3.93.66 \(\int \frac {e^{6+x} (6750+7650 x+450 x^2-450 x^3)+e^6 (1620-90 x-120 x^2+30 x^3)+e^6 (900+120 x-60 x^2) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x (-150 x+30 x^2)+(e^x (150-30 x)+10 x-2 x^2) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {2 e^6 (3+x)^2}{e^x+\frac {1}{15} (x-\log (5-x))} \]

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Rubi [F]  time = 5.46, 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 {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(6 + x)*(6750 + 7650*x + 450*x^2 - 450*x^3) + E^6*(1620 - 90*x - 120*x^2 + 30*x^3) + E^6*(900 + 120*x -
60*x^2)*Log[5 - x])/(-5*x^2 + x^3 + E^(2*x)*(-1125 + 225*x) + E^x*(-150*x + 30*x^2) + (E^x*(150 - 30*x) + 10*
x - 2*x^2)*Log[5 - x] + (-5 + x)*Log[5 - x]^2),x]

[Out]

60*E^6*Defer[Int][(15*E^x + x - Log[5 - x])^(-2), x] + 1920*E^6*Defer[Int][1/((-5 + x)*(15*E^x + x - Log[5 - x
])^2), x] + 120*E^6*Defer[Int][x/(15*E^x + x - Log[5 - x])^2, x] + 150*E^6*Defer[Int][x^2/(15*E^x + x - Log[5
- x])^2, x] + 30*E^6*Defer[Int][x^3/(15*E^x + x - Log[5 - x])^2, x] - 90*E^6*Defer[Int][(15*E^x + x - Log[5 -
x])^(-1), x] - 120*E^6*Defer[Int][x/(15*E^x + x - Log[5 - x]), x] - 30*E^6*Defer[Int][x^2/(15*E^x + x - Log[5
- x]), x] - 270*E^6*Defer[Int][Log[5 - x]/(15*E^x + x - Log[5 - x])^2, x] - 180*E^6*Defer[Int][(x*Log[5 - x])/
(15*E^x + x - Log[5 - x])^2, x] - 30*E^6*Defer[Int][(x^2*Log[5 - x])/(15*E^x + x - Log[5 - x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {30 e^6 (3+x) \left (-18+7 x-x^2+15 e^x \left (-5-4 x+x^2\right )+2 (-5+x) \log (5-x)\right )}{(5-x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx\\ &=\left (30 e^6\right ) \int \frac {(3+x) \left (-18+7 x-x^2+15 e^x \left (-5-4 x+x^2\right )+2 (-5+x) \log (5-x)\right )}{(5-x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx\\ &=\left (30 e^6\right ) \int \left (-\frac {3+4 x+x^2}{15 e^x+x-\log (5-x)}+\frac {(3+x)^2 \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx\\ &=-\left (\left (30 e^6\right ) \int \frac {3+4 x+x^2}{15 e^x+x-\log (5-x)} \, dx\right )+\left (30 e^6\right ) \int \frac {(3+x)^2 \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx\\ &=-\left (\left (30 e^6\right ) \int \left (\frac {3}{15 e^x+x-\log (5-x)}+\frac {4 x}{15 e^x+x-\log (5-x)}+\frac {x^2}{15 e^x+x-\log (5-x)}\right ) \, dx\right )+\left (30 e^6\right ) \int \left (\frac {11 \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {64 \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}+\frac {x \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{\left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx\\ &=-\left (\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx\right )+\left (30 e^6\right ) \int \frac {x \left (6-6 x+x^2+5 \log (5-x)-x \log (5-x)\right )}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (330 e^6\right ) \int \frac {6-6 x+x^2+5 \log (5-x)-x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1920 e^6\right ) \int \frac {6-6 x+x^2+5 \log (5-x)-x \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx\\ &=-\left (\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx\right )+\left (30 e^6\right ) \int \left (\frac {6 x}{\left (15 e^x+x-\log (5-x)\right )^2}-\frac {6 x^2}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {x^3}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {5 x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}-\frac {x^2 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (330 e^6\right ) \int \left (\frac {6}{\left (15 e^x+x-\log (5-x)\right )^2}-\frac {6 x}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {5 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}-\frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx+\left (1920 e^6\right ) \int \left (\frac {6}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}-\frac {6 x}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}+\frac {x^2}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}+\frac {5 \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}-\frac {x \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx\\ &=\left (30 e^6\right ) \int \frac {x^3}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx-\left (30 e^6\right ) \int \frac {x^2 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (150 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (180 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (180 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (330 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (330 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1650 e^6\right ) \int \frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1920 e^6\right ) \int \frac {x^2}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1920 e^6\right ) \int \frac {x \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1980 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1980 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (9600 e^6\right ) \int \frac {\log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (11520 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (11520 e^6\right ) \int \frac {x}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx\\ &=\left (30 e^6\right ) \int \frac {x^3}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx-\left (30 e^6\right ) \int \frac {x^2 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (150 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (180 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (180 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (330 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (330 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1650 e^6\right ) \int \frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1920 e^6\right ) \int \left (\frac {5}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {25}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}+\frac {x}{\left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx-\left (1920 e^6\right ) \int \left (\frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {5 \log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx+\left (1980 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1980 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (9600 e^6\right ) \int \frac {\log (5-x)}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (11520 e^6\right ) \int \left (\frac {1}{\left (15 e^x+x-\log (5-x)\right )^2}+\frac {5}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2}\right ) \, dx+\left (11520 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx\\ &=\left (30 e^6\right ) \int \frac {x^3}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (30 e^6\right ) \int \frac {x^2}{15 e^x+x-\log (5-x)} \, dx-\left (30 e^6\right ) \int \frac {x^2 \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (90 e^6\right ) \int \frac {1}{15 e^x+x-\log (5-x)} \, dx-\left (120 e^6\right ) \int \frac {x}{15 e^x+x-\log (5-x)} \, dx+\left (150 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (180 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (180 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (330 e^6\right ) \int \frac {x^2}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (330 e^6\right ) \int \frac {x \log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1650 e^6\right ) \int \frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1920 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1920 e^6\right ) \int \frac {\log (5-x)}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (1980 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (1980 e^6\right ) \int \frac {x}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (9600 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (11520 e^6\right ) \int \frac {1}{\left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (11520 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx+\left (48000 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx-\left (57600 e^6\right ) \int \frac {1}{(-5+x) \left (15 e^x+x-\log (5-x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.90, size = 27, normalized size = 0.90 \begin {gather*} -\frac {30 e^6 (3+x)^2}{-15 e^x-x+\log (5-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(6 + x)*(6750 + 7650*x + 450*x^2 - 450*x^3) + E^6*(1620 - 90*x - 120*x^2 + 30*x^3) + E^6*(900 + 1
20*x - 60*x^2)*Log[5 - x])/(-5*x^2 + x^3 + E^(2*x)*(-1125 + 225*x) + E^x*(-150*x + 30*x^2) + (E^x*(150 - 30*x)
+ 10*x - 2*x^2)*Log[5 - x] + (-5 + x)*Log[5 - x]^2),x]

