3.62.2 \(\int \frac {-5000 x^2+1250 e^{5+x} x^2+(15000 x^2-3700 e^{5+x} x^2) \log (x)+(10000 x-2450 e^{5+x} x) \log ^2(x)+\log (\frac {1}{4} (4-e^{5+x})) (200 x-50 e^{5+x} x+(-400 x+100 e^{5+x} x) \log (x)+(-200+50 e^{5+x}) \log ^2(x))}{-2500 x^6+625 e^{5+x} x^6+(-5000 x^5+1250 e^{5+x} x^5) \log (x)+(-2500 x^4+625 e^{5+x} x^4) \log ^2(x)+\log ^2(\frac {1}{4} (4-e^{5+x})) (-4 x^4+e^{5+x} x^4+(-8 x^3+2 e^{5+x} x^3) \log (x)+(-4 x^2+e^{5+x} x^2) \log ^2(x))+\log (\frac {1}{4} (4-e^{5+x})) (200 x^5-50 e^{5+x} x^5+(400 x^4-100 e^{5+x} x^4) \log (x)+(200 x^3-50 e^{5+x} x^3) \log ^2(x))} \, dx\)

Optimal. Leaf size=35 \[ \frac {2 \log (x)}{x \left (x-\frac {1}{25} \log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )\right ) (x+\log (x))} \]

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Rubi [F]  time = 13.96, 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 {-5000 x^2+1250 e^{5+x} x^2+\left (15000 x^2-3700 e^{5+x} x^2\right ) \log (x)+\left (10000 x-2450 e^{5+x} x\right ) \log ^2(x)+\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right ) \left (200 x-50 e^{5+x} x+\left (-400 x+100 e^{5+x} x\right ) \log (x)+\left (-200+50 e^{5+x}\right ) \log ^2(x)\right )}{-2500 x^6+625 e^{5+x} x^6+\left (-5000 x^5+1250 e^{5+x} x^5\right ) \log (x)+\left (-2500 x^4+625 e^{5+x} x^4\right ) \log ^2(x)+\log ^2\left (\frac {1}{4} \left (4-e^{5+x}\right )\right ) \left (-4 x^4+e^{5+x} x^4+\left (-8 x^3+2 e^{5+x} x^3\right ) \log (x)+\left (-4 x^2+e^{5+x} x^2\right ) \log ^2(x)\right )+\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right ) \left (200 x^5-50 e^{5+x} x^5+\left (400 x^4-100 e^{5+x} x^4\right ) \log (x)+\left (200 x^3-50 e^{5+x} x^3\right ) \log ^2(x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5000*x^2 + 1250*E^(5 + x)*x^2 + (15000*x^2 - 3700*E^(5 + x)*x^2)*Log[x] + (10000*x - 2450*E^(5 + x)*x)*L
og[x]^2 + Log[(4 - E^(5 + x))/4]*(200*x - 50*E^(5 + x)*x + (-400*x + 100*E^(5 + x)*x)*Log[x] + (-200 + 50*E^(5
+ x))*Log[x]^2))/(-2500*x^6 + 625*E^(5 + x)*x^6 + (-5000*x^5 + 1250*E^(5 + x)*x^5)*Log[x] + (-2500*x^4 + 625*
E^(5 + x)*x^4)*Log[x]^2 + Log[(4 - E^(5 + x))/4]^2*(-4*x^4 + E^(5 + x)*x^4 + (-8*x^3 + 2*E^(5 + x)*x^3)*Log[x]
+ (-4*x^2 + E^(5 + x)*x^2)*Log[x]^2) + Log[(4 - E^(5 + x))/4]*(200*x^5 - 50*E^(5 + x)*x^5 + (400*x^4 - 100*E^
(5 + x)*x^4)*Log[x] + (200*x^3 - 50*E^(5 + x)*x^3)*Log[x]^2)),x]

[Out]

