3.9.84 \(\int \frac {684-102 x+45 x^2+6 x^3+e^2 (120-24 x+12 x^2)}{-114 x^2+17 x^3+x^4+e^2 (-20 x^2+4 x^3)} \, dx\)

Optimal. Leaf size=31 \[ 3 \left (\frac {2}{x}+\log \left (2+\left (3+2 \left (4+e^2+\frac {x}{4}\right )\right ) (5-x)\right )\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 34, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 2, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2074, 628} \begin {gather*} 3 \log \left (-x^2-\left (17+4 e^2\right ) x+2 \left (57+10 e^2\right )\right )+\frac {6}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(684 - 102*x + 45*x^2 + 6*x^3 + E^2*(120 - 24*x + 12*x^2))/(-114*x^2 + 17*x^3 + x^4 + E^2*(-20*x^2 + 4*x^3
)),x]

[Out]

6/x + 3*Log[2*(57 + 10*E^2) - (17 + 4*E^2)*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 {6}{x^2}+\frac {3 \left (-17-4 e^2-2 x\right )}{2 \left (57+10 e^2\right )-\left (17+4 e^2\right ) x-x^2}\right ) \, dx\\ &=\frac {6}{x}+3 \int \frac {-17-4 e^2-2 x}{2 \left (57+10 e^2\right )+\left (-17-4 e^2\right ) x-x^2} \, dx\\ &=\frac {6}{x}+3 \log \left (2 \left (57+10 e^2\right )-\left (17+4 e^2\right ) x-x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 30, normalized size = 0.97 \begin {gather*} 3 \left (\frac {2}{x}+\log \left (114+20 e^2-17 x-4 e^2 x-x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(684 - 102*x + 45*x^2 + 6*x^3 + E^2*(120 - 24*x + 12*x^2))/(-114*x^2 + 17*x^3 + x^4 + E^2*(-20*x^2 +
4*x^3)),x]

[Out]

3*(2/x + Log[114 + 20*E^2 - 17*x - 4*E^2*x - x^2])

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fricas [A]  time = 1.08, size = 25, normalized size = 0.81 \begin {gather*} \frac {3 \, {\left (x \log \left (x^{2} + 4 \, {\left (x - 5\right )} e^{2} + 17 \, x - 114\right ) + 2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-24*x+120)*exp(2)+6*x^3+45*x^2-102*x+684)/((4*x^3-20*x^2)*exp(2)+x^4+17*x^3-114*x^2),x, algo
rithm="fricas")

[Out]

3*(x*log(x^2 + 4*(x - 5)*e^2 + 17*x - 114) + 2)/x

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giac [A]  time = 0.49, size = 27, normalized size = 0.87 \begin {gather*} \frac {6}{x} + 3 \, \log \left ({\left | x^{2} + 4 \, x e^{2} + 17 \, x - 20 \, e^{2} - 114 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-24*x+120)*exp(2)+6*x^3+45*x^2-102*x+684)/((4*x^3-20*x^2)*exp(2)+x^4+17*x^3-114*x^2),x, algo
rithm="giac")

[Out]

6/x + 3*log(abs(x^2 + 4*x*e^2 + 17*x - 20*e^2 - 114))

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maple [A]  time = 0.06, size = 27, normalized size = 0.87

method	result	size
default	\(\frac {6}{x}+3 \ln \left (4 \,{\mathrm e}^{2} x +x^{2}-20 \,{\mathrm e}^{2}+17 x -114\right )\)	\(27\)
norman	\(\frac {6}{x}+3 \ln \left (4 \,{\mathrm e}^{2} x +x^{2}-20 \,{\mathrm e}^{2}+17 x -114\right )\)	\(27\)
risch	\(\frac {6}{x}+3 \ln \left (x^{2}+\left (4 \,{\mathrm e}^{2}+17\right ) x -20 \,{\mathrm e}^{2}-114\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^2-24*x+120)*exp(2)+6*x^3+45*x^2-102*x+684)/((4*x^3-20*x^2)*exp(2)+x^4+17*x^3-114*x^2),x,method=_RET
URNVERBOSE)

[Out]

6/x+3*ln(4*exp(2)*x+x^2-20*exp(2)+17*x-114)

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maxima [A]  time = 0.35, size = 26, normalized size = 0.84 \begin {gather*} \frac {6}{x} + 3 \, \log \left (x^{2} + x {\left (4 \, e^{2} + 17\right )} - 20 \, e^{2} - 114\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-24*x+120)*exp(2)+6*x^3+45*x^2-102*x+684)/((4*x^3-20*x^2)*exp(2)+x^4+17*x^3-114*x^2),x, algo
rithm="maxima")

[Out]

6/x + 3*log(x^2 + x*(4*e^2 + 17) - 20*e^2 - 114)

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mupad [B]  time = 0.17, size = 26, normalized size = 0.84 \begin {gather*} 3\,\ln \left (x^2+\left (4\,{\mathrm {e}}^2+17\right )\,x-20\,{\mathrm {e}}^2-114\right )+\frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2)*(12*x^2 - 24*x + 120) - 102*x + 45*x^2 + 6*x^3 + 684)/(exp(2)*(20*x^2 - 4*x^3) + 114*x^2 - 17*x^3
- x^4),x)

[Out]

3*log(x^2 - 20*exp(2) + x*(4*exp(2) + 17) - 114) + 6/x

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sympy [A]  time = 0.75, size = 24, normalized size = 0.77 \begin {gather*} 3 \log {\left (x^{2} + x \left (17 + 4 e^{2}\right ) - 20 e^{2} - 114 \right )} + \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**2-24*x+120)*exp(2)+6*x**3+45*x**2-102*x+684)/((4*x**3-20*x**2)*exp(2)+x**4+17*x**3-114*x**2)
,x)

[Out]

3*log(x**2 + x*(17 + 4*exp(2)) - 20*exp(2) - 114) + 6/x

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