3.68.17 \(\int \frac {2375+4 e^4+e^2 (-195-8 x)+200 x+4 x^2}{2500+2575 x+199 x^2+4 x^3+e^4 (4+4 x)+e^2 (-200-203 x-8 x^2)} \, dx\)

Optimal. Leaf size=22 \[ 5+\log \left (-4+x-5 \left (x+\frac {x}{-25+e^2-x}\right )\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 37, normalized size of antiderivative = 1.68, number of steps used = 3, number of rules used = 2, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2074, 628} \begin {gather*} \log \left (4 x^2+\left (99-4 e^2\right ) x+4 \left (25-e^2\right )\right )-\log \left (x-e^2+25\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2375 + 4*E^4 + E^2*(-195 - 8*x) + 200*x + 4*x^2)/(2500 + 2575*x + 199*x^2 + 4*x^3 + E^4*(4 + 4*x) + E^2*(
-200 - 203*x - 8*x^2)),x]

[Out]

-Log[25 - E^2 + x] + Log[4*(25 - E^2) + (99 - 4*E^2)*x + 4*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 {1}{-25+e^2-x}+\frac {99-4 e^2+8 x}{4 \left (25-e^2\right )+\left (99-4 e^2\right ) x+4 x^2}\right ) \, dx\\ &=-\log \left (25-e^2+x\right )+\int \frac {99-4 e^2+8 x}{4 \left (25-e^2\right )+\left (99-4 e^2\right ) x+4 x^2} \, dx\\ &=-\log \left (25-e^2+x\right )+\log \left (4 \left (25-e^2\right )+\left (99-4 e^2\right ) x+4 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 34, normalized size = 1.55 \begin {gather*} -\log \left (25-e^2+x\right )+\log \left (100-4 e^2+99 x-4 e^2 x+4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2375 + 4*E^4 + E^2*(-195 - 8*x) + 200*x + 4*x^2)/(2500 + 2575*x + 199*x^2 + 4*x^3 + E^4*(4 + 4*x) +
E^2*(-200 - 203*x - 8*x^2)),x]

[Out]

-Log[25 - E^2 + x] + Log[100 - 4*E^2 + 99*x - 4*E^2*x + 4*x^2]

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fricas [A]  time = 0.60, size = 29, normalized size = 1.32 \begin {gather*} \log \left (4 \, x^{2} - 4 \, {\left (x + 1\right )} e^{2} + 99 \, x + 100\right ) - \log \left (x - e^{2} + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4*x+4)*exp(2)^2+(-8*x^2-203*x-200)*exp(2)+4*x^3+19
9*x^2+2575*x+2500),x, algorithm="fricas")

[Out]

log(4*x^2 - 4*(x + 1)*e^2 + 99*x + 100) - log(x - e^2 + 25)

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giac [B]  time = 0.22, size = 63, normalized size = 2.86 \begin {gather*} -1.33333333334167 \, \log \left (x + 17.6109439011000\right ) + 0.666666666673333 \, \log \left (x + 16.2791329762000\right ) + 0.666666666666667 \, \log \left (x + 1.08181092487000\right ) + \frac {1}{3} \, \log \left ({\left | 4 \, x^{3} - 8 \, x^{2} e^{2} + 199 \, x^{2} + 4 \, x e^{4} - 203 \, x e^{2} + 2575 \, x + 4 \, e^{4} - 200 \, e^{2} + 2500 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4*x+4)*exp(2)^2+(-8*x^2-203*x-200)*exp(2)+4*x^3+19
9*x^2+2575*x+2500),x, algorithm="giac")

[Out]

-1.33333333334167*log(x + 17.6109439011000) + 0.666666666673333*log(x + 16.2791329762000) + 0.666666666666667*
log(x + 1.08181092487000) + 1/3*log(abs(4*x^3 - 8*x^2*e^2 + 199*x^2 + 4*x*e^4 - 203*x*e^2 + 2575*x + 4*e^4 - 2
00*e^2 + 2500))

