3.1.89 \(\int \frac {e^{14 x/5} (-360-165 x-15 x^2)+(45 x+15 x^2+e^{14 x/5} (888 x+447 x^2+42 x^3)) \log (x)+(-360-165 x-15 x^2+(-120 x-15 x^2) \log (x)) \log (8+x)}{(360 x+285 x^2+70 x^3+5 x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {3 \left (e^{14 x/5}+\log (8+x)\right )}{(3+x) \log (x)} \]

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Rubi [F]  time = 4.64, 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^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((14*x)/5)*(-360 - 165*x - 15*x^2) + (45*x + 15*x^2 + E^((14*x)/5)*(888*x + 447*x^2 + 42*x^3))*Log[x] +
(-360 - 165*x - 15*x^2 + (-120*x - 15*x^2)*Log[x])*Log[8 + x])/((360*x + 285*x^2 + 70*x^3 + 5*x^4)*Log[x]^2),
x]

[Out]

-Defer[Int][E^((14*x)/5)/(x*Log[x]^2), x] + Defer[Int][E^((14*x)/5)/((3 + x)*Log[x]^2), x] - 3*Defer[Int][E^((
14*x)/5)/((3 + x)^2*Log[x]), x] + (42*Defer[Int][E^((14*x)/5)/((3 + x)*Log[x]), x])/5 + 9*Defer[Int][1/((3 + x
)^2*(8 + x)*Log[x]), x] + 3*Defer[Int][x/((3 + x)^2*(8 + x)*Log[x]), x] - Defer[Int][Log[8 + x]/(x*Log[x]^2),
x] + Defer[Int][Log[8 + x]/((3 + x)*Log[x]^2), x] - 3*Defer[Int][Log[8 + x]/((3 + x)^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{x \left (360+285 x+70 x^2+5 x^3\right ) \log ^2(x)} \, dx\\ &=\int \left (\frac {9}{(3+x)^2 (8+x) \log (x)}+\frac {3 x}{(3+x)^2 (8+x) \log (x)}+\frac {3 e^{14 x/5} \left (-15-5 x+37 x \log (x)+14 x^2 \log (x)\right )}{5 x (3+x)^2 \log ^2(x)}-\frac {3 (3+x+x \log (x)) \log (8+x)}{x (3+x)^2 \log ^2(x)}\right ) \, dx\\ &=\frac {3}{5} \int \frac {e^{14 x/5} \left (-15-5 x+37 x \log (x)+14 x^2 \log (x)\right )}{x (3+x)^2 \log ^2(x)} \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx-3 \int \frac {(3+x+x \log (x)) \log (8+x)}{x (3+x)^2 \log ^2(x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx\\ &=\frac {3}{5} \int \left (-\frac {5 e^{14 x/5}}{x (3+x) \log ^2(x)}+\frac {e^{14 x/5} (37+14 x)}{(3+x)^2 \log (x)}\right ) \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx-3 \int \left (\frac {(3+x+x \log (x)) \log (8+x)}{9 x \log ^2(x)}-\frac {(3+x+x \log (x)) \log (8+x)}{3 (3+x)^2 \log ^2(x)}-\frac {(3+x+x \log (x)) \log (8+x)}{9 (3+x) \log ^2(x)}\right ) \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx\\ &=-\left (\frac {1}{3} \int \frac {(3+x+x \log (x)) \log (8+x)}{x \log ^2(x)} \, dx\right )+\frac {1}{3} \int \frac {(3+x+x \log (x)) \log (8+x)}{(3+x) \log ^2(x)} \, dx+\frac {3}{5} \int \frac {e^{14 x/5} (37+14 x)}{(3+x)^2 \log (x)} \, dx-3 \int \frac {e^{14 x/5}}{x (3+x) \log ^2(x)} \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx+\int \frac {(3+x+x \log (x)) \log (8+x)}{(3+x)^2 \log ^2(x)} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {\log (8+x)}{\log ^2(x)}+\frac {3 \log (8+x)}{x \log ^2(x)}+\frac {\log (8+x)}{\log (x)}\right ) \, dx\right )+\frac {1}{3} \int \left (\frac {3 \log (8+x)}{(3+x) \log ^2(x)}+\frac {x \log (8+x)}{(3+x) \log ^2(x)}+\frac {x \log (8+x)}{(3+x) \log (x)}\right ) \, dx+\frac {3}{5} \int \left (-\frac {5 e^{14 x/5}}{(3+x)^2 \log (x)}+\frac {14 e^{14 x/5}}{(3+x) \log (x)}\right ) \, dx-3 \int \left (\frac {e^{14 x/5}}{3 x \log ^2(x)}-\frac {e^{14 x/5}}{3 (3+x) \log ^2(x)}\right ) \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx+\int \left (\frac {3 \log (8+x)}{(3+x)^2 \log ^2(x)}+\frac {x \log (8+x)}{(3+x)^2 \log ^2(x)}+\frac {x \log (8+x)}{(3+x)^2 \log (x)}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {\log (8+x)}{\log ^2(x)} \, dx\right )+\frac {1}{3} \int \frac {x \log (8+x)}{(3+x) \log ^2(x)} \, dx-\frac {1}{3} \int \frac {\log (8+x)}{\log (x)} \, dx+\frac {1}{3} \int \frac {x \log (8+x)}{(3+x) \log (x)} \, dx-3 \int \frac {e^{14 x/5}}{(3+x)^2 \log (x)} \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx+3 \int \frac {\log (8+x)}{(3+x)^2 \log ^2(x)} \, dx+\frac {42}{5} \int \frac {e^{14 x/5}}{(3+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx-\int \frac {e^{14 x/5}}{x \log ^2(x)} \, dx+\int \frac {e^{14 x/5}}{(3+x) \log ^2(x)} \, dx-\int \frac {\log (8+x)}{x \log ^2(x)} \, dx+\int \frac {x \log (8+x)}{(3+x)^2 \log ^2(x)} \, dx+\int \frac {\log (8+x)}{(3+x) \log ^2(x)} \, dx+\int \frac {x \log (8+x)}{(3+x)^2 \log (x)} \, dx\\ &=-\left (\frac {1}{3} \int \frac {\log (8+x)}{\log ^2(x)} \, dx\right )-\frac {1}{3} \int \frac {\log (8+x)}{\log (x)} \, dx+\frac {1}{3} \int \left (\frac {\log (8+x)}{\log ^2(x)}-\frac {3 \log (8+x)}{(3+x) \log ^2(x)}\right ) \, dx+\frac {1}{3} \int \left (\frac {\log (8+x)}{\log (x)}-\frac {3 \log (8+x)}{(3+x) \log (x)}\right ) \, dx-3 \int \frac {e^{14 x/5}}{(3+x)^2 \log (x)} \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx+3 \int \frac {\log (8+x)}{(3+x)^2 \log ^2(x)} \, dx+\frac {42}{5} \int \frac {e^{14 x/5}}{(3+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx-\int \frac {e^{14 x/5}}{x \log ^2(x)} \, dx+\int \frac {e^{14 x/5}}{(3+x) \log ^2(x)} \, dx-\int \frac {\log (8+x)}{x \log ^2(x)} \, dx+\int \frac {\log (8+x)}{(3+x) \log ^2(x)} \, dx+\int \left (-\frac {3 \log (8+x)}{(3+x)^2 \log ^2(x)}+\frac {\log (8+x)}{(3+x) \log ^2(x)}\right ) \, dx+\int \left (-\frac {3 \log (8+x)}{(3+x)^2 \log (x)}+\frac {\log (8+x)}{(3+x) \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {e^{14 x/5}}{(3+x)^2 \log (x)} \, dx\right )+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx-3 \int \frac {\log (8+x)}{(3+x)^2 \log (x)} \, dx+\frac {42}{5} \int \frac {e^{14 x/5}}{(3+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx-\int \frac {e^{14 x/5}}{x \log ^2(x)} \, dx+\int \frac {e^{14 x/5}}{(3+x) \log ^2(x)} \, dx-\int \frac {\log (8+x)}{x \log ^2(x)} \, dx+\int \frac {\log (8+x)}{(3+x) \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.51, size = 23, normalized size = 1.00 \begin {gather*} \frac {3 \left (e^{14 x/5}+\log (8+x)\right )}{(3+x) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((14*x)/5)*(-360 - 165*x - 15*x^2) + (45*x + 15*x^2 + E^((14*x)/5)*(888*x + 447*x^2 + 42*x^3))*Lo
g[x] + (-360 - 165*x - 15*x^2 + (-120*x - 15*x^2)*Log[x])*Log[8 + x])/((360*x + 285*x^2 + 70*x^3 + 5*x^4)*Log[
x]^2),x]

