3.11.23 \(\int \frac {e^2 (31250 x-25000 x^2+7500 x^3-950 x^4+40 x^5)}{31250-25000 x-2375 x^2+6700 x^3-1518 x^4-230 x^5+138 x^6-20 x^7+x^8} \, dx\) [1023]

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

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 53, normalized size of antiderivative = 1.71, number of steps used = 5, number of rules used = 3, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {12, 2099, 1601} \begin {gather*} 5 e^2 \log \left (-x^4+10 x^3-17 x^2-100 x+250\right )-5 e^2 \log \left (-x^4+10 x^3-21 x^2-50 x+125\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^2*(31250*x - 25000*x^2 + 7500*x^3 - 950*x^4 + 40*x^5))/(31250 - 25000*x - 2375*x^2 + 6700*x^3 - 1518*x^
4 - 230*x^5 + 138*x^6 - 20*x^7 + x^8),x]

[Out]

-5*E^2*Log[125 - 50*x - 21*x^2 + 10*x^3 - x^4] + 5*E^2*Log[250 - 100*x - 17*x^2 + 10*x^3 - x^4]

Rule 12

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

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2099

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} &=e^2 \int \frac {31250 x-25000 x^2+7500 x^3-950 x^4+40 x^5}{31250-25000 x-2375 x^2+6700 x^3-1518 x^4-230 x^5+138 x^6-20 x^7+x^8} \, dx\\ &=e^2 \int \left (\frac {10 \left (50+17 x-15 x^2+2 x^3\right )}{-250+100 x+17 x^2-10 x^3+x^4}-\frac {10 \left (25+21 x-15 x^2+2 x^3\right )}{-125+50 x+21 x^2-10 x^3+x^4}\right ) \, dx\\ &=\left (10 e^2\right ) \int \frac {50+17 x-15 x^2+2 x^3}{-250+100 x+17 x^2-10 x^3+x^4} \, dx-\left (10 e^2\right ) \int \frac {25+21 x-15 x^2+2 x^3}{-125+50 x+21 x^2-10 x^3+x^4} \, dx\\ &=-5 e^2 \log \left (125-50 x-21 x^2+10 x^3-x^4\right )+5 e^2 \log \left (250-100 x-17 x^2+10 x^3-x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 56, normalized size = 1.81 \begin {gather*} 10 e^2 \left (-\frac {1}{2} \log \left (125-50 x-21 x^2+10 x^3-x^4\right )+\frac {1}{2} \log \left (250-100 x-17 x^2+10 x^3-x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^2*(31250*x - 25000*x^2 + 7500*x^3 - 950*x^4 + 40*x^5))/(31250 - 25000*x - 2375*x^2 + 6700*x^3 - 1
518*x^4 - 230*x^5 + 138*x^6 - 20*x^7 + x^8),x]

[Out]

10*E^2*(-1/2*Log[125 - 50*x - 21*x^2 + 10*x^3 - x^4] + Log[250 - 100*x - 17*x^2 + 10*x^3 - x^4]/2)

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Maple [A]
time = 0.05, size = 48, normalized size = 1.55

method	result	size
default	\(10 \,{\mathrm e}^{2} \left (\frac {\ln \left (x^{4}-10 x^{3}+17 x^{2}+100 x -250\right )}{2}-\frac {\ln \left (x^{4}-10 x^{3}+21 x^{2}+50 x -125\right )}{2}\right )\)	\(48\)
norman	\(5 \,{\mathrm e}^{2} \ln \left (x^{4}-10 x^{3}+17 x^{2}+100 x -250\right )-5 \,{\mathrm e}^{2} \ln \left (x^{4}-10 x^{3}+21 x^{2}+50 x -125\right )\)	\(48\)
risch	\(5 \,{\mathrm e}^{2} \ln \left (x^{4}-10 x^{3}+17 x^{2}+100 x -250\right )-5 \,{\mathrm e}^{2} \ln \left (x^{4}-10 x^{3}+21 x^{2}+50 x -125\right )\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x^5-950*x^4+7500*x^3-25000*x^2+31250*x)*exp(2)/(x^8-20*x^7+138*x^6-230*x^5-1518*x^4+6700*x^3-2375*x^2-
25000*x+31250),x,method=_RETURNVERBOSE)

[Out]