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maple [A]  time = 0.08, size = 32, normalized size = 1.45

method	result	size
norman	\(-\ln \left ({\mathrm e}^{2}-x -25\right )+\ln \left (4 \,{\mathrm e}^{2} x -4 x^{2}+4 \,{\mathrm e}^{2}-99 x -100\right )\)	\(32\)
risch	\(-\ln \left (-{\mathrm e}^{2}+x +25\right )+\ln \left (4 x^{2}+\left (-4 \,{\mathrm e}^{2}+99\right ) x -4 \,{\mathrm e}^{2}+100\right )\)	\(32\)
default	\(-\left (\munderset {\textit {\_R} =\RootOf \left (4 \textit {\_Z}^{3}+\left (-8 \,{\mathrm e}^{2}+199\right ) \textit {\_Z}^{2}+\left (-203 \,{\mathrm e}^{2}+4 \,{\mathrm e}^{4}+2575\right ) \textit {\_Z} -200 \,{\mathrm e}^{2}+4 \,{\mathrm e}^{4}+2500\right )}{\sum }\frac {\left (8 \,{\mathrm e}^{2} \textit {\_R} -4 \textit {\_R}^{2}+195 \,{\mathrm e}^{2}-4 \,{\mathrm e}^{4}-200 \textit {\_R} -2375\right ) \ln \left (x -\textit {\_R} \right )}{4 \,{\mathrm e}^{4}-16 \,{\mathrm e}^{2} \textit {\_R} +12 \textit {\_R}^{2}-203 \,{\mathrm e}^{2}+398 \textit {\_R} +2575}\right )\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4*x+4)*exp(2)^2+(-8*x^2-203*x-200)*exp(2)+4*x^3+199*x^2+
2575*x+2500),x,method=_RETURNVERBOSE)

[Out]

-ln(exp(2)-x-25)+ln(4*exp(2)*x-4*x^2+4*exp(2)-99*x-100)

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maxima [A]  time = 0.35, size = 32, normalized size = 1.45 \begin {gather*} \log \left (4 \, x^{2} - x {\left (4 \, e^{2} - 99\right )} - 4 \, e^{2} + 100\right ) - \log \left (x - e^{2} + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(2)^2+(-8*x-195)*exp(2)+4*x^2+200*x+2375)/((4*x+4)*exp(2)^2+(-8*x^2-203*x-200)*exp(2)+4*x^3+19
9*x^2+2575*x+2500),x, algorithm="maxima")

[Out]

log(4*x^2 - x*(4*e^2 - 99) - 4*e^2 + 100) - log(x - e^2 + 25)

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mupad [B]  time = 4.41, size = 29, normalized size = 1.32 \begin {gather*} \ln \left (\frac {99\,x}{4}-{\mathrm {e}}^2-x\,{\mathrm {e}}^2+x^2+25\right )-\ln \left (x-{\mathrm {e}}^2+25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((200*x + 4*exp(4) + 4*x^2 - exp(2)*(8*x + 195) + 2375)/(2575*x - exp(2)*(203*x + 8*x^2 + 200) + 199*x^2 +
4*x^3 + exp(4)*(4*x + 4) + 2500),x)

[Out]

log((99*x)/4 - exp(2) - x*exp(2) + x^2 + 25) - log(x - exp(2) + 25)

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sympy [A]  time = 0.68, size = 26, normalized size = 1.18 \begin {gather*} - \log {\left (x - e^{2} + 25 \right )} + \log {\left (x^{2} + x \left (\frac {99}{4} - e^{2}\right ) - e^{2} + 25 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(2)**2+(-8*x-195)*exp(2)+4*x**2+200*x+2375)/((4*x+4)*exp(2)**2+(-8*x**2-203*x-200)*exp(2)+4*x*
*3+199*x**2+2575*x+2500),x)

[Out]

-log(x - exp(2) + 25) + log(x**2 + x*(99/4 - exp(2)) - exp(2) + 25)

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