3.77.50 \(\int \frac {e^3 (5 x-16 x^2+x^3+2 x^4+(-15-5 x) \log (9+3 x)+\frac {(-x^2+\log (9+3 x)) (e^3 (-75 x^2+5 x^3+7 x^4-x^5)+e^3 (75-5 x-7 x^2+x^3) \log (9+3 x))}{e^3})}{(-x^2+\log (9+3 x)) (-75 x^2+5 x^3+7 x^4-x^5+(75-5 x-7 x^2+x^3) \log (9+3 x))} \, dx\)

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

________________________________________________________________________________________

Rubi [F]  time = 50.46, 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^3 \left (5 x-16 x^2+x^3+2 x^4+(-15-5 x) \log (9+3 x)+\frac {\left (-x^2+\log (9+3 x)\right ) \left (e^3 \left (-75 x^2+5 x^3+7 x^4-x^5\right )+e^3 \left (75-5 x-7 x^2+x^3\right ) \log (9+3 x)\right )}{e^3}\right )}{\left (-x^2+\log (9+3 x)\right ) \left (-75 x^2+5 x^3+7 x^4-x^5+\left (75-5 x-7 x^2+x^3\right ) \log (9+3 x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^3*(5*x - 16*x^2 + x^3 + 2*x^4 + (-15 - 5*x)*Log[9 + 3*x] + ((-x^2 + Log[9 + 3*x])*(E^3*(-75*x^2 + 5*x^3
+ 7*x^4 - x^5) + E^3*(75 - 5*x - 7*x^2 + x^3)*Log[9 + 3*x]))/E^3))/((-x^2 + Log[9 + 3*x])*(-75*x^2 + 5*x^3 +
7*x^4 - x^5 + (75 - 5*x - 7*x^2 + x^3)*Log[9 + 3*x])),x]

[Out]

E^3*x - (68645*E^3*Defer[Int][(x^2 - Log[3*(3 + x)])^(-2), x])/64 - (47375*E^3*Defer[Int][1/((-5 + x)^2*(x^2 -
Log[3*(3 + x)])^2), x])/4 - (494825*E^3*Defer[Int][1/((-5 + x)*(x^2 - Log[3*(3 + x)])^2), x])/64 - (7665*E^3*
Defer[Int][x/(x^2 - Log[3*(3 + x)])^2, x])/64 - (325*E^3*Defer[Int][x^2/(x^2 - Log[3*(3 + x)])^2, x])/64 + (17
5*E^3*Defer[Int][x^3/(x^2 - Log[3*(3 + x)])^2, x])/64 + (35*E^3*Defer[Int][x^4/(x^2 - Log[3*(3 + x)])^2, x])/6
4 + (7*E^3*Defer[Int][x^5/(x^2 - Log[3*(3 + x)])^2, x])/64 - (2025*E^3*Defer[Int][(x^2 - Log[3*(3 + x)])^(-1),
x])/32 - (605*E^3*Defer[Int][1/((-5 + x)^2*(x^2 - Log[3*(3 + x)])), x])/4 - (22225*E^3*Defer[Int][1/((-5 + x)
*(x^2 - Log[3*(3 + x)])), x])/64 - (365*E^3*Defer[Int][x/(x^2 - Log[3*(3 + x)]), x])/32 - (9*E^3*Defer[Int][x^
2/(x^2 - Log[3*(3 + x)]), x])/32 - (3*E^3*Defer[Int][x^3/(x^2 - Log[3*(3 + x)]), x])/8 - (E^3*Defer[Int][x^4/(
x^2 - Log[3*(3 + x)]), x])/32 + (69285*E^3*Defer[Int][(x^2 - Log[9 + 3*x])^(-2), x])/64 + (47375*E^3*Defer[Int
][1/((-5 + x)^2*(x^2 - Log[9 + 3*x])^2), x])/4 + (497985*E^3*Defer[Int][1/((-5 + x)*(x^2 - Log[9 + 3*x])^2), x
])/64 + (7793*E^3*Defer[Int][x/(x^2 - Log[9 + 3*x])^2, x])/64 + (325*E^3*Defer[Int][x^2/(x^2 - Log[9 + 3*x])^2
, x])/64 - (175*E^3*Defer[Int][x^3/(x^2 - Log[9 + 3*x])^2, x])/64 - (35*E^3*Defer[Int][x^4/(x^2 - Log[9 + 3*x]
)^2, x])/64 - (7*E^3*Defer[Int][x^5/(x^2 - Log[9 + 3*x])^2, x])/64 - (3*E^3*Defer[Int][1/((3 + x)*(x^2 - Log[9
+ 3*x])^2), x])/8 + (2025*E^3*Defer[Int][(x^2 - Log[9 + 3*x])^(-1), x])/32 + (625*E^3*Defer[Int][1/((-5 + x)^
2*(x^2 - Log[9 + 3*x])), x])/4 + (22225*E^3*Defer[Int][1/((-5 + x)*(x^2 - Log[9 + 3*x])), x])/64 + (365*E^3*De
fer[Int][x/(x^2 - Log[9 + 3*x]), x])/32 + (9*E^3*Defer[Int][x^2/(x^2 - Log[9 + 3*x]), x])/32 + (3*E^3*Defer[In
t][x^3/(x^2 - Log[9 + 3*x]), x])/8 + (E^3*Defer[Int][x^4/(x^2 - Log[9 + 3*x]), x])/32

