3.78.72 \(\int \frac {-800-168 x+e^x (400+164 x+16 x^2)+(-200-40 x+e^x (100+40 x+4 x^2)) \log (5+x)}{40000 x^6+8000 x^7+e^x (-40000 x^6-8000 x^7)+e^{2 x} (10000 x^6+2000 x^7)+(20000 x^6+4000 x^7+e^x (-20000 x^6-4000 x^7)+e^{2 x} (5000 x^6+1000 x^7)) \log (5+x)+(2500 x^6+500 x^7+e^x (-2500 x^6-500 x^7)+e^{2 x} (625 x^6+125 x^7)) \log ^2(5+x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {4}{125 \left (2-e^x\right ) x^5 (4+\log (5+x))} \]

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Rubi [F]  time = 19.50, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-800 - 168*x + E^x*(400 + 164*x + 16*x^2) + (-200 - 40*x + E^x*(100 + 40*x + 4*x^2))*Log[5 + x])/(40000*x
^6 + 8000*x^7 + E^x*(-40000*x^6 - 8000*x^7) + E^(2*x)*(10000*x^6 + 2000*x^7) + (20000*x^6 + 4000*x^7 + E^x*(-2
0000*x^6 - 4000*x^7) + E^(2*x)*(5000*x^6 + 1000*x^7))*Log[5 + x] + (2500*x^6 + 500*x^7 + E^x*(-2500*x^6 - 500*
x^7) + E^(2*x)*(625*x^6 + 125*x^7))*Log[5 + x]^2),x]

[Out]

(16*Defer[Int][1/((-2 + E^x)*x^6*(4 + Log[5 + x])^2), x])/25 + (84*Defer[Int][1/((-2 + E^x)*x^5*(4 + Log[5 + x
])^2), x])/625 - (4*Defer[Int][1/((-2 + E^x)*x^4*(4 + Log[5 + x])^2), x])/3125 + (4*Defer[Int][1/((-2 + E^x)*x
^3*(4 + Log[5 + x])^2), x])/15625 - (4*Defer[Int][1/((-2 + E^x)*x^2*(4 + Log[5 + x])^2), x])/78125 + (4*Defer[
Int][1/((-2 + E^x)*x*(4 + Log[5 + x])^2), x])/390625 - (4*Defer[Int][1/((-2 + E^x)*(5 + x)*(4 + Log[5 + x])^2)
, x])/390625 + (4*Defer[Int][Log[5 + x]/((-2 + E^x)*x^6*(4 + Log[5 + x])^2), x])/25 + (4*Defer[Int][Log[5 + x]
/((-2 + E^x)*x^5*(4 + Log[5 + x])^2), x])/125 + (8*Defer[Int][1/((-2 + E^x)^2*x^5*(4 + Log[5 + x])), x])/125

