3.6.63 \(\int a^x b^x x^2 \, dx\) [563]

Optimal. Leaf size=49 \[ \frac {2 a^x b^x}{(\log (a)+\log (b))^3}-\frac {2 a^x b^x x}{(\log (a)+\log (b))^2}+\frac {a^x b^x x^2}{\log (a)+\log (b)} \]

[Out]

2*a^x*b^x/(ln(a)+ln(b))^3-2*a^x*b^x*x/(ln(a)+ln(b))^2+a^x*b^x*x^2/(ln(a)+ln(b))

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Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2325, 2207, 2225} \begin {gather*} \frac {x^2 a^x b^x}{\log (a)+\log (b)}-\frac {2 x a^x b^x}{(\log (a)+\log (b))^2}+\frac {2 a^x b^x}{(\log (a)+\log (b))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a^x*b^x*x^2,x]

[Out]

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

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]

Rule 2325

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 35, normalized size = 0.71 \begin {gather*} \frac {a^x b^x \left (2-2 x (\log (a)+\log (b))+x^2 (\log (a)+\log (b))^2\right )}{(\log (a)+\log (b))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a^x*b^x*x^2,x]

[Out]

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

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Maple [A]
time = 0.03, size = 52, normalized size = 1.06

method	result	size
risch	\(\frac {\left (\ln \left (b \right )^{2} x^{2}+2 \ln \left (b \right ) \ln \left (a \right ) x^{2}+\ln \left (a \right )^{2} x^{2}-2 \ln \left (b \right ) x -2 \ln \left (a \right ) x +2\right ) a^{x} b^{x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{3}}\)	\(52\)
gosper	\(\frac {\left (\ln \left (b \right )^{2} x^{2}+2 \ln \left (b \right ) \ln \left (a \right ) x^{2}+\ln \left (a \right )^{2} x^{2}-2 \ln \left (b \right ) x -2 \ln \left (a \right ) x +2\right ) a^{x} b^{x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right ) \left (\ln \left (b \right )^{2}+2 \ln \left (b \right ) \ln \left (a \right )+\ln \left (a \right )^{2}\right )}\)	\(69\)
meijerg	\(-\frac {2-\frac {\left (3 x^{2} \ln \left (b \right )^{2} \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )^{2}-6 x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )+6\right ) {\mathrm e}^{x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}}{3}}{\ln \left (b \right )^{3} \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )^{3}}\)	\(72\)
norman	\(\frac {x^{2} {\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\ln \left (a \right )+\ln \left (b \right )}-\frac {2 x \,{\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\ln \left (b \right )^{2}+2 \ln \left (b \right ) \ln \left (a \right )+\ln \left (a \right )^{2}}+\frac {2 \,{\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\left (\ln \left (b \right )^{2}+2 \ln \left (b \right ) \ln \left (a \right )+\ln \left (a \right )^{2}\right ) \left (\ln \left (a \right )+\ln \left (b \right )\right )}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln(b)^2*x^2+2*ln(b)*ln(a)*x^2+ln(a)^2*x^2-2*ln(b)*x-2*ln(a)*x+2)*a^x/(ln(a)+ln(b))^3*b^x

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Maxima [A]
time = 0.28, size = 67, normalized size = 1.37 \begin {gather*} \frac {{\left ({\left (\log \left (a\right )^{2} + 2 \, \log \left (a\right ) \log \left (b\right ) + \log \left (b\right )^{2}\right )} x^{2} - 2 \, x {\left (\log \left (a\right ) + \log \left (b\right )\right )} + 2\right )} e^{\left (x \log \left (a\right ) + x \log \left (b\right )\right )}}{\log \left (a\right )^{3} + 3 \, \log \left (a\right )^{2} \log \left (b\right ) + 3 \, \log \left (a\right ) \log \left (b\right )^{2} + \log \left (b\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((log(a)^2 + 2*log(a)*log(b) + log(b)^2)*x^2 - 2*x*(log(a) + log(b)) + 2)*e^(x*log(a) + x*log(b))/(log(a)^3 +
3*log(a)^2*log(b) + 3*log(a)*log(b)^2 + log(b)^3)

