3.13.9 \(\int \frac {-3+770 e+5 e^4}{243-3 x+e^4 (-405+5 x)+e (3240-850 x+10 x^2)} \, dx\)

Optimal. Leaf size=28 \[ \log \left (\frac {-81+x}{4+\frac {\frac {3}{5}-e^4}{2 e}-x}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 24, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 1981, 616, 31} \begin {gather*} \log (81-x)-\log \left (-10 e x-5 e^4+40 e+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 770*E + 5*E^4)/(243 - 3*x + E^4*(-405 + 5*x) + E*(3240 - 850*x + 10*x^2)),x]

[Out]

Log[81 - x] - Log[3 + 40*E - 5*E^4 - 10*E*x]

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (-3+770 e+5 e^4\right ) \int \frac {1}{243-3 x+e^4 (-405+5 x)+e \left (3240-850 x+10 x^2\right )} \, dx\\ &=\left (-3+770 e+5 e^4\right ) \int \frac {1}{81 \left (3+40 e-5 e^4\right )-\left (3+850 e-5 e^4\right ) x+10 e x^2} \, dx\\ &=(10 e) \int \frac {1}{-810 e+10 e x} \, dx-(10 e) \int \frac {1}{-3-40 e+5 e^4+10 e x} \, dx\\ &=\log (81-x)-\log \left (3+40 e-5 e^4-10 e x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.82 \begin {gather*} -\log \left (3-5 e^4-10 e (-4+x)\right )+\log (81-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 770*E + 5*E^4)/(243 - 3*x + E^4*(-405 + 5*x) + E*(3240 - 850*x + 10*x^2)),x]

[Out]

-Log[3 - 5*E^4 - 10*E*(-4 + x)] + Log[81 - x]

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fricas [A]  time = 0.67, size = 21, normalized size = 0.75 \begin {gather*} -\log \left (10 \, {\left (x - 4\right )} e + 5 \, e^{4} - 3\right ) + \log \left (x - 81\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(4)+770*exp(1)-3)/((5*x-405)*exp(4)+(10*x^2-850*x+3240)*exp(1)-3*x+243),x, algorithm="fricas")

[Out]

-log(10*(x - 4)*e + 5*e^4 - 3) + log(x - 81)

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giac [B]  time = 0.34, size = 67, normalized size = 2.39 \begin {gather*} -{\left (\frac {e \log \left ({\left | 10 \, x e + 5 \, e^{4} - 40 \, e - 3 \right |}\right )}{5 \, e^{5} + 770 \, e^{2} - 3 \, e} - \frac {\log \left ({\left | x - 81 \right |}\right )}{5 \, e^{4} + 770 \, e - 3}\right )} {\left (5 \, e^{4} + 770 \, e - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(4)+770*exp(1)-3)/((5*x-405)*exp(4)+(10*x^2-850*x+3240)*exp(1)-3*x+243),x, algorithm="giac")

[Out]

-(e*log(abs(10*x*e + 5*e^4 - 40*e - 3))/(5*e^5 + 770*e^2 - 3*e) - log(abs(x - 81))/(5*e^4 + 770*e - 3))*(5*e^4
+ 770*e - 3)

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maple [A]  time = 0.28, size = 24, normalized size = 0.86

method	result	size
norman	\(-\ln \left (10 x \,{\mathrm e}-40 \,{\mathrm e}+5 \,{\mathrm e}^{4}-3\right )+\ln \left (x -81\right )\)	\(24\)
risch	\(-\ln \left (10 x \,{\mathrm e}-40 \,{\mathrm e}+5 \,{\mathrm e}^{4}-3\right )+\ln \left (81-x \right )\)	\(26\)
default	\(\left (5 \,{\mathrm e}^{4}+770 \,{\mathrm e}-3\right ) \left (\frac {\ln \left (x -81\right )}{5 \,{\mathrm e}^{4}+770 \,{\mathrm e}-3}-\frac {\ln \left (10 x \,{\mathrm e}-40 \,{\mathrm e}+5 \,{\mathrm e}^{4}-3\right )}{5 \,{\mathrm e}^{4}+770 \,{\mathrm e}-3}\right )\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*exp(4)+770*exp(1)-3)/((5*x-405)*exp(4)+(10*x^2-850*x+3240)*exp(1)-3*x+243),x,method=_RETURNVERBOSE)

[Out]