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^3 \int \frac {5 x-16 x^2+x^3+2 x^4+(-15-5 x) \log (9+3 x)+\frac {\left (-x^2+\log (9+3 x)\right ) \left (e^3 \left (-75 x^2+5 x^3+7 x^4-x^5\right )+e^3 \left (75-5 x-7 x^2+x^3\right ) \log (9+3 x)\right )}{e^3}}{\left (-x^2+\log (9+3 x)\right ) \left (-75 x^2+5 x^3+7 x^4-x^5+\left (75-5 x-7 x^2+x^3\right ) \log (9+3 x)\right )} \, dx\\ &=e^3 \int \frac {x \left (5-16 x+x^2+77 x^3-5 x^4-7 x^5+x^6\right )-\left (15+5 x+150 x^2-10 x^3-14 x^4+2 x^5\right ) \log (3 (3+x))+(-5+x)^2 (3+x) \log ^2(3 (3+x))}{(5-x)^2 (3+x) \left (x^2-\log (9+3 x)\right )^2} \, dx\\ &=e^3 \int \left (\frac {-5 x+16 x^2-x^3-77 x^4+5 x^5+7 x^6-x^7+15 \log (3 (3+x))+5 x \log (3 (3+x))+150 x^2 \log (3 (3+x))-10 x^3 \log (3 (3+x))-14 x^4 \log (3 (3+x))+2 x^5 \log (3 (3+x))-75 \log ^2(3 (3+x))+5 x \log ^2(3 (3+x))+7 x^2 \log ^2(3 (3+x))-x^3 \log ^2(3 (3+x))}{64 (-5+x) \left (x^2-\log (9+3 x)\right )^2}+\frac {5 x-16 x^2+x^3+77 x^4-5 x^5-7 x^6+x^7-15 \log (3 (3+x))-5 x \log (3 (3+x))-150 x^2 \log (3 (3+x))+10 x^3 \log (3 (3+x))+14 x^4 \log (3 (3+x))-2 x^5 \log (3 (3+x))+75 \log ^2(3 (3+x))-5 x \log ^2(3 (3+x))-7 x^2 \log ^2(3 (3+x))+x^3 \log ^2(3 (3+x))}{8 (-5+x)^2 \left (x^2-\log (9+3 x)\right )^2}+\frac {5 x-16 x^2+x^3+77 x^4-5 x^5-7 x^6+x^7-15 \log (3 (3+x))-5 x \log (3 (3+x))-150 x^2 \log (3 (3+x))+10 x^3 \log (3 (3+x))+14 x^4 \log (3 (3+x))-2 x^5 \log (3 (3+x))+75 \log ^2(3 (3+x))-5 x \log ^2(3 (3+x))-7 x^2 \log ^2(3 (3+x))+x^3 \log ^2(3 (3+x))}{64 (3+x) \left (x^2-\log (9+3 x)\right )^2}\right ) \, dx\\ &=\frac {1}{64} e^3 \int \frac {-5 x+16 x^2-x^3-77 x^4+5 x^5+7 x^6-x^7+15 \log (3 (3+x))+5 x \log (3 (3+x))+150 x^2 \log (3 (3+x))-10 x^3 \log (3 (3+x))-14 x^4 \log (3 (3+x))+2 x^5 \log (3 (3+x))-75 \log ^2(3 (3+x))+5 x \log ^2(3 (3+x))+7 x^2 \log ^2(3 (3+x))-x^3 \log ^2(3 (3+x))}{(-5+x) \left (x^2-\log (9+3 x)\right )^2} \, dx+\frac {1}{64} e^3 \int \frac {5 x-16 x^2+x^3+77 x^4-5 x^5-7 x^6+x^7-15 \log (3 (3+x))-5 x \log (3 (3+x))-150 x^2 \log (3 (3+x))+10 x^3 \log (3 (3+x))+14 x^4 \log (3 (3+x))-2 x^5 \log (3 (3+x))+75 \log ^2(3 (3+x))-5 x \log ^2(3 (3+x))-7 x^2 \log ^2(3 (3+x))+x^3 \log ^2(3 (3+x))}{(3+x) \left (x^2-\log (9+3 x)\right )^2} \, dx+\frac {1}{8} e^3 \int \frac {5 x-16 x^2+x^3+77 x^4-5 x^5-7 x^6+x^7-15 \log (3 (3+x))-5 x \log (3 (3+x))-150 x^2 \log (3 (3+x))+10 x^3 \log (3 (3+x))+14 x^4 \log (3 (3+x))-2 x^5 \log (3 (3+x))+75 \log ^2(3 (3+x))-5 x \log ^2(3 (3+x))-7 x^2 \log ^2(3 (3+x))+x^3 \log ^2(3 (3+x))}{(-5+x)^2 \left (x^2-\log (9+3 x)\right )^2} \, dx\\ &=\frac {1}{64} e^3 \int \frac {x \left (5-16 x+x^2+77 x^3-5 x^4-7 x^5+x^6\right )-\left (15+5 x+150 x^2-10 x^3-14 x^4+2 x^5\right ) \log (3 (3+x))+(-5+x)^2 (3+x) \log ^2(3 (3+x))}{(5-x) \left (x^2-\log (9+3 x)\right )^2} \, dx+\frac {1}{64} e^3 \int \frac {x \left (5-16 x+x^2+77 x^3-5 x^4-7 x^5+x^6\right )-\left (15+5 x+150 x^2-10 x^3-14 x^4+2 x^5\right ) \log (3 (3+x))+(-5+x)^2 (3+x) \log ^2(3 (3+x))}{(3+x) \left (x^2-\log (9+3 x)\right )^2} \, dx+\frac {1}{8} e^3 \int \frac {x \left (5-16 x+x^2+77 x^3-5 x^4-7 x^5+x^6\right )-\left (15+5 x+150 x^2-10 x^3-14 x^4+2 x^5\right ) \log (3 (3+x))+(-5+x)^2 (3+x) \log ^2(3 (3+x))}{(5-x)^2 \left (x^2-\log (9+3 x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 27, normalized size = 0.90 \begin {gather*} e^3 \left (x+\frac {x}{(-5+x) \left (-x^2+\log (3 (3+x))\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^3*(5*x - 16*x^2 + x^3 + 2*x^4 + (-15 - 5*x)*Log[9 + 3*x] + ((-x^2 + Log[9 + 3*x])*(E^3*(-75*x^2 +
5*x^3 + 7*x^4 - x^5) + E^3*(75 - 5*x - 7*x^2 + x^3)*Log[9 + 3*x]))/E^3))/((-x^2 + Log[9 + 3*x])*(-75*x^2 + 5*
x^3 + 7*x^4 - x^5 + (75 - 5*x - 7*x^2 + x^3)*Log[9 + 3*x])),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 57, normalized size = 1.90 \begin {gather*} -\frac {{\left (x^{2} - 5 \, x\right )} e^{3} \log \left (3 \, x + 9\right ) - {\left (x^{4} - 5 \, x^{3} - x\right )} e^{3}}{x^{3} - 5 \, x^{2} - {\left (x - 5\right )} \log \left (3 \, x + 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-7*x^2-5*x+75)*exp(3)*log(3*x+9)+(-x^5+7*x^4+5*x^3-75*x^2)*exp(3))*exp(log(log(3*x+9)-x^2)-3)+
(-5*x-15)*log(3*x+9)+2*x^4+x^3-16*x^2+5*x)/((x^3-7*x^2-5*x+75)*log(3*x+9)-x^5+7*x^4+5*x^3-75*x^2)/exp(log(log(
3*x+9)-x^2)-3),x, algorithm="fricas")

