3.92.10 \(\int \frac {4000-4000 x+e (960 x^2-320 x^3-400 x^4+160 x^5)+(-400 x+400 x^2+e (-16 x^4+16 x^5-4 x^6)) \log (4)+(10 x^2-10 x^3) \log ^2(4)}{1600 x^2-1600 x^3+400 x^4+(-160 x^3+160 x^4-40 x^5) \log (4)+(4 x^4-4 x^5+x^6) \log ^2(4)} \, dx\)

Optimal. Leaf size=31 \[ \frac {5}{(-2+x) x}-\frac {e (-3-x) x}{5-\frac {1}{4} x \log (4)} \]

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Rubi [A]  time = 0.16, antiderivative size = 47, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 1, integrand size = 131, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2074} \begin {gather*} -\frac {5}{2 (2-x)}-\frac {5}{2 x}+\frac {80 e (20+\log (64))}{\log ^2(4) (20-x \log (4))}-\frac {4 e x}{\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4000 - 4000*x + E*(960*x^2 - 320*x^3 - 400*x^4 + 160*x^5) + (-400*x + 400*x^2 + E*(-16*x^4 + 16*x^5 - 4*x
^6))*Log[4] + (10*x^2 - 10*x^3)*Log[4]^2)/(1600*x^2 - 1600*x^3 + 400*x^4 + (-160*x^3 + 160*x^4 - 40*x^5)*Log[4
] + (4*x^4 - 4*x^5 + x^6)*Log[4]^2),x]

[Out]

-5/(2*(2 - x)) - 5/(2*x) - (4*E*x)/Log[4] + (80*E*(20 + Log[64]))/(Log[4]^2*(20 - x*Log[4]))

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 {5}{2 (-2+x)^2}+\frac {5}{2 x^2}-\frac {4 e}{\log (4)}+\frac {80 e (20+\log (64))}{\log (4) (-20+x \log (4))^2}\right ) \, dx\\ &=-\frac {5}{2 (2-x)}-\frac {5}{2 x}-\frac {4 e x}{\log (4)}+\frac {80 e (20+\log (64))}{\log ^2(4) (20-x \log (4))}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.16, size = 79, normalized size = 2.55 \begin {gather*} -2 \left (\frac {5}{4 x}+\frac {e x \log (16)}{\log ^2(4)}+5 \left (\frac {1}{8-4 x}+\frac {8 e \left (-500 \log ^2(4)-40 \log ^3(4)+3 \log ^4(4)+1000 \log (16)+200 \log (4) \log (16)\right )}{(-10+\log (4))^2 \log ^3(4) (-20+x \log (4))}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4000 - 4000*x + E*(960*x^2 - 320*x^3 - 400*x^4 + 160*x^5) + (-400*x + 400*x^2 + E*(-16*x^4 + 16*x^5
- 4*x^6))*Log[4] + (10*x^2 - 10*x^3)*Log[4]^2)/(1600*x^2 - 1600*x^3 + 400*x^4 + (-160*x^3 + 160*x^4 - 40*x^5)
*Log[4] + (4*x^4 - 4*x^5 + x^6)*Log[4]^2),x]

[Out]

-2*(5/(4*x) + (E*x*Log[16])/Log[4]^2 + 5*((8 - 4*x)^(-1) + (8*E*(-500*Log[4]^2 - 40*Log[4]^3 + 3*Log[4]^4 + 10
00*Log[16] + 200*Log[4]*Log[16]))/((-10 + Log[4])^2*Log[4]^3*(-20 + x*Log[4]))))

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fricas [B]  time = 1.15, size = 88, normalized size = 2.84 \begin {gather*} \frac {5 \, x \log \relax (2)^{3} + 20 \, {\left (x^{3} - 5 \, x^{2} + 6 \, x\right )} e \log \relax (2) - 2 \, {\left ({\left (x^{4} - 2 \, x^{3}\right )} e + 25\right )} \log \relax (2)^{2} - 200 \, {\left (x^{2} - 2 \, x\right )} e}{{\left (x^{3} - 2 \, x^{2}\right )} \log \relax (2)^{3} - 10 \, {\left (x^{2} - 2 \, x\right )} \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-10*x^3+10*x^2)*log(2)^2+2*((-4*x^6+16*x^5-16*x^4)*exp(1)+400*x^2-400*x)*log(2)+(160*x^5-400*x^4
-320*x^3+960*x^2)*exp(1)-4000*x+4000)/(4*(x^6-4*x^5+4*x^4)*log(2)^2+2*(-40*x^5+160*x^4-160*x^3)*log(2)+400*x^4
-1600*x^3+1600*x^2),x, algorithm="fricas")