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Fricas [A]
time = 0.36, size = 71, normalized size = 1.45 \begin {gather*} \frac {{\left (x^{2} \log \left (a\right )^{2} + x^{2} \log \left (b\right )^{2} - 2 \, x \log \left (a\right ) + 2 \, {\left (x^{2} \log \left (a\right ) - x\right )} \log \left (b\right ) + 2\right )} a^{x} b^{x}}{\log \left (a\right )^{3} + 3 \, \log \left (a\right )^{2} \log \left (b\right ) + 3 \, \log \left (a\right ) \log \left (b\right )^{2} + \log \left (b\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*log(a)^2 + x^2*log(b)^2 - 2*x*log(a) + 2*(x^2*log(a) - x)*log(b) + 2)*a^x*b^x/(log(a)^3 + 3*log(a)^2*log(
b) + 3*log(a)*log(b)^2 + log(b)^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (51) = 102\).
time = 0.52, size = 583, normalized size = 11.90 \begin {gather*} \begin {cases} \frac {a^{x} b^{x} x^{2} \log {\left (a \right )}^{2}}{\log {\left (a \right )}^{3} + 3 \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 \log {\left (a \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} + \frac {2 a^{x} b^{x} x^{2} \log {\left (a \right )} \log {\left (b \right )}}{\log {\left (a \right )}^{3} + 3 \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 \log {\left (a \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} + \frac {a^{x} b^{x} x^{2} \log {\left (b \right )}^{2}}{\log {\left (a \right )}^{3} + 3 \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 \log {\left (a \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} - \frac {2 a^{x} b^{x} x \log {\left (a \right )}}{\log {\left (a \right )}^{3} + 3 \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 \log {\left (a \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} - \frac {2 a^{x} b^{x} x \log {\left (b \right )}}{\log {\left (a \right )}^{3} + 3 \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 \log {\left (a \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} + \frac {2 a^{x} b^{x}}{\log {\left (a \right )}^{3} + 3 \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 \log {\left (a \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} & \text {for}\: a \neq \frac {1}{b} \\\frac {b^{x} x^{2} \left (\frac {1}{b}\right )^{x} \log {\left (\frac {1}{b} \right )}^{2}}{\log {\left (\frac {1}{b} \right )}^{3} + 3 \log {\left (\frac {1}{b} \right )}^{2} \log {\left (b \right )} + 3 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} + \frac {2 b^{x} x^{2} \left (\frac {1}{b}\right )^{x} \log {\left (\frac {1}{b} \right )} \log {\left (b \right )}}{\log {\left (\frac {1}{b} \right )}^{3} + 3 \log {\left (\frac {1}{b} \right )}^{2} \log {\left (b \right )} + 3 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} + \frac {b^{x} x^{2} \left (\frac {1}{b}\right )^{x} \log {\left (b \right )}^{2}}{\log {\left (\frac {1}{b} \right )}^{3} + 3 \log {\left (\frac {1}{b} \right )}^{2} \log {\left (b \right )} + 3 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} - \frac {2 b^{x} x \left (\frac {1}{b}\right )^{x} \log {\left (\frac {1}{b} \right )}}{\log {\left (\frac {1}{b} \right )}^{3} + 3 \log {\left (\frac {1}{b} \right )}^{2} \log {\left (b \right )} + 3 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} - \frac {2 b^{x} x \left (\frac {1}{b}\right )^{x} \log {\left (b \right )}}{\log {\left (\frac {1}{b} \right )}^{3} + 3 \log {\left (\frac {1}{b} \right )}^{2} \log {\left (b \right )} + 3 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} + \frac {2 b^{x} \left (\frac {1}{b}\right )^{x}}{\log {\left (\frac {1}{b} \right )}^{3} + 3 \log {\left (\frac {1}{b} \right )}^{2} \log {\left (b \right )} + 3 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )}^{2} + \log {\left (b \right )}^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**x*b**x*x**2*log(a)**2/(log(a)**3 + 3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3) + 2*a**x
*b**x*x**2*log(a)*log(b)/(log(a)**3 + 3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3) + a**x*b**x*x**2*lo
g(b)**2/(log(a)**3 + 3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3) - 2*a**x*b**x*x*log(a)/(log(a)**3 +
3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3) - 2*a**x*b**x*x*log(b)/(log(a)**3 + 3*log(a)**2*log(b) +
3*log(a)*log(b)**2 + log(b)**3) + 2*a**x*b**x/(log(a)**3 + 3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3
), Ne(a, 1/b)), (b**x*x**2*(1/b)**x*log(1/b)**2/(log(1/b)**3 + 3*log(1/b)**2*log(b) + 3*log(1/b)*log(b)**2 + l
og(b)**3) + 2*b**x*x**2*(1/b)**x*log(1/b)*log(b)/(log(1/b)**3 + 3*log(1/b)**2*log(b) + 3*log(1/b)*log(b)**2 +
log(b)**3) + b**x*x**2*(1/b)**x*log(b)**2/(log(1/b)**3 + 3*log(1/b)**2*log(b) + 3*log(1/b)*log(b)**2 + log(b)*
*3) - 2*b**x*x*(1/b)**x*log(1/b)/(log(1/b)**3 + 3*log(1/b)**2*log(b) + 3*log(1/b)*log(b)**2 + log(b)**3) - 2*b
**x*x*(1/b)**x*log(b)/(log(1/b)**3 + 3*log(1/b)**2*log(b) + 3*log(1/b)*log(b)**2 + log(b)**3) + 2*b**x*(1/b)**
x/(log(1/b)**3 + 3*log(1/b)**2*log(b) + 3*log(1/b)*log(b)**2 + log(b)**3), True))