[Out]

-((x^2 - 5*x)*e^3*log(3*x + 9) - (x^4 - 5*x^3 - x)*e^3)/(x^3 - 5*x^2 - (x - 5)*log(3*x + 9))

________________________________________________________________________________________

giac [B]  time = 0.20, size = 89, normalized size = 2.97 \begin {gather*} \frac {x^{4} e^{3} - 5 \, x^{3} e^{3} - x^{2} e^{3} \log \relax (3) - x^{2} e^{3} \log \left (x + 3\right ) + 5 \, x e^{3} \log \relax (3) + 5 \, x e^{3} \log \left (x + 3\right ) - x e^{3}}{x^{3} - 5 \, x^{2} - x \log \relax (3) - x \log \left (x + 3\right ) + 5 \, \log \relax (3) + 5 \, \log \left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-7*x^2-5*x+75)*exp(3)*log(3*x+9)+(-x^5+7*x^4+5*x^3-75*x^2)*exp(3))*exp(log(log(3*x+9)-x^2)-3)+
(-5*x-15)*log(3*x+9)+2*x^4+x^3-16*x^2+5*x)/((x^3-7*x^2-5*x+75)*log(3*x+9)-x^5+7*x^4+5*x^3-75*x^2)/exp(log(log(
3*x+9)-x^2)-3),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.41, size = 30, normalized size = 1.00

method	result	size
risch	\(x \,{\mathrm e}^{3}-\frac {x \,{\mathrm e}^{3}}{\left (x^{2}-\ln \left (3 x +9\right )\right ) \left (x -5\right )}\)	\(30\)
default	\(\frac {\left (3 x +9\right ) {\mathrm e}^{3}}{3}-\frac {x \,{\mathrm e}^{3}}{\left (x^{2}-\ln \left (3 x +9\right )\right ) \left (x -5\right )}\)	\(35\)
norman	\(\frac {x^{4} {\mathrm e}^{3}-25 x^{2} {\mathrm e}^{3}+25 \,{\mathrm e}^{3} \ln \left (3 x +9\right )-{\mathrm e}^{3} x^{2} \ln \left (3 x +9\right )-x \,{\mathrm e}^{3}}{\left (x^{2}-\ln \left (3 x +9\right )\right ) \left (x -5\right )}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3-7*x^2-5*x+75)*exp(3)*ln(3*x+9)+(-x^5+7*x^4+5*x^3-75*x^2)*exp(3))*exp(ln(ln(3*x+9)-x^2)-3)+(-5*x-15)
*ln(3*x+9)+2*x^4+x^3-16*x^2+5*x)/((x^3-7*x^2-5*x+75)*ln(3*x+9)-x^5+7*x^4+5*x^3-75*x^2)/exp(ln(ln(3*x+9)-x^2)-3
),x,method=_RETURNVERBOSE)