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-200-42 x+e^x \left (100+41 x+4 x^2\right )+(5+x) \left (-10+e^x (5+x)\right ) \log (5+x)\right )}{125 \left (2-e^x\right )^2 x^6 (5+x) (4+\log (5+x))^2} \, dx\\ &=\frac {4}{125} \int \frac {-200-42 x+e^x \left (100+41 x+4 x^2\right )+(5+x) \left (-10+e^x (5+x)\right ) \log (5+x)}{\left (2-e^x\right )^2 x^6 (5+x) (4+\log (5+x))^2} \, dx\\ &=\frac {4}{125} \int \left (\frac {2}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))}+\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^6 (5+x) (4+\log (5+x))^2}\right ) \, dx\\ &=\frac {4}{125} \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^6 (5+x) (4+\log (5+x))^2} \, dx+\frac {8}{125} \int \frac {1}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))} \, dx\\ &=\frac {4}{125} \int \left (\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{5 \left (-2+e^x\right ) x^6 (4+\log (5+x))^2}-\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{25 \left (-2+e^x\right ) x^5 (4+\log (5+x))^2}+\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{125 \left (-2+e^x\right ) x^4 (4+\log (5+x))^2}-\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{625 \left (-2+e^x\right ) x^3 (4+\log (5+x))^2}+\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{3125 \left (-2+e^x\right ) x^2 (4+\log (5+x))^2}-\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{15625 \left (-2+e^x\right ) x (4+\log (5+x))^2}+\frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{15625 \left (-2+e^x\right ) (5+x) (4+\log (5+x))^2}\right ) \, dx+\frac {8}{125} \int \frac {1}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))} \, dx\\ &=-\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x (4+\log (5+x))^2} \, dx}{1953125}+\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) (5+x) (4+\log (5+x))^2} \, dx}{1953125}+\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^2 (4+\log (5+x))^2} \, dx}{390625}-\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^3 (4+\log (5+x))^2} \, dx}{78125}+\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^4 (4+\log (5+x))^2} \, dx}{15625}-\frac {4 \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^5 (4+\log (5+x))^2} \, dx}{3125}+\frac {4}{625} \int \frac {100+41 x+4 x^2+25 \log (5+x)+10 x \log (5+x)+x^2 \log (5+x)}{\left (-2+e^x\right ) x^6 (4+\log (5+x))^2} \, dx+\frac {8}{125} \int \frac {1}{\left (-2+e^x\right )^2 x^5 (4+\log (5+x))} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.79, size = 22, normalized size = 0.92 \begin {gather*} -\frac {4}{125 \left (-2+e^x\right ) x^5 (4+\log (5+x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-800 - 168*x + E^x*(400 + 164*x + 16*x^2) + (-200 - 40*x + E^x*(100 + 40*x + 4*x^2))*Log[5 + x])/(4
0000*x^6 + 8000*x^7 + E^x*(-40000*x^6 - 8000*x^7) + E^(2*x)*(10000*x^6 + 2000*x^7) + (20000*x^6 + 4000*x^7 + E
^x*(-20000*x^6 - 4000*x^7) + E^(2*x)*(5000*x^6 + 1000*x^7))*Log[5 + x] + (2500*x^6 + 500*x^7 + E^x*(-2500*x^6
- 500*x^7) + E^(2*x)*(625*x^6 + 125*x^7))*Log[5 + x]^2),x]

[Out]

-4/(125*(-2 + E^x)*x^5*(4 + Log[5 + x]))

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fricas [A]  time = 1.10, size = 34, normalized size = 1.42 \begin {gather*} -\frac {4}{125 \, {\left (4 \, x^{5} e^{x} - 8 \, x^{5} + {\left (x^{5} e^{x} - 2 \, x^{5}\right )} \log \left (x + 5\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+40*x+100)*exp(x)-40*x-200)*log(5+x)+(16*x^2+164*x+400)*exp(x)-168*x-800)/(((125*x^7+625*x^6
)*exp(x)^2+(-500*x^7-2500*x^6)*exp(x)+500*x^7+2500*x^6)*log(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-20
000*x^6)*exp(x)+4000*x^7+20000*x^6)*log(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+(-8000*x^7-40000*x^6)*exp(x)+8000*x
^7+40000*x^6),x, algorithm="fricas")

[Out]

-4/125/(4*x^5*e^x - 8*x^5 + (x^5*e^x - 2*x^5)*log(x + 5))

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giac [A]  time = 0.23, size = 36, normalized size = 1.50 \begin {gather*} -\frac {4}{125 \, {\left (x^{5} e^{x} \log \left (x + 5\right ) + 4 \, x^{5} e^{x} - 2 \, x^{5} \log \left (x + 5\right ) - 8 \, x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+40*x+100)*exp(x)-40*x-200)*log(5+x)+(16*x^2+164*x+400)*exp(x)-168*x-800)/(((125*x^7+625*x^6
)*exp(x)^2+(-500*x^7-2500*x^6)*exp(x)+500*x^7+2500*x^6)*log(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-20
000*x^6)*exp(x)+4000*x^7+20000*x^6)*log(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+(-8000*x^7-40000*x^6)*exp(x)+8000*x
^7+40000*x^6),x, algorithm="giac")

[Out]

-4/125/(x^5*e^x*log(x + 5) + 4*x^5*e^x - 2*x^5*log(x + 5) - 8*x^5)

