\(\int a^{3 x} x^2 \, dx\) [537]

Optimal result

Integrand size = 9, antiderivative size = 44 \[ \int a^{3 x} x^2 \, dx=\frac {2 a^{3 x}}{27 \log ^3(a)}-\frac {2 a^{3 x} x}{9 \log ^2(a)}+\frac {a^{3 x} x^2}{3 \log (a)} \]

[Out]

2/27*a^(3*x)/ln(a)^3-2/9*a^(3*x)*x/ln(a)^2+1/3*a^(3*x)*x^2/ln(a)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 2225} \[ \int a^{3 x} x^2 \, dx=\frac {x^2 a^{3 x}}{3 \log (a)}+\frac {2 a^{3 x}}{27 \log ^3(a)}-\frac {2 x a^{3 x}}{9 \log ^2(a)} \]

[In]

Int[a^(3*x)*x^2,x]

[Out]

(2*a^(3*x))/(27*Log[a]^3) - (2*a^(3*x)*x)/(9*Log[a]^2) + (a^(3*x)*x^2)/(3*Log[a])

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

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 {a^{3 x} x^2}{3 \log (a)}-\frac {2 \int a^{3 x} x \, dx}{3 \log (a)} \\ & = -\frac {2 a^{3 x} x}{9 \log ^2(a)}+\frac {a^{3 x} x^2}{3 \log (a)}+\frac {2 \int a^{3 x} \, dx}{9 \log ^2(a)} \\ & = \frac {2 a^{3 x}}{27 \log ^3(a)}-\frac {2 a^{3 x} x}{9 \log ^2(a)}+\frac {a^{3 x} x^2}{3 \log (a)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int a^{3 x} x^2 \, dx=\frac {a^{3 x} \left (2-6 x \log (a)+9 x^2 \log ^2(a)\right )}{27 \log ^3(a)} \]

[In]

Integrate[a^(3*x)*x^2,x]

[Out]

(a^(3*x)*(2 - 6*x*Log[a] + 9*x^2*Log[a]^2))/(27*Log[a]^3)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64

method	result	size
gosper	\(\frac {\left (9 x^{2} \ln \left (a \right )^{2}-6 x \ln \left (a \right )+2\right ) a^{3 x}}{27 \ln \left (a \right )^{3}}\)	\(28\)
risch	\(\frac {\left (9 x^{2} \ln \left (a \right )^{2}-6 x \ln \left (a \right )+2\right ) a^{3 x}}{27 \ln \left (a \right )^{3}}\)	\(28\)
meijerg	\(-\frac {2-\frac {\left (27 x^{2} \ln \left (a \right )^{2}-18 x \ln \left (a \right )+6\right ) {\mathrm e}^{3 x \ln \left (a \right )}}{3}}{27 \ln \left (a \right )^{3}}\)	\(33\)
parallelrisch	\(\frac {9 x^{2} a^{3 x} \ln \left (a \right )^{2}-6 x \,a^{3 x} \ln \left (a \right )+2 a^{3 x}}{27 \ln \left (a \right )^{3}}\)	\(39\)
norman	\(\frac {2 \,{\mathrm e}^{3 x \ln \left (a \right )}}{27 \ln \left (a \right )^{3}}-\frac {2 x \,{\mathrm e}^{3 x \ln \left (a \right )}}{9 \ln \left (a \right )^{2}}+\frac {x^{2} {\mathrm e}^{3 x \ln \left (a \right )}}{3 \ln \left (a \right )}\)	\(42\)

[In]

int(a^(3*x)*x^2,x,method=_RETURNVERBOSE)

[Out]

1/27*(9*x^2*ln(a)^2-6*x*ln(a)+2)*a^(3*x)/ln(a)^3

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61 \[ \int a^{3 x} x^2 \, dx=\frac {{\left (9 \, x^{2} \log \left (a\right )^{2} - 6 \, x \log \left (a\right ) + 2\right )} a^{3 \, x}}{27 \, \log \left (a\right )^{3}} \]

[In]

integrate(a^(3*x)*x^2,x, algorithm="fricas")

[Out]

1/27*(9*x^2*log(a)^2 - 6*x*log(a) + 2)*a^(3*x)/log(a)^3

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int a^{3 x} x^2 \, dx=\begin {cases} \frac {a^{3 x} \left (9 x^{2} \log {\left (a \right )}^{2} - 6 x \log {\left (a \right )} + 2\right )}{27 \log {\left (a \right )}^{3}} & \text {for}\: \log {\left (a \right )}^{3} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \]

[In]

integrate(a**(3*x)*x**2,x)

[Out]

Piecewise((a**(3*x)*(9*x**2*log(a)**2 - 6*x*log(a) + 2)/(27*log(a)**3), Ne(log(a)**3, 0)), (x**3/3, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61 \[ \int a^{3 x} x^2 \, dx=\frac {{\left (9 \, x^{2} \log \left (a\right )^{2} - 6 \, x \log \left (a\right ) + 2\right )} a^{3 \, x}}{27 \, \log \left (a\right )^{3}} \]

[In]

integrate(a^(3*x)*x^2,x, algorithm="maxima")

