\(\int (6 e^{3 x}+\frac {6 x^2}{e^5}) \, dx\) [6899]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \left (6 e^{3 x}+\frac {6 x^2}{e^5}\right ) \, dx=2 \left (1+e^{3 x}+\frac {x^3}{e^5}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2225} \[ \int \left (6 e^{3 x}+\frac {6 x^2}{e^5}\right ) \, dx=\frac {2 x^3}{e^5}+2 e^{3 x} \]

[In]

Int[6*E^(3*x) + (6*x^2)/E^5,x]

[Out]

2*E^(3*x) + (2*x^3)/E^5

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {2 x^3}{e^5}+6 \int e^{3 x} \, dx \\ & = 2 e^{3 x}+\frac {2 x^3}{e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \left (6 e^{3 x}+\frac {6 x^2}{e^5}\right ) \, dx=6 \left (\frac {e^{3 x}}{3}+\frac {x^3}{3 e^5}\right ) \]

[In]

Integrate[6*E^(3*x) + (6*x^2)/E^5,x]

[Out]

6*(E^(3*x)/3 + x^3/(3*E^5))

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
risch	\(2 \,{\mathrm e}^{-5} x^{3}+2 \,{\mathrm e}^{3 x}\)	\(15\)
norman	\(2 \,{\mathrm e}^{-5} x^{3}+2 \,{\mathrm e}^{3 x}\)	\(17\)
default	\(2 \,{\mathrm e}^{\ln \left (x \right )-5} x^{2}+2 \,{\mathrm e}^{3 x}\)	\(18\)
parts	\(2 \,{\mathrm e}^{\ln \left (x \right )-5} x^{2}+2 \,{\mathrm e}^{3 x}\)	\(18\)
parallelrisch	\(-\frac {-2 x^{3} {\mathrm e}^{\ln \left (x \right )-5}-2 x \,{\mathrm e}^{3 x}}{x}\)	\(24\)

[In]

int(6*x*exp(ln(x)-5)+6*exp(3*x),x,method=_RETURNVERBOSE)

[Out]

2*exp(-5)*x^3+2*exp(3*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (6 e^{3 x}+\frac {6 x^2}{e^5}\right ) \, dx=2 \, {\left (x^{3} + e^{\left (3 \, x + 5\right )}\right )} e^{\left (-5\right )} \]

[In]

integrate(6*x*exp(log(x)-5)+6*exp(3*x),x, algorithm="fricas")

[Out]

2*(x^3 + e^(3*x + 5))*e^(-5)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (6 e^{3 x}+\frac {6 x^2}{e^5}\right ) \, dx=\frac {2 x^{3}}{e^{5}} + 2 e^{3 x} \]

[In]

integrate(6*x*exp(ln(x)-5)+6*exp(3*x),x)

[Out]

2*x**3*exp(-5) + 2*exp(3*x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (6 e^{3 x}+\frac {6 x^2}{e^5}\right ) \, dx=2 \, x^{3} e^{\left (-5\right )} + 2 \, e^{\left (3 \, x\right )} \]

[In]

integrate(6*x*exp(log(x)-5)+6*exp(3*x),x, algorithm="maxima")

[Out]

2*x^3*e^(-5) + 2*e^(3*x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (6 e^{3 x}+\frac {6 x^2}{e^5}\right ) \, dx=2 \, x^{3} e^{\left (-5\right )} + 2 \, e^{\left (3 \, x\right )} \]

[In]

integrate(6*x*exp(log(x)-5)+6*exp(3*x),x, algorithm="giac")

[Out]

2*x^3*e^(-5) + 2*e^(3*x)

Mupad [B] (verification not implemented)

Time = 11.74 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (6 e^{3 x}+\frac {6 x^2}{e^5}\right ) \, dx=2\,{\mathrm {e}}^{3\,x}+2\,x^3\,{\mathrm {e}}^{-5} \]

[In]

int(6*exp(3*x) + 6*x*exp(log(x) - 5),x)

[Out]

2*exp(3*x) + 2*x^3*exp(-5)