3.537 \(\int a^{3 x} x^2 \, dx\)
Optimal. Leaf size=44 \[ \frac{x^2 a^{3 x}}{3 \log (a)}-\frac{2 x a^{3 x}}{9 \log ^2(a)}+\frac{2 a^{3 x}}{27 \log ^3(a)} \]
[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])
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Rubi [A] time = 0.0215737, antiderivative size = 44, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used =
{2176, 2194} \[ \frac{x^2 a^{3 x}}{3 \log (a)}-\frac{2 x a^{3 x}}{9 \log ^2(a)}+\frac{2 a^{3 x}}{27 \log ^3(a)} \]
Antiderivative was successfully verified.
[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 2176
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] && !$UseGamma === True
Rule 2194
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*} \int a^{3 x} x^2 \, dx &=\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] time = 0.010054, size = 29, normalized size = 0.66 \[ \frac{a^{3 x} \left (9 x^2 \log ^2(a)-6 x \log (a)+2\right )}{27 \log ^3(a)} \]
Antiderivative was successfully verified.
[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)
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Maple [A] time = 0.006, size = 28, normalized size = 0.6 \begin{align*}{\frac{ \left ( 9\,{x}^{2} \left ( \ln \left ( a \right ) \right ) ^{2}-6\,x\ln \left ( a \right ) +2 \right ){a}^{3\,x}}{27\, \left ( \ln \left ( a \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(a^(3*x)*x^2,x)
[Out]
1/27*(9*x^2*ln(a)^2-6*x*ln(a)+2)*a^(3*x)/ln(a)^3
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Maxima [A] time = 0.94159, size = 36, normalized size = 0.82 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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Fricas [A] time = 1.90109, size = 77, normalized size = 1.75 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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Sympy [A] time = 0.108499, size = 39, normalized size = 0.89 \begin{align*} \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}\: 27 \log{\left (a \right )}^{3} \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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(27*log(a)**3, 0)), (x**3/3, True))
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Giac [C] time = 1.18254, size = 1115, normalized size = 25.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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) + 1/2*I*abs(a)^(3*x)*((36*I*pi^2*x^2*sgn(a) - 72*pi*x^2*log
(abs(a))*sgn(a) - 36*I*pi^2*x^2 + 72*pi*x^2*log(abs(a)) + 72*I*x^2*log(abs(a))^2 + 24*pi*x*sgn(a) - 24*pi*x -
48*I*x*log(abs(a)) + 16*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) - (36*I*pi^2*x^2*sgn(a) + 72*pi*x^2*log(abs(a))*sgn(a) - 36*I*pi^2*x^2 - 72*pi*x^2*log(abs(a)) + 72*I*x^
2*log(abs(a))^2 - 24*pi*x*sgn(a) + 24*pi*x - 48*I*x*log(abs(a)) + 16*I)*e^(-3/2*I*pi*x*sgn(a) + 3/2*I*pi*x)/(1
08*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))