[Out]

1/27*(9*x^2*log(a)^2 - 6*x*log(a) + 2)*a^(3*x)/log(a)^3

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 826, normalized size of antiderivative = 18.77 \[ \int a^{3 x} x^2 \, dx=\text {Too large to display} \]

[In]

integrate(a^(3*x)*x^2,x, algorithm="giac")

[Out]

-1/27*((6*(3*pi*x^2*log(abs(a))*sgn(a) - 3*pi*x^2*log(abs(a)) - pi*x*sgn(a) + pi*x)*(pi^3*sgn(a) - 3*pi*log(ab
s(a))^2*sgn(a) - pi^3 + 3*pi*log(abs(a))^2)/((pi^3*sgn(a) - 3*pi*log(abs(a))^2*sgn(a) - pi^3 + 3*pi*log(abs(a)
)^2)^2 + (3*pi^2*log(abs(a))*sgn(a) - 3*pi^2*log(abs(a)) + 2*log(abs(a))^3)^2) - (9*pi^2*x^2*sgn(a) - 9*pi^2*x
^2 + 18*x^2*log(abs(a))^2 - 12*x*log(abs(a)) + 4)*(3*pi^2*log(abs(a))*sgn(a) - 3*pi^2*log(abs(a)) + 2*log(abs(
a))^3)/((pi^3*sgn(a) - 3*pi*log(abs(a))^2*sgn(a) - pi^3 + 3*pi*log(abs(a))^2)^2 + (3*pi^2*log(abs(a))*sgn(a) -
3*pi^2*log(abs(a)) + 2*log(abs(a))^3)^2))*cos(-3/2*pi*x*sgn(a) + 3/2*pi*x) - ((9*pi^2*x^2*sgn(a) - 9*pi^2*x^2
+ 18*x^2*log(abs(a))^2 - 12*x*log(abs(a)) + 4)*(pi^3*sgn(a) - 3*pi*log(abs(a))^2*sgn(a) - pi^3 + 3*pi*log(abs
(a))^2)/((pi^3*sgn(a) - 3*pi*log(abs(a))^2*sgn(a) - pi^3 + 3*pi*log(abs(a))^2)^2 + (3*pi^2*log(abs(a))*sgn(a)
- 3*pi^2*log(abs(a)) + 2*log(abs(a))^3)^2) + 6*(3*pi*x^2*log(abs(a))*sgn(a) - 3*pi*x^2*log(abs(a)) - pi*x*sgn(
a) + pi*x)*(3*pi^2*log(abs(a))*sgn(a) - 3*pi^2*log(abs(a)) + 2*log(abs(a))^3)/((pi^3*sgn(a) - 3*pi*log(abs(a))
^2*sgn(a) - pi^3 + 3*pi*log(abs(a))^2)^2 + (3*pi^2*log(abs(a))*sgn(a) - 3*pi^2*log(abs(a)) + 2*log(abs(a))^3)^
2))*sin(-3/2*pi*x*sgn(a) + 3/2*pi*x))*abs(a)^(3*x) - 2*I*abs(a)^(3*x)*((-9*I*pi^2*x^2*sgn(a) + 18*pi*x^2*log(a
bs(a))*sgn(a) + 9*I*pi^2*x^2 - 18*pi*x^2*log(abs(a)) - 18*I*x^2*log(abs(a))^2 - 6*pi*x*sgn(a) + 6*pi*x + 12*I*
x*log(abs(a)) - 4*I)*e^(3/2*I*pi*x*sgn(a) - 3/2*I*pi*x)/(-108*I*pi^3*sgn(a) + 324*pi^2*log(abs(a))*sgn(a) + 32
4*I*pi*log(abs(a))^2*sgn(a) + 108*I*pi^3 - 324*pi^2*log(abs(a)) - 324*I*pi*log(abs(a))^2 + 216*log(abs(a))^3)
- (-9*I*pi^2*x^2*sgn(a) - 18*pi*x^2*log(abs(a))*sgn(a) + 9*I*pi^2*x^2 + 18*pi*x^2*log(abs(a)) - 18*I*x^2*log(a
bs(a))^2 + 6*pi*x*sgn(a) - 6*pi*x + 12*I*x*log(abs(a)) - 4*I)*e^(-3/2*I*pi*x*sgn(a) + 3/2*I*pi*x)/(108*I*pi^3*
sgn(a) + 324*pi^2*log(abs(a))*sgn(a) - 324*I*pi*log(abs(a))^2*sgn(a) - 108*I*pi^3 - 324*pi^2*log(abs(a)) + 324
*I*pi*log(abs(a))^2 + 216*log(abs(a))^3))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61 \[ \int a^{3 x} x^2 \, dx=\frac {a^{3\,x}\,\left (9\,x^2\,{\ln \left (a\right )}^2-6\,x\,\ln \left (a\right )+2\right )}{27\,{\ln \left (a\right )}^3} \]

[In]

int(a^(3*x)*x^2,x)

[Out]

(a^(3*x)*(9*x^2*log(a)^2 - 6*x*log(a) + 2))/(27*log(a)^3)