-2450*Defer[Int][1/(x*(25*x - Log[1 - E^(5 + x)/4])^2), x] + 50*Defer[Int][Log[(4 - E^(5 + x))/4]/(x^2*(25*x -
Log[1 - E^(5 + x)/4])^2), x] + 1250*Defer[Int][1/((25*x - Log[1 - E^(5 + x)/4])^2*(x + Log[x])^2), x] + 1250*
Defer[Int][x/((25*x - Log[1 - E^(5 + x)/4])^2*(x + Log[x])^2), x] - 50*Defer[Int][Log[(4 - E^(5 + x))/4]/((25*
x - Log[1 - E^(5 + x)/4])^2*(x + Log[x])^2), x] - 50*Defer[Int][Log[(4 - E^(5 + x))/4]/(x*(25*x - Log[1 - E^(5
+ x)/4])^2*(x + Log[x])^2), x] + 1200*Defer[Int][1/((25*x - Log[1 - E^(5 + x)/4])^2*(x + Log[x])), x] + 200*D
efer[Int][Log[x]/((-4 + E^(5 + x))*x*(25*x - Log[1 - E^(5 + x)/4])^2*(x + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-50 \left (-4+e^{5+x}\right ) \log \left (1-\frac {e^{5+x}}{4}\right ) \left (-x+2 x \log (x)+\log ^2(x)\right )+50 x \left (-25 \left (-4+e^{5+x}\right ) x+2 \left (-150+37 e^{5+x}\right ) x \log (x)+\left (-200+49 e^{5+x}\right ) \log ^2(x)\right )}{\left (4-e^{5+x}\right ) x^2 \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx\\ &=\int \left (-\frac {2450 \log ^2(x)}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}+\frac {200 \log (x)}{\left (-4+e^{5+x}\right ) x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))}-\frac {50 (-25+74 \log (x))}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}+\frac {50 \log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right ) \left (-x+2 x \log (x)+\log ^2(x)\right )}{x^2 \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}\right ) \, dx\\ &=-\left (50 \int \frac {-25+74 \log (x)}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx\right )+50 \int \frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right ) \left (-x+2 x \log (x)+\log ^2(x)\right )}{x^2 \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx+200 \int \frac {\log (x)}{\left (-4+e^{5+x}\right ) x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx-2450 \int \frac {\log ^2(x)}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx\\ &=50 \int \left (\frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{x^2 \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2}+\frac {(-1-x) \log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}\right ) \, dx-50 \int \left (\frac {-25-74 x}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}+\frac {74}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))}\right ) \, dx+200 \int \frac {\log (x)}{\left (-4+e^{5+x}\right ) x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx-2450 \int \left (\frac {1}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2}+\frac {x}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}-\frac {2}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))}\right ) \, dx\\ &=50 \int \frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{x^2 \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2} \, dx-50 \int \frac {-25-74 x}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx+50 \int \frac {(-1-x) \log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx+200 \int \frac {\log (x)}{\left (-4+e^{5+x}\right ) x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx-2450 \int \frac {1}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2} \, dx-2450 \int \frac {x}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx-3700 \int \frac {1}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx+4900 \int \frac {1}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx\\ &=50 \int \frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{x^2 \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2} \, dx-50 \int \left (-\frac {25}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}-\frac {74 x}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}\right ) \, dx+50 \int \left (-\frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}-\frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2}\right ) \, dx+200 \int \frac {\log (x)}{\left (-4+e^{5+x}\right ) x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx-2450 \int \frac {1}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2} \, dx-2450 \int \frac {x}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx-3700 \int \frac {1}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx+4900 \int \frac {1}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx\\ &=50 \int \frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{x^2 \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2} \, dx-50 \int \frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx-50 \int \frac {\log \left (\frac {1}{4} \left (4-e^{5+x}\right )\right )}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx+200 \int \frac {\log (x)}{\left (-4+e^{5+x}\right ) x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx+1250 \int \frac {1}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx-2450 \int \frac {1}{x \left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2} \, dx-2450 \int \frac {x}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx+3700 \int \frac {x}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))^2} \, dx-3700 \int \frac {1}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx+4900 \int \frac {1}{\left (25 x-\log \left (1-\frac {e^{5+x}}{4}\right )\right )^2 (x+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 31, normalized size = 0.89 \begin {gather*} -\frac {50 \log (x)}{x \left (-25 x+\log \left (1-\frac {e^{5+x}}{4}\right )\right ) (x+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5000*x^2 + 1250*E^(5 + x)*x^2 + (15000*x^2 - 3700*E^(5 + x)*x^2)*Log[x] + (10000*x - 2450*E^(5 + x
)*x)*Log[x]^2 + Log[(4 - E^(5 + x))/4]*(200*x - 50*E^(5 + x)*x + (-400*x + 100*E^(5 + x)*x)*Log[x] + (-200 + 5
0*E^(5 + x))*Log[x]^2))/(-2500*x^6 + 625*E^(5 + x)*x^6 + (-5000*x^5 + 1250*E^(5 + x)*x^5)*Log[x] + (-2500*x^4
+ 625*E^(5 + x)*x^4)*Log[x]^2 + Log[(4 - E^(5 + x))/4]^2*(-4*x^4 + E^(5 + x)*x^4 + (-8*x^3 + 2*E^(5 + x)*x^3)*
Log[x] + (-4*x^2 + E^(5 + x)*x^2)*Log[x]^2) + Log[(4 - E^(5 + x))/4]*(200*x^5 - 50*E^(5 + x)*x^5 + (400*x^4 -
100*E^(5 + x)*x^4)*Log[x] + (200*x^3 - 50*E^(5 + x)*x^3)*Log[x]^2)),x]