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maple [A]  time = 0.04, size = 20, normalized size = 0.83

method	result	size
risch	\(-\frac {4}{125 \left ({\mathrm e}^{x}-2\right ) x^{5} \left (\ln \left (5+x \right )+4\right )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^2+40*x+100)*exp(x)-40*x-200)*ln(5+x)+(16*x^2+164*x+400)*exp(x)-168*x-800)/(((125*x^7+625*x^6)*exp(x
)^2+(-500*x^7-2500*x^6)*exp(x)+500*x^7+2500*x^6)*ln(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-20000*x^6)
*exp(x)+4000*x^7+20000*x^6)*ln(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+(-8000*x^7-40000*x^6)*exp(x)+8000*x^7+40000*
x^6),x,method=_RETURNVERBOSE)

[Out]

-4/125/(exp(x)-2)/x^5/(ln(5+x)+4)

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maxima [A]  time = 0.48, size = 34, normalized size = 1.42 \begin {gather*} -\frac {4}{125 \, {\left (4 \, x^{5} e^{x} - 8 \, x^{5} + {\left (x^{5} e^{x} - 2 \, x^{5}\right )} \log \left (x + 5\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+40*x+100)*exp(x)-40*x-200)*log(5+x)+(16*x^2+164*x+400)*exp(x)-168*x-800)/(((125*x^7+625*x^6
)*exp(x)^2+(-500*x^7-2500*x^6)*exp(x)+500*x^7+2500*x^6)*log(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-20
000*x^6)*exp(x)+4000*x^7+20000*x^6)*log(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+(-8000*x^7-40000*x^6)*exp(x)+8000*x
^7+40000*x^6),x, algorithm="maxima")

[Out]

-4/125/(4*x^5*e^x - 8*x^5 + (x^5*e^x - 2*x^5)*log(x + 5))

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mupad [B]  time = 5.23, size = 19, normalized size = 0.79 \begin {gather*} -\frac {4}{125\,x^5\,\left ({\mathrm {e}}^x-2\right )\,\left (\ln \left (x+5\right )+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(168*x + log(x + 5)*(40*x - exp(x)*(40*x + 4*x^2 + 100) + 200) - exp(x)*(164*x + 16*x^2 + 400) + 800)/(lo
g(x + 5)^2*(exp(2*x)*(625*x^6 + 125*x^7) - exp(x)*(2500*x^6 + 500*x^7) + 2500*x^6 + 500*x^7) - exp(x)*(40000*x
^6 + 8000*x^7) + exp(2*x)*(10000*x^6 + 2000*x^7) + 40000*x^6 + 8000*x^7 + log(x + 5)*(exp(2*x)*(5000*x^6 + 100
0*x^7) - exp(x)*(20000*x^6 + 4000*x^7) + 20000*x^6 + 4000*x^7)),x)

[Out]

-4/(125*x^5*(exp(x) - 2)*(log(x + 5) + 4))

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sympy [B]  time = 0.40, size = 36, normalized size = 1.50 \begin {gather*} - \frac {4}{- 250 x^{5} \log {\left (x + 5 \right )} - 1000 x^{5} + \left (125 x^{5} \log {\left (x + 5 \right )} + 500 x^{5}\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**2+40*x+100)*exp(x)-40*x-200)*ln(5+x)+(16*x**2+164*x+400)*exp(x)-168*x-800)/(((125*x**7+625*x
**6)*exp(x)**2+(-500*x**7-2500*x**6)*exp(x)+500*x**7+2500*x**6)*ln(5+x)**2+((1000*x**7+5000*x**6)*exp(x)**2+(-
4000*x**7-20000*x**6)*exp(x)+4000*x**7+20000*x**6)*ln(5+x)+(2000*x**7+10000*x**6)*exp(x)**2+(-8000*x**7-40000*
x**6)*exp(x)+8000*x**7+40000*x**6),x)

[Out]

-4/(-250*x**5*log(x + 5) - 1000*x**5 + (125*x**5*log(x + 5) + 500*x**5)*exp(x))

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