[Out]

x*exp(3)-x*exp(3)/(x^2-ln(3*x+9))/(x-5)

________________________________________________________________________________________

maxima [B]  time = 0.48, size = 69, normalized size = 2.30 \begin {gather*} \frac {{\left (x^{4} - 5 \, x^{3} - x^{2} \log \relax (3) + x {\left (5 \, \log \relax (3) - 1\right )} - {\left (x^{2} - 5 \, x\right )} \log \left (x + 3\right )\right )} e^{3}}{x^{3} - 5 \, x^{2} - x \log \relax (3) - {\left (x - 5\right )} \log \left (x + 3\right ) + 5 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-7*x^2-5*x+75)*exp(3)*log(3*x+9)+(-x^5+7*x^4+5*x^3-75*x^2)*exp(3))*exp(log(log(3*x+9)-x^2)-3)+
(-5*x-15)*log(3*x+9)+2*x^4+x^3-16*x^2+5*x)/((x^3-7*x^2-5*x+75)*log(3*x+9)-x^5+7*x^4+5*x^3-75*x^2)/exp(log(log(
3*x+9)-x^2)-3),x, algorithm="maxima")

[Out]

(x^4 - 5*x^3 - x^2*log(3) + x*(5*log(3) - 1) - (x^2 - 5*x)*log(x + 3))*e^3/(x^3 - 5*x^2 - x*log(3) - (x - 5)*l
og(x + 3) + 5*log(3))

________________________________________________________________________________________

mupad [B]  time = 4.79, size = 51, normalized size = 1.70 \begin {gather*} \frac {x\,{\mathrm {e}}^3\,\left (x\,\ln \left (3\,x+9\right )-5\,\ln \left (3\,x+9\right )+5\,x^2-x^3+1\right )}{\left (\ln \left (3\,x+9\right )-x^2\right )\,\left (x-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3 - log(log(3*x + 9) - x^2))*(5*x - log(3*x + 9)*(5*x + 15) - exp(log(log(3*x + 9) - x^2) - 3)*(exp(
3)*(75*x^2 - 5*x^3 - 7*x^4 + x^5) + exp(3)*log(3*x + 9)*(5*x + 7*x^2 - x^3 - 75)) - 16*x^2 + x^3 + 2*x^4))/(lo
g(3*x + 9)*(5*x + 7*x^2 - x^3 - 75) + 75*x^2 - 5*x^3 - 7*x^4 + x^5),x)

[Out]

(x*exp(3)*(x*log(3*x + 9) - 5*log(3*x + 9) + 5*x^2 - x^3 + 1))/((log(3*x + 9) - x^2)*(x - 5))

________________________________________________________________________________________

sympy [A]  time = 0.23, size = 27, normalized size = 0.90 \begin {gather*} x e^{3} + \frac {x e^{3}}{- x^{3} + 5 x^{2} + \left (x - 5\right ) \log {\left (3 x + 9 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3-7*x**2-5*x+75)*exp(3)*ln(3*x+9)+(-x**5+7*x**4+5*x**3-75*x**2)*exp(3))*exp(ln(ln(3*x+9)-x**2)
-3)+(-5*x-15)*ln(3*x+9)+2*x**4+x**3-16*x**2+5*x)/((x**3-7*x**2-5*x+75)*ln(3*x+9)-x**5+7*x**4+5*x**3-75*x**2)/e
xp(ln(ln(3*x+9)-x**2)-3),x)

[Out]

x*exp(3) + x*exp(3)/(-x**3 + 5*x**2 + (x - 5)*log(3*x + 9))

________________________________________________________________________________________