[Out]

(-50*Log[x])/(x*(-25*x + Log[1 - E^(5 + x)/4])*(x + Log[x]))

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fricas [A]  time = 0.55, size = 38, normalized size = 1.09 \begin {gather*} \frac {50 \, \log \relax (x)}{25 \, x^{3} + 25 \, x^{2} \log \relax (x) - {\left (x^{2} + x \log \relax (x)\right )} \log \left (-\frac {1}{4} \, e^{\left (x + 5\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*exp(5+x)-200)*log(x)^2+(100*x*exp(5+x)-400*x)*log(x)-50*x*exp(5+x)+200*x)*log(-1/4*exp(5+x)+1)
+(-2450*x*exp(5+x)+10000*x)*log(x)^2+(-3700*x^2*exp(5+x)+15000*x^2)*log(x)+1250*x^2*exp(5+x)-5000*x^2)/(((x^2*
exp(5+x)-4*x^2)*log(x)^2+(2*x^3*exp(5+x)-8*x^3)*log(x)+x^4*exp(5+x)-4*x^4)*log(-1/4*exp(5+x)+1)^2+((-50*x^3*ex
p(5+x)+200*x^3)*log(x)^2+(-100*x^4*exp(5+x)+400*x^4)*log(x)-50*x^5*exp(5+x)+200*x^5)*log(-1/4*exp(5+x)+1)+(625
*x^4*exp(5+x)-2500*x^4)*log(x)^2+(1250*x^5*exp(5+x)-5000*x^5)*log(x)+625*x^6*exp(5+x)-2500*x^6),x, algorithm="
fricas")

[Out]

50*log(x)/(25*x^3 + 25*x^2*log(x) - (x^2 + x*log(x))*log(-1/4*e^(x + 5) + 1))

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giac [B]  time = 0.76, size = 61, normalized size = 1.74 \begin {gather*} \frac {50 \, \log \relax (x)}{25 \, x^{3} + 2 \, x^{2} \log \relax (2) + 25 \, x^{2} \log \relax (x) + 2 \, x \log \relax (2) \log \relax (x) - x^{2} \log \left (-e^{\left (x + 5\right )} + 4\right ) - x \log \relax (x) \log \left (-e^{\left (x + 5\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*exp(5+x)-200)*log(x)^2+(100*x*exp(5+x)-400*x)*log(x)-50*x*exp(5+x)+200*x)*log(-1/4*exp(5+x)+1)
+(-2450*x*exp(5+x)+10000*x)*log(x)^2+(-3700*x^2*exp(5+x)+15000*x^2)*log(x)+1250*x^2*exp(5+x)-5000*x^2)/(((x^2*
exp(5+x)-4*x^2)*log(x)^2+(2*x^3*exp(5+x)-8*x^3)*log(x)+x^4*exp(5+x)-4*x^4)*log(-1/4*exp(5+x)+1)^2+((-50*x^3*ex
p(5+x)+200*x^3)*log(x)^2+(-100*x^4*exp(5+x)+400*x^4)*log(x)-50*x^5*exp(5+x)+200*x^5)*log(-1/4*exp(5+x)+1)+(625
*x^4*exp(5+x)-2500*x^4)*log(x)^2+(1250*x^5*exp(5+x)-5000*x^5)*log(x)+625*x^6*exp(5+x)-2500*x^6),x, algorithm="
giac")