[Out]

(5*x*log(2)^3 + 20*(x^3 - 5*x^2 + 6*x)*e*log(2) - 2*((x^4 - 2*x^3)*e + 25)*log(2)^2 - 200*(x^2 - 2*x)*e)/((x^3
- 2*x^2)*log(2)^3 - 10*(x^2 - 2*x)*log(2)^2)

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giac [B]  time = 0.15, size = 82, normalized size = 2.65 \begin {gather*} -\frac {2 \, x e}{\log \relax (2)} - \frac {5 \, {\left (12 \, x^{2} e \log \relax (2) - x \log \relax (2)^{3} + 40 \, x^{2} e - 24 \, x e \log \relax (2) - 80 \, x e + 10 \, \log \relax (2)^{2}\right )}}{{\left (x^{3} \log \relax (2) - 2 \, x^{2} \log \relax (2) - 10 \, x^{2} + 20 \, x\right )} \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-10*x^3+10*x^2)*log(2)^2+2*((-4*x^6+16*x^5-16*x^4)*exp(1)+400*x^2-400*x)*log(2)+(160*x^5-400*x^4
-320*x^3+960*x^2)*exp(1)-4000*x+4000)/(4*(x^6-4*x^5+4*x^4)*log(2)^2+2*(-40*x^5+160*x^4-160*x^3)*log(2)+400*x^4
-1600*x^3+1600*x^2),x, algorithm="giac")

[Out]

-2*x*e/log(2) - 5*(12*x^2*e*log(2) - x*log(2)^3 + 40*x^2*e - 24*x*e*log(2) - 80*x*e + 10*log(2)^2)/((x^3*log(2
) - 2*x^2*log(2) - 10*x^2 + 20*x)*log(2)^2)

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maple [A]  time = 0.15, size = 45, normalized size = 1.45

method	result	size
default	\(-\frac {2 \,{\mathrm e} x}{\ln \relax (2)}+\frac {5}{2 \left (x -2\right )}-\frac {20 \,{\mathrm e} \left (3 \ln \relax (2)+10\right )}{\ln \relax (2)^{2} \left (x \ln \relax (2)-10\right )}-\frac {5}{2 x}\)	\(45\)
norman	\(\frac {-50+\left (-\frac {\ln \relax (2)^{2}}{4}-2 \,{\mathrm e}\right ) x^{3}+\left (\frac {\ln \relax (2)^{2}}{2}+12 \,{\mathrm e}+\frac {5 \ln \relax (2)}{2}\right ) x^{2}-2 x^{4} {\mathrm e}}{x \left (x -2\right ) \left (x \ln \relax (2)-10\right )}\)	\(61\)
gosper	\(-\frac {8 x^{4} {\mathrm e}+x^{3} \ln \relax (2)^{2}+8 x^{3} {\mathrm e}-2 x^{2} \ln \relax (2)^{2}-48 x^{2} {\mathrm e}-10 x^{2} \ln \relax (2)+200}{4 x \left (x^{2} \ln \relax (2)-2 x \ln \relax (2)-10 x +20\right )}\)	\(71\)
risch	\(-\frac {2 \,{\mathrm e} x}{\ln \relax (2)}+\frac {-\frac {20 \,{\mathrm e} \left (3 \ln \relax (2)+10\right ) x^{2}}{\ln \relax (2)}+\frac {5 \left (\ln \relax (2)^{3}+24 \,{\mathrm e} \ln \relax (2)+80 \,{\mathrm e}\right ) x}{\ln \relax (2)}-50 \ln \relax (2)}{\ln \relax (2) x \left (x^{2} \ln \relax (2)-2 x \ln \relax (2)-10 x +20\right )}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(-10*x^3+10*x^2)*ln(2)^2+2*((-4*x^6+16*x^5-16*x^4)*exp(1)+400*x^2-400*x)*ln(2)+(160*x^5-400*x^4-320*x^3
+960*x^2)*exp(1)-4000*x+4000)/(4*(x^6-4*x^5+4*x^4)*ln(2)^2+2*(-40*x^5+160*x^4-160*x^3)*ln(2)+400*x^4-1600*x^3+
1600*x^2),x,method=_RETURNVERBOSE)

[Out]

