3.514 \(\int \left (1+a^{m x}\right )^3 \, dx\)

Optimal. Leaf size=50 \[ \frac{3 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{2 m \log (a)}+\frac{a^{3 m x}}{3 m \log (a)}+x \]

[Out]

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

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Rubi [A]  time = 0.036049, antiderivative size = 50, 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 \[ \frac{3 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{2 m \log (a)}+\frac{a^{3 m x}}{3 m \log (a)}+x \]

Antiderivative was successfully verified.

[In]  Int[(1 + a^(m*x))^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3 m x}}{3 m \log{\left (a \right )}} + \frac{3 a^{m x}}{m \log{\left (a \right )}} + \frac{\log{\left (a^{m x} \right )}}{m \log{\left (a \right )}} + \frac{3 \int ^{a^{m x}} x\, dx}{m \log{\left (a \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+a**(m*x))**3,x)

[Out]

a**(3*m*x)/(3*m*log(a)) + 3*a**(m*x)/(m*log(a)) + log(a**(m*x))/(m*log(a)) + 3*I
ntegral(x, (x, a**(m*x)))/(m*log(a))

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Mathematica [A]  time = 0.0194399, size = 41, normalized size = 0.82 \[ \frac{18 a^{m x}+9 a^{2 m x}+2 a^{3 m x}+6 m x \log (a)}{6 m \log (a)} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + a^(m*x))^3,x]

[Out]

(18*a^(m*x) + 9*a^(2*m*x) + 2*a^(3*m*x) + 6*m*x*Log[a])/(6*m*Log[a])

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Maple [A]  time = 0.003, size = 62, normalized size = 1.2 \[{\frac{ \left ({a}^{mx} \right ) ^{3}}{3\,m\ln \left ( a \right ) }}+{\frac{3\, \left ({a}^{mx} \right ) ^{2}}{2\,m\ln \left ( a \right ) }}+3\,{\frac{{a}^{mx}}{m\ln \left ( a \right ) }}+{\frac{\ln \left ({a}^{mx} \right ) }{m\ln \left ( a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+a^(m*x))^3,x)

[Out]

1/3/m/ln(a)*(a^(m*x))^3+3/2/m/ln(a)*(a^(m*x))^2+3*a^(m*x)/m/ln(a)+1/m/ln(a)*ln(a
^(m*x))

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Maxima [A]  time = 1.37683, size = 62, normalized size = 1.24 \[ x + \frac{a^{3 \, m x}}{3 \, m \log \left (a\right )} + \frac{3 \, a^{2 \, m x}}{2 \, m \log \left (a\right )} + \frac{3 \, a^{m x}}{m \log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(m*x) + 1)^3,x, algorithm="maxima")

[Out]

x + 1/3*a^(3*m*x)/(m*log(a)) + 3/2*a^(2*m*x)/(m*log(a)) + 3*a^(m*x)/(m*log(a))

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Fricas [A]  time = 0.241489, size = 53, normalized size = 1.06 \[ \frac{6 \, m x \log \left (a\right ) + 2 \, a^{3 \, m x} + 9 \, a^{2 \, m x} + 18 \, a^{m x}}{6 \, m \log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(m*x) + 1)^3,x, algorithm="fricas")

[Out]

1/6*(6*m*x*log(a) + 2*a^(3*m*x) + 9*a^(2*m*x) + 18*a^(m*x))/(m*log(a))

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Sympy [A]  time = 0.14797, size = 71, normalized size = 1.42 \[ x + \begin{cases} \frac{2 a^{3 m x} m^{2} \log{\left (a \right )}^{2} + 9 a^{2 m x} m^{2} \log{\left (a \right )}^{2} + 18 a^{m x} m^{2} \log{\left (a \right )}^{2}}{6 m^{3} \log{\left (a \right )}^{3}} & \text{for}\: 6 m^{3} \log{\left (a \right )}^{3} \neq 0 \\7 x & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+a**(m*x))**3,x)

[Out]

x + Piecewise(((2*a**(3*m*x)*m**2*log(a)**2 + 9*a**(2*m*x)*m**2*log(a)**2 + 18*a
**(m*x)*m**2*log(a)**2)/(6*m**3*log(a)**3), Ne(6*m**3*log(a)**3, 0)), (7*x, True
))

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GIAC/XCAS [A]  time = 0.200246, size = 54, normalized size = 1.08 \[ \frac{6 \, m x{\rm ln}\left ({\left | a \right |}\right ) + 2 \, a^{3 \, m x} + 9 \, a^{2 \, m x} + 18 \, a^{m x}}{6 \, m{\rm ln}\left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(m*x) + 1)^3,x, algorithm="giac")

[Out]

1/6*(6*m*x*ln(abs(a)) + 2*a^(3*m*x) + 9*a^(2*m*x) + 18*a^(m*x))/(m*ln(a))