[Out]

50*log(x)/(25*x^3 + 2*x^2*log(2) + 25*x^2*log(x) + 2*x*log(2)*log(x) - x^2*log(-e^(x + 5) + 4) - x*log(x)*log(
-e^(x + 5) + 4))

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maple [A]  time = 0.08, size = 31, normalized size = 0.89

method	result	size
risch	\(\frac {50 \ln \relax (x )}{\left (x +\ln \relax (x )\right ) x \left (25 x -\ln \left (-\frac {{\mathrm e}^{5+x}}{4}+1\right )\right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((50*exp(5+x)-200)*ln(x)^2+(100*x*exp(5+x)-400*x)*ln(x)-50*x*exp(5+x)+200*x)*ln(-1/4*exp(5+x)+1)+(-2450*x
*exp(5+x)+10000*x)*ln(x)^2+(-3700*x^2*exp(5+x)+15000*x^2)*ln(x)+1250*x^2*exp(5+x)-5000*x^2)/(((x^2*exp(5+x)-4*
x^2)*ln(x)^2+(2*x^3*exp(5+x)-8*x^3)*ln(x)+x^4*exp(5+x)-4*x^4)*ln(-1/4*exp(5+x)+1)^2+((-50*x^3*exp(5+x)+200*x^3
)*ln(x)^2+(-100*x^4*exp(5+x)+400*x^4)*ln(x)-50*x^5*exp(5+x)+200*x^5)*ln(-1/4*exp(5+x)+1)+(625*x^4*exp(5+x)-250
0*x^4)*ln(x)^2+(1250*x^5*exp(5+x)-5000*x^5)*ln(x)+625*x^6*exp(5+x)-2500*x^6),x,method=_RETURNVERBOSE)

[Out]

50*ln(x)/(x+ln(x))/x/(25*x-ln(-1/4*exp(5+x)+1))

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maxima [A]  time = 0.94, size = 52, normalized size = 1.49 \begin {gather*} \frac {50 \, \log \relax (x)}{25 \, x^{3} + 2 \, x^{2} \log \relax (2) + {\left (25 \, x^{2} + 2 \, x \log \relax (2)\right )} \log \relax (x) - {\left (x^{2} + x \log \relax (x)\right )} \log \left (-e^{\left (x + 5\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*exp(5+x)-200)*log(x)^2+(100*x*exp(5+x)-400*x)*log(x)-50*x*exp(5+x)+200*x)*log(-1/4*exp(5+x)+1)
+(-2450*x*exp(5+x)+10000*x)*log(x)^2+(-3700*x^2*exp(5+x)+15000*x^2)*log(x)+1250*x^2*exp(5+x)-5000*x^2)/(((x^2*
exp(5+x)-4*x^2)*log(x)^2+(2*x^3*exp(5+x)-8*x^3)*log(x)+x^4*exp(5+x)-4*x^4)*log(-1/4*exp(5+x)+1)^2+((-50*x^3*ex
p(5+x)+200*x^3)*log(x)^2+(-100*x^4*exp(5+x)+400*x^4)*log(x)-50*x^5*exp(5+x)+200*x^5)*log(-1/4*exp(5+x)+1)+(625
*x^4*exp(5+x)-2500*x^4)*log(x)^2+(1250*x^5*exp(5+x)-5000*x^5)*log(x)+625*x^6*exp(5+x)-2500*x^6),x, algorithm="
maxima")