-2*exp(1)/ln(2)*x+5/2/(x-2)-20*exp(1)*(3*ln(2)+10)/ln(2)^2/(x*ln(2)-10)-5/2/x

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maxima [B]  time = 0.35, size = 87, normalized size = 2.81 \begin {gather*} -\frac {2 \, x e}{\log \relax (2)} - \frac {5 \, {\left (4 \, {\left (3 \, e \log \relax (2) + 10 \, e\right )} x^{2} - {\left (\log \relax (2)^{3} + 24 \, e \log \relax (2) + 80 \, e\right )} x + 10 \, \log \relax (2)^{2}\right )}}{x^{3} \log \relax (2)^{3} - 2 \, {\left (\log \relax (2)^{3} + 5 \, \log \relax (2)^{2}\right )} x^{2} + 20 \, x \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-10*x^3+10*x^2)*log(2)^2+2*((-4*x^6+16*x^5-16*x^4)*exp(1)+400*x^2-400*x)*log(2)+(160*x^5-400*x^4
-320*x^3+960*x^2)*exp(1)-4000*x+4000)/(4*(x^6-4*x^5+4*x^4)*log(2)^2+2*(-40*x^5+160*x^4-160*x^3)*log(2)+400*x^4
-1600*x^3+1600*x^2),x, algorithm="maxima")

[Out]

-2*x*e/log(2) - 5*(4*(3*e*log(2) + 10*e)*x^2 - (log(2)^3 + 24*e*log(2) + 80*e)*x + 10*log(2)^2)/(x^3*log(2)^3
- 2*(log(2)^3 + 5*log(2)^2)*x^2 + 20*x*log(2)^2)

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mupad [B]  time = 8.09, size = 91, normalized size = 2.94 \begin {gather*} -\frac {\frac {20\,\left (10\,\mathrm {e}+3\,\mathrm {e}\,\ln \relax (2)\right )\,x^2}{\ln \relax (2)}-\frac {5\,\left (80\,\mathrm {e}+24\,\mathrm {e}\,\ln \relax (2)+{\ln \relax (2)}^3\right )\,x}{\ln \relax (2)}+50\,\ln \relax (2)}{{\ln \relax (2)}^2\,x^3+\left (-10\,\ln \relax (2)-2\,{\ln \relax (2)}^2\right )\,x^2+20\,\ln \relax (2)\,x}-\frac {2\,x\,\mathrm {e}}{\ln \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(2)^2*(10*x^2 - 10*x^3) - 2*log(2)*(400*x + exp(1)*(16*x^4 - 16*x^5 + 4*x^6) - 400*x^2) - 4000*x + e
xp(1)*(960*x^2 - 320*x^3 - 400*x^4 + 160*x^5) + 4000)/(4*log(2)^2*(4*x^4 - 4*x^5 + x^6) - 2*log(2)*(160*x^3 -
160*x^4 + 40*x^5) + 1600*x^2 - 1600*x^3 + 400*x^4),x)

[Out]

- (50*log(2) + (20*x^2*(10*exp(1) + 3*exp(1)*log(2)))/log(2) - (5*x*(80*exp(1) + 24*exp(1)*log(2) + log(2)^3))
/log(2))/(x^3*log(2)^2 + 20*x*log(2) - x^2*(10*log(2) + 2*log(2)^2)) - (2*x*exp(1))/log(2)

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sympy [B]  time = 6.63, size = 94, normalized size = 3.03 \begin {gather*} - \frac {2 e x}{\log {\relax (2 )}} - \frac {x^{2} \left (60 e \log {\relax (2 )} + 200 e\right ) + x \left (- 400 e - 120 e \log {\relax (2 )} - 5 \log {\relax (2 )}^{3}\right ) + 50 \log {\relax (2 )}^{2}}{x^{3} \log {\relax (2 )}^{3} + x^{2} \left (- 10 \log {\relax (2 )}^{2} - 2 \log {\relax (2 )}^{3}\right ) + 20 x \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-10*x**3+10*x**2)*ln(2)**2+2*((-4*x**6+16*x**5-16*x**4)*exp(1)+400*x**2-400*x)*ln(2)+(160*x**5-4
00*x**4-320*x**3+960*x**2)*exp(1)-4000*x+4000)/(4*(x**6-4*x**5+4*x**4)*ln(2)**2+2*(-40*x**5+160*x**4-160*x**3)
*ln(2)+400*x**4-1600*x**3+1600*x**2),x)

[Out]

-2*E*x/log(2) - (x**2*(60*E*log(2) + 200*E) + x*(-400*E - 120*E*log(2) - 5*log(2)**3) + 50*log(2)**2)/(x**3*lo
g(2)**3 + x**2*(-10*log(2)**2 - 2*log(2)**3) + 20*x*log(2)**2)

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