[Out]

(3*(E^((14*x)/5) + Log[8 + x]))/((3 + x)*Log[x])

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fricas [A]  time = 0.72, size = 20, normalized size = 0.87 \begin {gather*} \frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{{\left (x + 3\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447*x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15
*x^2+45*x)*log(x)+(-15*x^2-165*x-360)*exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorith
m="fricas")

[Out]

3*(e^(14/5*x) + log(x + 8))/((x + 3)*log(x))

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giac [A]  time = 0.42, size = 22, normalized size = 0.96 \begin {gather*} \frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{x \log \relax (x) + 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447*x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15
*x^2+45*x)*log(x)+(-15*x^2-165*x-360)*exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorith
m="giac")

[Out]

3*(e^(14/5*x) + log(x + 8))/(x*log(x) + 3*log(x))

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maple [A]  time = 0.05, size = 32, normalized size = 1.39

method	result	size
risch	\(\frac {3 \ln \left (x +8\right )}{\left (3+x \right ) \ln \relax (x )}+\frac {3 \,{\mathrm e}^{\frac {14 x}{5}}}{\left (3+x \right ) \ln \relax (x )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-15*x^2-120*x)*ln(x)-15*x^2-165*x-360)*ln(x+8)+((42*x^3+447*x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*
x)*ln(x)+(-15*x^2-165*x-360)*exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/ln(x)^2,x,method=_RETURNVERBO
SE)

[Out]

3/(3+x)/ln(x)*ln(x+8)+3*exp(14/5*x)/(3+x)/ln(x)

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maxima [A]  time = 0.71, size = 20, normalized size = 0.87 \begin {gather*} \frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{{\left (x + 3\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447*x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15
*x^2+45*x)*log(x)+(-15*x^2-165*x-360)*exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorith
m="maxima")

[Out]

3*(e^(14/5*x) + log(x + 8))/((x + 3)*log(x))

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mupad [B]  time = 0.70, size = 20, normalized size = 0.87 \begin {gather*} \frac {3\,\left ({\mathrm {e}}^{\frac {14\,x}{5}}+\ln \left (x+8\right )\right )}{\ln \relax (x)\,\left (x+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x + 8)*(165*x + log(x)*(120*x + 15*x^2) + 15*x^2 + 360) - log(x)*(45*x + 15*x^2 + exp(2*x)*exp((4*x)
/5)*(888*x + 447*x^2 + 42*x^3)) + exp(2*x)*exp((4*x)/5)*(165*x + 15*x^2 + 360))/(log(x)^2*(360*x + 285*x^2 + 7
0*x^3 + 5*x^4)),x)

[Out]

(3*(exp((14*x)/5) + log(x + 8)))/(log(x)*(x + 3))

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sympy [A]  time = 0.61, size = 34, normalized size = 1.48 \begin {gather*} \frac {3 e^{\frac {14 x}{5}}}{x \log {\relax (x )} + 3 \log {\relax (x )}} + \frac {3 \log {\left (x + 8 \right )}}{x \log {\relax (x )} + 3 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x**2-120*x)*ln(x)-15*x**2-165*x-360)*ln(x+8)+((42*x**3+447*x**2+888*x)*exp(2/5*x)**2*exp(x)**
2+15*x**2+45*x)*ln(x)+(-15*x**2-165*x-360)*exp(2/5*x)**2*exp(x)**2)/(5*x**4+70*x**3+285*x**2+360*x)/ln(x)**2,x
)

[Out]

3*exp(14*x/5)/(x*log(x) + 3*log(x)) + 3*log(x + 8)/(x*log(x) + 3*log(x))

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