3.515 \(\int \left (1+a^{m x}\right )^4 \, dx\)

Optimal. Leaf size=65 \[ \frac{4 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{m \log (a)}+\frac{4 a^{3 m x}}{3 m \log (a)}+\frac{a^{4 m x}}{4 m \log (a)}+x \]

[Out]

x + (4*a^(m*x))/(m*Log[a]) + (3*a^(2*m*x))/(m*Log[a]) + (4*a^(3*m*x))/(3*m*Log[a
]) + a^(4*m*x)/(4*m*Log[a])

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Rubi [A]  time = 0.0425654, antiderivative size = 65, 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 \[ \frac{4 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{m \log (a)}+\frac{4 a^{3 m x}}{3 m \log (a)}+\frac{a^{4 m x}}{4 m \log (a)}+x \]

Antiderivative was successfully verified.

[In]  Int[(1 + a^(m*x))^4,x]

[Out]

x + (4*a^(m*x))/(m*Log[a]) + (3*a^(2*m*x))/(m*Log[a]) + (4*a^(3*m*x))/(3*m*Log[a
]) + a^(4*m*x)/(4*m*Log[a])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{4 m x}}{4 m \log{\left (a \right )}} + \frac{4 a^{3 m x}}{3 m \log{\left (a \right )}} + \frac{4 a^{m x}}{m \log{\left (a \right )}} + \frac{\log{\left (a^{m x} \right )}}{m \log{\left (a \right )}} + \frac{6 \int ^{a^{m x}} x\, dx}{m \log{\left (a \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+a**(m*x))**4,x)

[Out]

a**(4*m*x)/(4*m*log(a)) + 4*a**(3*m*x)/(3*m*log(a)) + 4*a**(m*x)/(m*log(a)) + lo
g(a**(m*x))/(m*log(a)) + 6*Integral(x, (x, a**(m*x)))/(m*log(a))

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Mathematica [A]  time = 0.0231901, size = 49, normalized size = 0.75 \[ \frac{48 a^{m x}+36 a^{2 m x}+16 a^{3 m x}+3 a^{4 m x}+12 m x \log (a)}{12 m \log (a)} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + a^(m*x))^4,x]

[Out]

(48*a^(m*x) + 36*a^(2*m*x) + 16*a^(3*m*x) + 3*a^(4*m*x) + 12*m*x*Log[a])/(12*m*L
og[a])

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Maple [A]  time = 0.004, size = 78, normalized size = 1.2 \[{\frac{ \left ({a}^{mx} \right ) ^{4}}{4\,m\ln \left ( a \right ) }}+{\frac{4\, \left ({a}^{mx} \right ) ^{3}}{3\,m\ln \left ( a \right ) }}+3\,{\frac{ \left ({a}^{mx} \right ) ^{2}}{m\ln \left ( a \right ) }}+4\,{\frac{{a}^{mx}}{m\ln \left ( a \right ) }}+{\frac{\ln \left ({a}^{mx} \right ) }{m\ln \left ( a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+a^(m*x))^4,x)

[Out]

1/4/m/ln(a)*(a^(m*x))^4+4/3/m/ln(a)*(a^(m*x))^3+3/m/ln(a)*(a^(m*x))^2+4*a^(m*x)/
m/ln(a)+1/m/ln(a)*ln(a^(m*x))

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Maxima [A]  time = 1.34543, size = 82, normalized size = 1.26 \[ x + \frac{a^{4 \, m x}}{4 \, m \log \left (a\right )} + \frac{4 \, a^{3 \, m x}}{3 \, m \log \left (a\right )} + \frac{3 \, a^{2 \, m x}}{m \log \left (a\right )} + \frac{4 \, a^{m x}}{m \log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(m*x) + 1)^4,x, algorithm="maxima")

[Out]

x + 1/4*a^(4*m*x)/(m*log(a)) + 4/3*a^(3*m*x)/(m*log(a)) + 3*a^(2*m*x)/(m*log(a))
+ 4*a^(m*x)/(m*log(a))

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Fricas [A]  time = 0.216538, size = 63, normalized size = 0.97 \[ \frac{12 \, m x \log \left (a\right ) + 3 \, a^{4 \, m x} + 16 \, a^{3 \, m x} + 36 \, a^{2 \, m x} + 48 \, a^{m x}}{12 \, m \log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(m*x) + 1)^4,x, algorithm="fricas")

[Out]

1/12*(12*m*x*log(a) + 3*a^(4*m*x) + 16*a^(3*m*x) + 36*a^(2*m*x) + 48*a^(m*x))/(m
*log(a))

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Sympy [A]  time = 0.174139, size = 88, normalized size = 1.35 \[ x + \begin{cases} \frac{3 a^{4 m x} m^{3} \log{\left (a \right )}^{3} + 16 a^{3 m x} m^{3} \log{\left (a \right )}^{3} + 36 a^{2 m x} m^{3} \log{\left (a \right )}^{3} + 48 a^{m x} m^{3} \log{\left (a \right )}^{3}}{12 m^{4} \log{\left (a \right )}^{4}} & \text{for}\: 12 m^{4} \log{\left (a \right )}^{4} \neq 0 \\15 x & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+a**(m*x))**4,x)

[Out]

x + Piecewise(((3*a**(4*m*x)*m**3*log(a)**3 + 16*a**(3*m*x)*m**3*log(a)**3 + 36*
a**(2*m*x)*m**3*log(a)**3 + 48*a**(m*x)*m**3*log(a)**3)/(12*m**4*log(a)**4), Ne(
12*m**4*log(a)**4, 0)), (15*x, True))

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GIAC/XCAS [A]  time = 0.205564, size = 65, normalized size = 1. \[ \frac{12 \, m x{\rm ln}\left ({\left | a \right |}\right ) + 3 \, a^{4 \, m x} + 16 \, a^{3 \, m x} + 36 \, a^{2 \, m x} + 48 \, a^{m x}}{12 \, m{\rm ln}\left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(m*x) + 1)^4,x, algorithm="giac")

[Out]

1/12*(12*m*x*ln(abs(a)) + 3*a^(4*m*x) + 16*a^(3*m*x) + 36*a^(2*m*x) + 48*a^(m*x)
)/(m*ln(a))