3.81.4 \(\int \frac {51+15 e^{2+x}-3 x+(3 x-15 e^{2+x} x) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} (2720 x-160 x^2)} \, dx\)

Optimal. Leaf size=22 \[ \frac {3 \log (x)}{16 \left (2+5 \left (3+e^{2+x}\right )-x\right )} \]

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Rubi [F]  time = 1.19, 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 {51+15 e^{2+x}-3 x+\left (3 x-15 e^{2+x} x\right ) \log (x)}{4624 x+400 e^{4+2 x} x-544 x^2+16 x^3+e^{2+x} \left (2720 x-160 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(51 + 15*E^(2 + x) - 3*x + (3*x - 15*E^(2 + x)*x)*Log[x])/(4624*x + 400*E^(4 + 2*x)*x - 544*x^2 + 16*x^3 +
E^(2 + x)*(2720*x - 160*x^2)),x]

[Out]

(27*Log[x]*Defer[Int][(17 + 5*E^(2 + x) - x)^(-2), x])/8 - (3*Log[x]*Defer[Int][(17 + 5*E^(2 + x) - x)^(-1), x
])/16 + (3*Defer[Int][1/((17 + 5*E^(2 + x) - x)*x), x])/16 - (3*Log[x]*Defer[Int][x/(17 + 5*E^(2 + x) - x)^2,
x])/16 + (3*Defer[Int][Defer[Int][(17 + 5*E^(2 + x) - x)^(-1), x]/x, x])/16 - (27*Defer[Int][Defer[Int][(-17 -
5*E^(2 + x) + x)^(-2), x]/x, x])/8 + (3*Defer[Int][Defer[Int][x/(-17 - 5*E^(2 + x) + x)^2, x]/x, x])/16

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (17+5 e^{2+x}-x-\left (-1+5 e^{2+x}\right ) x \log (x)\right )}{16 \left (17+5 e^{2+x}-x\right )^2 x} \, dx\\ &=\frac {3}{16} \int \frac {17+5 e^{2+x}-x-\left (-1+5 e^{2+x}\right ) x \log (x)}{\left (17+5 e^{2+x}-x\right )^2 x} \, dx\\ &=\frac {3}{16} \int \left (-\frac {(-18+x) \log (x)}{\left (17+5 e^{2+x}-x\right )^2}-\frac {-1+x \log (x)}{\left (17+5 e^{2+x}-x\right ) x}\right ) \, dx\\ &=-\left (\frac {3}{16} \int \frac {(-18+x) \log (x)}{\left (17+5 e^{2+x}-x\right )^2} \, dx\right )-\frac {3}{16} \int \frac {-1+x \log (x)}{\left (17+5 e^{2+x}-x\right ) x} \, dx\\ &=-\left (\frac {3}{16} \int \left (-\frac {1}{\left (17+5 e^{2+x}-x\right ) x}+\frac {\log (x)}{17+5 e^{2+x}-x}\right ) \, dx\right )+\frac {3}{16} \int \frac {-18 \int \frac {1}{\left (-17-5 e^{2+x}+x\right )^2} \, dx+\int \frac {x}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x} \, dx-\frac {1}{16} (3 \log (x)) \int \frac {x}{\left (17+5 e^{2+x}-x\right )^2} \, dx+\frac {1}{8} (27 \log (x)) \int \frac {1}{\left (17+5 e^{2+x}-x\right )^2} \, dx\\ &=\frac {3}{16} \int \frac {1}{\left (17+5 e^{2+x}-x\right ) x} \, dx-\frac {3}{16} \int \frac {\log (x)}{17+5 e^{2+x}-x} \, dx+\frac {3}{16} \int \left (-\frac {18 \int \frac {1}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x}+\frac {\int \frac {x}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x}\right ) \, dx-\frac {1}{16} (3 \log (x)) \int \frac {x}{\left (17+5 e^{2+x}-x\right )^2} \, dx+\frac {1}{8} (27 \log (x)) \int \frac {1}{\left (17+5 e^{2+x}-x\right )^2} \, dx\\ &=\frac {3}{16} \int \frac {1}{\left (17+5 e^{2+x}-x\right ) x} \, dx+\frac {3}{16} \int \frac {\int \frac {1}{17+5 e^{2+x}-x} \, dx}{x} \, dx+\frac {3}{16} \int \frac {\int \frac {x}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x} \, dx-\frac {27}{8} \int \frac {\int \frac {1}{\left (-17-5 e^{2+x}+x\right )^2} \, dx}{x} \, dx-\frac {1}{16} (3 \log (x)) \int \frac {1}{17+5 e^{2+x}-x} \, dx-\frac {1}{16} (3 \log (x)) \int \frac {x}{\left (17+5 e^{2+x}-x\right )^2} \, dx+\frac {1}{8} (27 \log (x)) \int \frac {1}{\left (17+5 e^{2+x}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 20, normalized size = 0.91 \begin {gather*} \frac {3 \log (x)}{16 \left (17+5 e^{2+x}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(51 + 15*E^(2 + x) - 3*x + (3*x - 15*E^(2 + x)*x)*Log[x])/(4624*x + 400*E^(4 + 2*x)*x - 544*x^2 + 16
*x^3 + E^(2 + x)*(2720*x - 160*x^2)),x]

