A]  time = 0.50, size = 26, normalized size = 1.04 \begin {gather*} -\frac {{\left (x + 1\right )} \log \left (x^{2} + 6 \, x + 6 \, e^{5} + 9\right ) - 3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(5)-2*x^3-13*x^2-32*x-33)/((6*x^2+12*x+6)*exp(5)+x^4+8*x^3+22*x^2+24*x+9),x, algorithm="fric
as")

[Out]

-((x + 1)*log(x^2 + 6*x + 6*e^5 + 9) - 3)/(x + 1)

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giac [A]  time = 0.30, size = 23, normalized size = 0.92 \begin {gather*} \frac {3}{x + 1} - \log \left (x^{2} + 6 \, x + 6 \, e^{5} + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(5)-2*x^3-13*x^2-32*x-33)/((6*x^2+12*x+6)*exp(5)+x^4+8*x^3+22*x^2+24*x+9),x, algorithm="giac
")

[Out]

3/(x + 1) - log(x^2 + 6*x + 6*e^5 + 9)

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maple [A]  time = 0.07, size = 24, normalized size = 0.96

method	result	size
default	\(-\ln \left (x^{2}+6 \,{\mathrm e}^{5}+6 x +9\right )+\frac {3}{x +1}\)	\(24\)
norman	\(-\ln \left (x^{2}+6 \,{\mathrm e}^{5}+6 x +9\right )+\frac {3}{x +1}\)	\(24\)
risch	\(-\ln \left (x^{2}+6 \,{\mathrm e}^{5}+6 x +9\right )+\frac {3}{x +1}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-18*exp(5)-2*x^3-13*x^2-32*x-33)/((6*x^2+12*x+6)*exp(5)+x^4+8*x^3+22*x^2+24*x+9),x,method=_RETURNVERBOSE)

[Out]

-ln(x^2+6*exp(5)+6*x+9)+3/(x+1)

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maxima [A]  time = 0.49, size = 23, normalized size = 0.92 \begin {gather*} \frac {3}{x + 1} - \log \left (x^{2} + 6 \, x + 6 \, e^{5} + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(5)-2*x^3-13*x^2-32*x-33)/((6*x^2+12*x+6)*exp(5)+x^4+8*x^3+22*x^2+24*x+9),x, algorithm="maxi
ma")

[Out]

3/(x + 1) - log(x^2 + 6*x + 6*e^5 + 9)

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mupad [B]  time = 0.11, size = 23, normalized size = 0.92 \begin {gather*} \frac {3}{x+1}-\ln \left (x^2+6\,x+6\,{\mathrm {e}}^5+9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x + 18*exp(5) + 13*x^2 + 2*x^3 + 33)/(24*x + exp(5)*(12*x + 6*x^2 + 6) + 22*x^2 + 8*x^3 + x^4 + 9),x)

[Out]

3/(x + 1) - log(6*x + 6*exp(5) + x^2 + 9)

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sympy [A]  time = 0.45, size = 19, normalized size = 0.76 \begin {gather*} - \log {\left (x^{2} + 6 x + 9 + 6 e^{5} \right )} + \frac {3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(5)-2*x**3-13*x**2-32*x-33)/((6*x**2+12*x+6)*exp(5)+x**4+8*x**3+22*x**2+24*x+9),x)

[Out]

-log(x**2 + 6*x + 9 + 6*exp(5)) + 3/(x + 1)

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