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Giac [C] Result contains complex when optimal does not.
time = 5.49, size = 2631, normalized size = 53.69 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*(pi*x^2*log(abs(a))*sgn(a) + pi*x^2*log(abs(b))*sgn(a) + pi*x^2*log(abs(a))*sgn(b) + pi*x^2*log(abs(b))*sg
n(b) - 2*pi*x^2*log(abs(a)) - 2*pi*x^2*log(abs(b)) - pi*x*sgn(a) - pi*x*sgn(b) + 2*pi*x)*(3*pi^3*sgn(a)*sgn(b)
- 4*pi^3*sgn(a) + 3*pi*log(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a)
- 4*pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b)
+ 5*pi^3 - 6*pi*log(abs(a))^2 - 12*pi*log(abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)/((3*pi^3*sgn(a)*sgn(b) -
4*pi^3*sgn(a) + 3*pi*log(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) -
4*pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) +
5*pi^3 - 6*pi*log(abs(a))^2 - 12*pi*log(abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)^2 + (3*pi^2*log(abs(a))*sgn(
a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*
log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6
*log(abs(a))^2*log(abs(b)) - 6*log(abs(a))*log(abs(b))^2 - 2*log(abs(b))^3)^2) + (pi^2*x^2*sgn(a)*sgn(b) - 2*p
i^2*x^2*sgn(a) - 2*pi^2*x^2*sgn(b) + 3*pi^2*x^2 - 2*x^2*log(abs(a))^2 - 4*x^2*log(abs(a))*log(abs(b)) - 2*x^2*
log(abs(b))^2 + 4*x*log(abs(a)) + 4*x*log(abs(b)) - 4)*(3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*
sgn(a)*sgn(b) - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log
(abs(b))*sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) - 6*
log(abs(a))*log(abs(b))^2 - 2*log(abs(b))^3)/((3*pi^3*sgn(a)*sgn(b) - 4*pi^3*sgn(a) + 3*pi*log(abs(a))^2*sgn(a
) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) - 4*pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b
) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) + 5*pi^3 - 6*pi*log(abs(a))^2 - 12*pi*log(
abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)^2 + (3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sg
n(b) - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*
sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) - 6*log(abs(a
))*log(abs(b))^2 - 2*log(abs(b))^3)^2))*cos(-1/2*pi*x*sgn(a) - 1/2*pi*x*sgn(b) + pi*x) + ((pi^2*x^2*sgn(a)*sgn
(b) - 2*pi^2*x^2*sgn(a) - 2*pi^2*x^2*sgn(b) + 3*pi^2*x^2 - 2*x^2*log(abs(a))^2 - 4*x^2*log(abs(a))*log(abs(b))
- 2*x^2*log(abs(b))^2 + 4*x*log(abs(a)) + 4*x*log(abs(b)) - 4)*(3*pi^3*sgn(a)*sgn(b) - 4*pi^3*sgn(a) + 3*pi*l
og(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) - 4*pi^3*sgn(b) + 3*pi*l