[Out]

(3*Log[x])/(16*(17 + 5*E^(2 + x) - x))

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fricas [A]  time = 0.52, size = 15, normalized size = 0.68 \begin {gather*} -\frac {3 \, \log \relax (x)}{16 \, {\left (x - 5 \, e^{\left (x + 2\right )} - 17\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x*exp(2+x)+3*x)*log(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)^2+(-160*x^2+2720*x)*exp(2+x)+16*x^3
-544*x^2+4624*x),x, algorithm="fricas")

[Out]

-3/16*log(x)/(x - 5*e^(x + 2) - 17)

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giac [A]  time = 0.14, size = 15, normalized size = 0.68 \begin {gather*} -\frac {3 \, \log \relax (x)}{16 \, {\left (x - 5 \, e^{\left (x + 2\right )} - 17\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x*exp(2+x)+3*x)*log(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)^2+(-160*x^2+2720*x)*exp(2+x)+16*x^3
-544*x^2+4624*x),x, algorithm="giac")

[Out]

-3/16*log(x)/(x - 5*e^(x + 2) - 17)

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maple [A]  time = 0.03, size = 16, normalized size = 0.73

method	result	size
risch	\(-\frac {3 \ln \relax (x )}{16 \left (x -5 \,{\mathrm e}^{2+x}-17\right )}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-15*x*exp(2+x)+3*x)*ln(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)^2+(-160*x^2+2720*x)*exp(2+x)+16*x^3-544*x^
2+4624*x),x,method=_RETURNVERBOSE)

[Out]

-3/16/(x-5*exp(2+x)-17)*ln(x)

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maxima [A]  time = 0.40, size = 15, normalized size = 0.68 \begin {gather*} -\frac {3 \, \log \relax (x)}{16 \, {\left (x - 5 \, e^{\left (x + 2\right )} - 17\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x*exp(2+x)+3*x)*log(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)^2+(-160*x^2+2720*x)*exp(2+x)+16*x^3
-544*x^2+4624*x),x, algorithm="maxima")

[Out]

-3/16*log(x)/(x - 5*e^(x + 2) - 17)

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mupad [B]  time = 6.11, size = 17, normalized size = 0.77 \begin {gather*} \frac {3\,\ln \relax (x)}{16\,\left (5\,{\mathrm {e}}^{x+2}-x+17\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((15*exp(x + 2) - 3*x + log(x)*(3*x - 15*x*exp(x + 2)) + 51)/(4624*x + exp(x + 2)*(2720*x - 160*x^2) + 400*
x*exp(2*x + 4) - 544*x^2 + 16*x^3),x)

[Out]

(3*log(x))/(16*(5*exp(x + 2) - x + 17))

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sympy [A]  time = 0.28, size = 15, normalized size = 0.68 \begin {gather*} \frac {3 \log {\relax (x )}}{- 16 x + 80 e^{x + 2} + 272} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x*exp(2+x)+3*x)*ln(x)+15*exp(2+x)-3*x+51)/(400*x*exp(2+x)**2+(-160*x**2+2720*x)*exp(2+x)+16*x*
*3-544*x**2+4624*x),x)

[Out]

3*log(x)/(-16*x + 80*exp(x + 2) + 272)

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