[Out]

50*log(x)/(25*x^3 + 2*x^2*log(2) + (25*x^2 + 2*x*log(2))*log(x) - (x^2 + x*log(x))*log(-e^(x + 5) + 4))

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mupad [B]  time = 6.66, size = 785, normalized size = 22.43 \begin {gather*} \frac {\frac {25\,\left (100\,x-25\,x\,{\mathrm {e}}^{x+5}-200\,{\ln \relax (x)}^2-300\,x\,\ln \relax (x)+49\,{\mathrm {e}}^{x+5}\,{\ln \relax (x)}^2+74\,x\,{\mathrm {e}}^{x+5}\,\ln \relax (x)\right )}{2\,x\,{\left (x+\ln \relax (x)\right )}^2\,\left (6\,{\mathrm {e}}^{x+5}-25\right )}-\frac {25\,\ln \left (1-\frac {{\mathrm {e}}^5\,{\mathrm {e}}^x}{4}\right )\,\left ({\mathrm {e}}^{x+5}-4\right )\,\left ({\ln \relax (x)}^2+2\,x\,\ln \relax (x)-x\right )}{2\,x^2\,{\left (x+\ln \relax (x)\right )}^2\,\left (6\,{\mathrm {e}}^{x+5}-25\right )}}{25\,x-\ln \left (1-\frac {{\mathrm {e}}^5\,{\mathrm {e}}^x}{4}\right )}-\frac {\frac {25\,\left (1825\,{\mathrm {e}}^{x+5}-7500\,x-444\,{\mathrm {e}}^{2\,x+10}+36\,{\mathrm {e}}^{3\,x+15}+5450\,x\,{\mathrm {e}}^{x+5}-1326\,x\,{\mathrm {e}}^{2\,x+10}+108\,x\,{\mathrm {e}}^{3\,x+15}+5375\,x^2\,{\mathrm {e}}^{x+5}-75\,x^3\,{\mathrm {e}}^{x+5}+25\,x^4\,{\mathrm {e}}^{x+5}+25\,x^5\,{\mathrm {e}}^{x+5}-1308\,x^2\,{\mathrm {e}}^{2\,x+10}+30\,x^3\,{\mathrm {e}}^{2\,x+10}+18\,x^4\,{\mathrm {e}}^{2\,x+10}+6\,x^5\,{\mathrm {e}}^{2\,x+10}+108\,x^2\,{\mathrm {e}}^{3\,x+15}-7500\,x^2-2500\right )}{4\,x\,{\left (x+1\right )}^3\,{\left (6\,{\mathrm {e}}^{x+5}-25\right )}^3}+\frac {25\,\ln \relax (x)\,\left (1825\,{\mathrm {e}}^{x+5}-5000\,x-444\,{\mathrm {e}}^{2\,x+10}+36\,{\mathrm {e}}^{3\,x+15}+3625\,x\,{\mathrm {e}}^{x+5}-882\,x\,{\mathrm {e}}^{2\,x+10}+72\,x\,{\mathrm {e}}^{3\,x+15}+50\,x^3\,{\mathrm {e}}^{x+5}+25\,x^4\,{\mathrm {e}}^{x+5}+12\,x^2\,{\mathrm {e}}^{2\,x+10}+12\,x^3\,{\mathrm {e}}^{2\,x+10}+6\,x^4\,{\mathrm {e}}^{2\,x+10}-2500\right )}{4\,x\,{\left (x+1\right )}^3\,{\left (6\,{\mathrm {e}}^{x+5}-25\right )}^3}}{x+\ln \relax (x)}+\frac {\frac {25\,\left (500\,x-98\,{\mathrm {e}}^{x+5}+12\,{\mathrm {e}}^{2\,x+10}-245\,x\,{\mathrm {e}}^{x+5}+30\,x\,{\mathrm {e}}^{2\,x+10}-97\,x^2\,{\mathrm {e}}^{x+5}+x^3\,{\mathrm {e}}^{x+5}+12\,x^2\,{\mathrm {e}}^{2\,x+10}+200\,x^2+200\right )}{4\,x\,\left (x+1\right )\,{\left (6\,{\mathrm {e}}^{x+5}-25\right )}^2}+\frac {25\,\ln \relax (x)\,\left (6\,{\mathrm {e}}^{2\,x+10}-49\,{\mathrm {e}}^{x+5}+x\,{\mathrm {e}}^{x+5}+x^2\,{\mathrm {e}}^{x+5}+100\right )}{4\,x\,\left (x+1\right )\,{\left (6\,{\mathrm {e}}^{x+5}-25\right )}^2}}{x^2+2\,x\,\ln \relax (x)+{\ln \relax (x)}^2}-\frac {\frac {25\,x^3}{12}+\frac {25\,x^2}{6}+\frac {125\,x}{24}+\frac {25}{12}}{x^5+3\,x^4+3\,x^3+x^2}+\frac {15625\,\left (x^7+3\,x^6+3\,x^5+x^4\right )}{12\,x^3\,{\left (x+1\right )}^4\,\left (11250\,{\mathrm {e}}^{x+5}-2700\,{\mathrm {e}}^{2\,x+10}+216\,{\mathrm {e}}^{3\,x+15}-15625\right )}-\frac {25\,\left (-x^7-3\,x^6-2\,x^5+3\,x^4+8\,x^3+7\,x^2+2\,x\right )}{24\,x^3\,{\left (x+1\right )}^4\,\left (6\,{\mathrm {e}}^{x+5}-25\right )}+\frac {625\,\left (3\,x^7+9\,x^6+10\,x^5+5\,x^4+x^3\right )}{24\,x^3\,{\left (x+1\right )}^4\,\left (36\,{\mathrm {e}}^{2\,x+10}-300\,{\mathrm {e}}^{x+5}+625\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(1 - exp(x + 5)/4)*(200*x - 50*x*exp(x + 5) - log(x)*(400*x - 100*x*exp(x + 5)) + log(x)^2*(50*exp(x +
5) - 200)) - log(x)*(3700*x^2*exp(x + 5) - 15000*x^2) + 1250*x^2*exp(x + 5) - 5000*x^2 + log(x)^2*(10000*x -
2450*x*exp(x + 5)))/(log(1 - exp(x + 5)/4)^2*(log(x)*(2*x^3*exp(x + 5) - 8*x^3) + x^4*exp(x + 5) + log(x)^2*(x
^2*exp(x + 5) - 4*x^2) - 4*x^4) + log(x)*(1250*x^5*exp(x + 5) - 5000*x^5) + 625*x^6*exp(x + 5) + log(x)^2*(625
*x^4*exp(x + 5) - 2500*x^4) - log(1 - exp(x + 5)/4)*(log(x)*(100*x^4*exp(x + 5) - 400*x^4) + 50*x^5*exp(x + 5)
+ log(x)^2*(50*x^3*exp(x + 5) - 200*x^3) - 200*x^5) - 2500*x^6),x)