og(abs(a))^2*sgn(b) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) + 5*pi^3 - 6*pi*log(abs(
a))^2 - 12*pi*log(abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)/((3*pi^3*sgn(a)*sgn(b) - 4*pi^3*sgn(a) + 3*pi*log(
abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) - 4*pi^3*sgn(b) + 3*pi*log(
abs(a))^2*sgn(b) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) + 5*pi^3 - 6*pi*log(abs(a))
^2 - 12*pi*log(abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)^2 + (3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(ab
s(b))*sgn(a)*sgn(b) - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi
^2*log(abs(b))*sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)
) - 6*log(abs(a))*log(abs(b))^2 - 2*log(abs(b))^3)^2) - 2*(pi*x^2*log(abs(a))*sgn(a) + pi*x^2*log(abs(b))*sgn(
a) + pi*x^2*log(abs(a))*sgn(b) + pi*x^2*log(abs(b))*sgn(b) - 2*pi*x^2*log(abs(a)) - 2*pi*x^2*log(abs(b)) - pi*
x*sgn(a) - pi*x*sgn(b) + 2*pi*x)*(3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 6*pi^2
*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*sgn(b) + 9*pi
^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) - 6*log(abs(a))*log(abs(b)
)^2 - 2*log(abs(b))^3)/((3*pi^3*sgn(a)*sgn(b) - 4*pi^3*sgn(a) + 3*pi*log(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*l
og(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) - 4*pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) + 6*pi*log(abs(a))*l
og(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) + 5*pi^3 - 6*pi*log(abs(a))^2 - 12*pi*log(abs(a))*log(abs(b)) -
6*pi*log(abs(b))^2)^2 + (3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 6*pi^2*log(abs(
a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*sgn(b) + 9*pi^2*log(ab
s(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) - 6*log(abs(a))*log(abs(b))^2 - 2*l
og(abs(b))^3)^2))*sin(-1/2*pi*x*sgn(a) - 1/2*pi*x*sgn(b) + pi*x))*e^(x*(log(abs(a)) + log(abs(b)))) - 2*I*((I*
pi^2*x^2*sgn(a)*sgn(b) - 2*I*pi^2*x^2*sgn(a) + 2*pi*x^2*log(abs(a))*sgn(a) + 2*pi*x^2*log(abs(b))*sgn(a) - 2*I
*pi^2*x^2*sgn(b) + 2*pi*x^2*log(abs(a))*sgn(b) ...

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Mupad [B]
time = 0.05, size = 35, normalized size = 0.71 \begin {gather*} \frac {a^x\,b^x\,\left (x^2\,{\left (\ln \left (a\right )+\ln \left (b\right )\right )}^2-2\,x\,\left (\ln \left (a\right )+\ln \left (b\right )\right )+2\right )}{{\left (\ln \left (a\right )+\ln \left (b\right )\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^x*b^x*(x^2*(log(a) + log(b))^2 - 2*x*(log(a) + log(b)) + 2))/(log(a) + log(b))^3

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