3.6 \(\int F^{c (a+b x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {F^{c (a+b x)}}{b c \log (F)} \]

[Out]

F^(c*(b*x+a))/b/c/ln(F)

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2194} \[ \frac {F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x)),x]

[Out]

F^(c*(a + b*x))/(b*c*Log[F])

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 F^{c (a+b x)} \, dx &=\frac {F^{c (a+b x)}}{b c \log (F)}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.05 \[ \frac {F^{a c+b c x}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x)),x]

[Out]

F^(a*c + b*c*x)/(b*c*Log[F])

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fricas [A]  time = 0.44, size = 21, normalized size = 1.05 \[ \frac {F^{b c x + a c}}{b c \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a)),x, algorithm="fricas")

[Out]

F^(b*c*x + a*c)/(b*c*log(F))

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giac [A]  time = 0.31, size = 21, normalized size = 1.05 \[ \frac {F^{b c x + a c}}{b c \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a)),x, algorithm="giac")

[Out]

F^(b*c*x + a*c)/(b*c*log(F))

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maple [A]  time = 0.00, size = 21, normalized size = 1.05 \[ \frac {F^{\left (b x +a \right ) c}}{b c \ln \relax (F )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^((b*x+a)*c),x)

[Out]

F^((b*x+a)*c)/b/c/ln(F)

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maxima [A]  time = 0.48, size = 20, normalized size = 1.00 \[ \frac {F^{{\left (b x + a\right )} c}}{b c \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a)),x, algorithm="maxima")

[Out]

F^((b*x + a)*c)/(b*c*log(F))

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mupad [B]  time = 3.43, size = 21, normalized size = 1.05 \[ \frac {F^{a\,c+b\,c\,x}}{b\,c\,\ln \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x)),x)

[Out]

F^(a*c + b*c*x)/(b*c*log(F))

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sympy [A]  time = 0.10, size = 20, normalized size = 1.00 \[ \begin {cases} \frac {F^{c \left (a + b x\right )}}{b c \log {\relax (F )}} & \text {for}\: b c \log {\relax (F )} \neq 0 \\x & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a)),x)

[Out]

Piecewise((F**(c*(a + b*x))/(b*c*log(F)), Ne(b*c*log(F), 0)), (x, True))

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