[Out]

((25*(100*x - 25*x*exp(x + 5) - 200*log(x)^2 - 300*x*log(x) + 49*exp(x + 5)*log(x)^2 + 74*x*exp(x + 5)*log(x))
)/(2*x*(x + log(x))^2*(6*exp(x + 5) - 25)) - (25*log(1 - (exp(5)*exp(x))/4)*(exp(x + 5) - 4)*(log(x)^2 - x + 2
*x*log(x)))/(2*x^2*(x + log(x))^2*(6*exp(x + 5) - 25)))/(25*x - log(1 - (exp(5)*exp(x))/4)) - ((25*(1825*exp(x
+ 5) - 7500*x - 444*exp(2*x + 10) + 36*exp(3*x + 15) + 5450*x*exp(x + 5) - 1326*x*exp(2*x + 10) + 108*x*exp(3
*x + 15) + 5375*x^2*exp(x + 5) - 75*x^3*exp(x + 5) + 25*x^4*exp(x + 5) + 25*x^5*exp(x + 5) - 1308*x^2*exp(2*x
+ 10) + 30*x^3*exp(2*x + 10) + 18*x^4*exp(2*x + 10) + 6*x^5*exp(2*x + 10) + 108*x^2*exp(3*x + 15) - 7500*x^2 -
2500))/(4*x*(x + 1)^3*(6*exp(x + 5) - 25)^3) + (25*log(x)*(1825*exp(x + 5) - 5000*x - 444*exp(2*x + 10) + 36*
exp(3*x + 15) + 3625*x*exp(x + 5) - 882*x*exp(2*x + 10) + 72*x*exp(3*x + 15) + 50*x^3*exp(x + 5) + 25*x^4*exp(
x + 5) + 12*x^2*exp(2*x + 10) + 12*x^3*exp(2*x + 10) + 6*x^4*exp(2*x + 10) - 2500))/(4*x*(x + 1)^3*(6*exp(x +
5) - 25)^3))/(x + log(x)) + ((25*(500*x - 98*exp(x + 5) + 12*exp(2*x + 10) - 245*x*exp(x + 5) + 30*x*exp(2*x +
10) - 97*x^2*exp(x + 5) + x^3*exp(x + 5) + 12*x^2*exp(2*x + 10) + 200*x^2 + 200))/(4*x*(x + 1)*(6*exp(x + 5)
- 25)^2) + (25*log(x)*(6*exp(2*x + 10) - 49*exp(x + 5) + x*exp(x + 5) + x^2*exp(x + 5) + 100))/(4*x*(x + 1)*(6
*exp(x + 5) - 25)^2))/(log(x)^2 + 2*x*log(x) + x^2) - ((125*x)/24 + (25*x^2)/6 + (25*x^3)/12 + 25/12)/(x^2 + 3
*x^3 + 3*x^4 + x^5) + (15625*(x^4 + 3*x^5 + 3*x^6 + x^7))/(12*x^3*(x + 1)^4*(11250*exp(x + 5) - 2700*exp(2*x +
10) + 216*exp(3*x + 15) - 15625)) - (25*(2*x + 7*x^2 + 8*x^3 + 3*x^4 - 2*x^5 - 3*x^6 - x^7))/(24*x^3*(x + 1)^
4*(6*exp(x + 5) - 25)) + (625*(x^3 + 5*x^4 + 10*x^5 + 9*x^6 + 3*x^7))/(24*x^3*(x + 1)^4*(36*exp(2*x + 10) - 30
0*exp(x + 5) + 625))

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sympy [A]  time = 0.50, size = 37, normalized size = 1.06 \begin {gather*} - \frac {50 \log {\relax (x )}}{- 25 x^{3} - 25 x^{2} \log {\relax (x )} + \left (x^{2} + x \log {\relax (x )}\right ) \log {\left (1 - \frac {e^{x + 5}}{4} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*exp(5+x)-200)*ln(x)**2+(100*x*exp(5+x)-400*x)*ln(x)-50*x*exp(5+x)+200*x)*ln(-1/4*exp(5+x)+1)+(
-2450*x*exp(5+x)+10000*x)*ln(x)**2+(-3700*x**2*exp(5+x)+15000*x**2)*ln(x)+1250*x**2*exp(5+x)-5000*x**2)/(((x**
2*exp(5+x)-4*x**2)*ln(x)**2+(2*x**3*exp(5+x)-8*x**3)*ln(x)+x**4*exp(5+x)-4*x**4)*ln(-1/4*exp(5+x)+1)**2+((-50*
x**3*exp(5+x)+200*x**3)*ln(x)**2+(-100*x**4*exp(5+x)+400*x**4)*ln(x)-50*x**5*exp(5+x)+200*x**5)*ln(-1/4*exp(5+
x)+1)+(625*x**4*exp(5+x)-2500*x**4)*ln(x)**2+(1250*x**5*exp(5+x)-5000*x**5)*ln(x)+625*x**6*exp(5+x)-2500*x**6)
,x)

[Out]

-50*log(x)/(-25*x**3 - 25*x**2*log(x) + (x**2 + x*log(x))*log(1 - exp(x + 5)/4))

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