10*exp(2)*(1/2*ln(x^4-10*x^3+17*x^2+100*x-250)-1/2*ln(x^4-10*x^3+21*x^2+50*x-125))

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Maxima [A]
time = 0.26, size = 45, normalized size = 1.45 \begin {gather*} -5 \, {\left (\log \left (x^{4} - 10 \, x^{3} + 21 \, x^{2} + 50 \, x - 125\right ) - \log \left (x^{4} - 10 \, x^{3} + 17 \, x^{2} + 100 \, x - 250\right )\right )} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^5-950*x^4+7500*x^3-25000*x^2+31250*x)*exp(2)/(x^8-20*x^7+138*x^6-230*x^5-1518*x^4+6700*x^3-237
5*x^2-25000*x+31250),x, algorithm="maxima")

[Out]

-5*(log(x^4 - 10*x^3 + 21*x^2 + 50*x - 125) - log(x^4 - 10*x^3 + 17*x^2 + 100*x - 250))*e^2

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Fricas [A]
time = 0.37, size = 47, normalized size = 1.52 \begin {gather*} -5 \, e^{2} \log \left (x^{4} - 10 \, x^{3} + 21 \, x^{2} + 50 \, x - 125\right ) + 5 \, e^{2} \log \left (x^{4} - 10 \, x^{3} + 17 \, x^{2} + 100 \, x - 250\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^5-950*x^4+7500*x^3-25000*x^2+31250*x)*exp(2)/(x^8-20*x^7+138*x^6-230*x^5-1518*x^4+6700*x^3-237
5*x^2-25000*x+31250),x, algorithm="fricas")

[Out]

-5*e^2*log(x^4 - 10*x^3 + 21*x^2 + 50*x - 125) + 5*e^2*log(x^4 - 10*x^3 + 17*x^2 + 100*x - 250)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
time = 0.32, size = 49, normalized size = 1.58 \begin {gather*} 5 e^{2} \log {\left (x^{4} - 10 x^{3} + 17 x^{2} + 100 x - 250 \right )} - 5 e^{2} \log {\left (x^{4} - 10 x^{3} + 21 x^{2} + 50 x - 125 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x**5-950*x**4+7500*x**3-25000*x**2+31250*x)*exp(2)/(x**8-20*x**7+138*x**6-230*x**5-1518*x**4+670
0*x**3-2375*x**2-25000*x+31250),x)

[Out]

5*exp(2)*log(x**4 - 10*x**3 + 17*x**2 + 100*x - 250) - 5*exp(2)*log(x**4 - 10*x**3 + 21*x**2 + 50*x - 125)

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Giac [A]
time = 0.39, size = 47, normalized size = 1.52 \begin {gather*} -5 \, {\left (\log \left ({\left | x^{4} - 10 \, x^{3} + 21 \, x^{2} + 50 \, x - 125 \right |}\right ) - \log \left ({\left | x^{4} - 10 \, x^{3} + 17 \, x^{2} + 100 \, x - 250 \right |}\right )\right )} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^5-950*x^4+7500*x^3-25000*x^2+31250*x)*exp(2)/(x^8-20*x^7+138*x^6-230*x^5-1518*x^4+6700*x^3-237
5*x^2-25000*x+31250),x, algorithm="giac")

[Out]

-5*(log(abs(x^4 - 10*x^3 + 21*x^2 + 50*x - 125)) - log(abs(x^4 - 10*x^3 + 17*x^2 + 100*x - 250)))*e^2

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Mupad [B]
time = 1.46, size = 48, normalized size = 1.55 \begin {gather*} -10\,{\mathrm {e}}^2\,\mathrm {atanh}\left (\frac {15254\,x^4-152540\,x^3+332970\,x^2+604750\,x-1511875}{21572\,x^4-215720\,x^3+440376\,x^2+1236550\,x-3091375}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2)*(31250*x - 25000*x^2 + 7500*x^3 - 950*x^4 + 40*x^5))/(25000*x + 2375*x^2 - 6700*x^3 + 1518*x^4 +
230*x^5 - 138*x^6 + 20*x^7 - x^8 - 31250),x)

[Out]

-10*exp(2)*atanh((604750*x + 332970*x^2 - 152540*x^3 + 15254*x^4 - 1511875)/(1236550*x + 440376*x^2 - 215720*x
^3 + 21572*x^4 - 3091375))

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