[Out]

(-30*E^6*(3 + x)^2)/(-15*E^x - x + Log[5 - x])

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fricas [A]  time = 0.58, size = 35, normalized size = 1.17 \begin {gather*} \frac {30 \, {\left (x^{2} + 6 \, x + 9\right )} e^{12}}{x e^{6} - e^{6} \log \left (-x + 5\right ) + 15 \, e^{\left (x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-60*x^2+120*x+900)*exp(6)*log(5-x)+(-450*x^3+450*x^2+7650*x+6750)*exp(6)*exp(x)+(30*x^3-120*x^2-90
*x+1620)*exp(6))/((x-5)*log(5-x)^2+((-30*x+150)*exp(x)-2*x^2+10*x)*log(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*
x)*exp(x)+x^3-5*x^2),x, algorithm="fricas")

[Out]

30*(x^2 + 6*x + 9)*e^12/(x*e^6 - e^6*log(-x + 5) + 15*e^(x + 6))

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giac [A]  time = 0.25, size = 34, normalized size = 1.13 \begin {gather*} \frac {30 \, {\left (x^{2} e^{6} + 6 \, x e^{6} + 9 \, e^{6}\right )}}{x + 15 \, e^{x} - \log \left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-60*x^2+120*x+900)*exp(6)*log(5-x)+(-450*x^3+450*x^2+7650*x+6750)*exp(6)*exp(x)+(30*x^3-120*x^2-90
*x+1620)*exp(6))/((x-5)*log(5-x)^2+((-30*x+150)*exp(x)-2*x^2+10*x)*log(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*
x)*exp(x)+x^3-5*x^2),x, algorithm="giac")

[Out]