-ln(10*x*exp(1)-40*exp(1)+5*exp(4)-3)+ln(x-81)

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maxima [B]  time = 0.40, size = 60, normalized size = 2.14 \begin {gather*} -{\left (\frac {\log \left (10 \, x e + 5 \, e^{4} - 40 \, e - 3\right )}{5 \, e^{4} + 770 \, e - 3} - \frac {\log \left (x - 81\right )}{5 \, e^{4} + 770 \, e - 3}\right )} {\left (5 \, e^{4} + 770 \, e - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(4)+770*exp(1)-3)/((5*x-405)*exp(4)+(10*x^2-850*x+3240)*exp(1)-3*x+243),x, algorithm="maxima")

[Out]

-(log(10*x*e + 5*e^4 - 40*e - 3)/(5*e^4 + 770*e - 3) - log(x - 81)/(5*e^4 + 770*e - 3))*(5*e^4 + 770*e - 3)

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mupad [B]  time = 0.18, size = 29, normalized size = 1.04 \begin {gather*} \mathrm {atan}\left (\frac {\mathrm {e}\,\left (x\,40{}\mathrm {i}-3240{}\mathrm {i}\right )}{1540\,\mathrm {e}+10\,{\mathrm {e}}^4-6}+1{}\mathrm {i}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((770*exp(1) + 5*exp(4) - 3)/(exp(1)*(10*x^2 - 850*x + 3240) - 3*x + exp(4)*(5*x - 405) + 243),x)

[Out]

atan((exp(1)*(x*40i - 3240i))/(1540*exp(1) + 10*exp(4) - 6) + 1i)*2i

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sympy [B]  time = 0.73, size = 272, normalized size = 9.71 \begin {gather*} \left (- \frac {\log {\left (x + \frac {- 850 e - \frac {4620 e}{-3 + 5 e^{4} + 770 e} - 3 - \frac {30 e^{4}}{-3 + 5 e^{4} + 770 e} + \frac {9}{-3 + 5 e^{4} + 770 e} + \frac {25 e^{8}}{-3 + 5 e^{4} + 770 e} + 5 e^{4} + \frac {7700 e^{5}}{-3 + 5 e^{4} + 770 e} + \frac {592900 e^{2}}{-3 + 5 e^{4} + 770 e}}{20 e} \right )}}{-3 + 5 e^{4} + 770 e} + \frac {\log {\left (x + \frac {- 850 e - \frac {592900 e^{2}}{-3 + 5 e^{4} + 770 e} - \frac {7700 e^{5}}{-3 + 5 e^{4} + 770 e} - \frac {25 e^{8}}{-3 + 5 e^{4} + 770 e} - 3 - \frac {9}{-3 + 5 e^{4} + 770 e} + \frac {30 e^{4}}{-3 + 5 e^{4} + 770 e} + \frac {4620 e}{-3 + 5 e^{4} + 770 e} + 5 e^{4}}{20 e} \right )}}{-3 + 5 e^{4} + 770 e}\right ) \left (-3 + 5 e^{4} + 770 e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*exp(4)+770*exp(1)-3)/((5*x-405)*exp(4)+(10*x**2-850*x+3240)*exp(1)-3*x+243),x)

[Out]

(-log(x + (-850*E - 4620*E/(-3 + 5*exp(4) + 770*E) - 3 - 30*exp(4)/(-3 + 5*exp(4) + 770*E) + 9/(-3 + 5*exp(4)
+ 770*E) + 25*exp(8)/(-3 + 5*exp(4) + 770*E) + 5*exp(4) + 7700*exp(5)/(-3 + 5*exp(4) + 770*E) + 592900*exp(2)/
(-3 + 5*exp(4) + 770*E))*exp(-1)/20)/(-3 + 5*exp(4) + 770*E) + log(x + (-850*E - 592900*exp(2)/(-3 + 5*exp(4)
+ 770*E) - 7700*exp(5)/(-3 + 5*exp(4) + 770*E) - 25*exp(8)/(-3 + 5*exp(4) + 770*E) - 3 - 9/(-3 + 5*exp(4) + 77
0*E) + 30*exp(4)/(-3 + 5*exp(4) + 770*E) + 4620*E/(-3 + 5*exp(4) + 770*E) + 5*exp(4))*exp(-1)/20)/(-3 + 5*exp(
4) + 770*E))*(-3 + 5*exp(4) + 770*E)

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