30*(x^2*e^6 + 6*x*e^6 + 9*e^6)/(x + 15*e^x - log(-x + 5))

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maple [A]  time = 0.06, size = 29, normalized size = 0.97

method	result	size
risch	\(\frac {30 \left (x^{2}+6 x +9\right ) {\mathrm e}^{6}}{15 \,{\mathrm e}^{x}-\ln \left (5-x \right )+x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-60*x^2+120*x+900)*exp(6)*ln(5-x)+(-450*x^3+450*x^2+7650*x+6750)*exp(6)*exp(x)+(30*x^3-120*x^2-90*x+1620
)*exp(6))/((x-5)*ln(5-x)^2+((-30*x+150)*exp(x)-2*x^2+10*x)*ln(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*x)*exp(x)
+x^3-5*x^2),x,method=_RETURNVERBOSE)

[Out]

30*(x^2+6*x+9)*exp(6)/(15*exp(x)-ln(5-x)+x)

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maxima [A]  time = 0.41, size = 34, normalized size = 1.13 \begin {gather*} \frac {30 \, {\left (x^{2} e^{6} + 6 \, x e^{6} + 9 \, e^{6}\right )}}{x + 15 \, e^{x} - \log \left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-60*x^2+120*x+900)*exp(6)*log(5-x)+(-450*x^3+450*x^2+7650*x+6750)*exp(6)*exp(x)+(30*x^3-120*x^2-90
*x+1620)*exp(6))/((x-5)*log(5-x)^2+((-30*x+150)*exp(x)-2*x^2+10*x)*log(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*
x)*exp(x)+x^3-5*x^2),x, algorithm="maxima")

[Out]

30*(x^2*e^6 + 6*x*e^6 + 9*e^6)/(x + 15*e^x - log(-x + 5))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^6\,{\mathrm {e}}^x\,\left (-450\,x^3+450\,x^2+7650\,x+6750\right )-{\mathrm {e}}^6\,\left (-30\,x^3+120\,x^2+90\,x-1620\right )+{\mathrm {e}}^6\,\ln \left (5-x\right )\,\left (-60\,x^2+120\,x+900\right )}{{\ln \left (5-x\right )}^2\,\left (x-5\right )-{\mathrm {e}}^x\,\left (150\,x-30\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (225\,x-1125\right )-5\,x^2+x^3-\ln \left (5-x\right )\,\left ({\mathrm {e}}^x\,\left (30\,x-150\right )-10\,x+2\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(6)*exp(x)*(7650*x + 450*x^2 - 450*x^3 + 6750) - exp(6)*(90*x + 120*x^2 - 30*x^3 - 1620) + exp(6)*log(
5 - x)*(120*x - 60*x^2 + 900))/(log(5 - x)^2*(x - 5) - exp(x)*(150*x - 30*x^2) + exp(2*x)*(225*x - 1125) - 5*x
^2 + x^3 - log(5 - x)*(exp(x)*(30*x - 150) - 10*x + 2*x^2)),x)

[Out]

int((exp(6)*exp(x)*(7650*x + 450*x^2 - 450*x^3 + 6750) - exp(6)*(90*x + 120*x^2 - 30*x^3 - 1620) + exp(6)*log(
5 - x)*(120*x - 60*x^2 + 900))/(log(5 - x)^2*(x - 5) - exp(x)*(150*x - 30*x^2) + exp(2*x)*(225*x - 1125) - 5*x
^2 + x^3 - log(5 - x)*(exp(x)*(30*x - 150) - 10*x + 2*x^2)), x)

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sympy [A]  time = 0.38, size = 32, normalized size = 1.07 \begin {gather*} \frac {2 x^{2} e^{6} + 12 x e^{6} + 18 e^{6}}{\frac {x}{15} + e^{x} - \frac {\log {\left (5 - x \right )}}{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-60*x**2+120*x+900)*exp(6)*ln(5-x)+(-450*x**3+450*x**2+7650*x+6750)*exp(6)*exp(x)+(30*x**3-120*x**
2-90*x+1620)*exp(6))/((x-5)*ln(5-x)**2+((-30*x+150)*exp(x)-2*x**2+10*x)*ln(5-x)+(225*x-1125)*exp(x)**2+(30*x**
2-150*x)*exp(x)+x**3-5*x**2),x)

[Out]

(2*x**2*exp(6) + 12*x*exp(6) + 18*exp(6))/(x/15 + exp(x) - log(5 - x)/15)

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