3.4.73 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx\) [373]

Optimal. Leaf size=27 \[ \frac {F^{a+b (c+d x)^n}}{b d n \log (F)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2240} \begin {gather*} \frac {F^{a+b (c+d x)^n}}{b d n \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + n),x]

[Out]

F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx &=\frac {F^{a+b (c+d x)^n}}{b d n \log (F)}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {F^{a+b (c+d x)^n}}{b d n \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + n),x]

[Out]

F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])

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Maple [A]
time = 0.06, size = 28, normalized size = 1.04

method	result	size
risch	\(\frac {F^{a +b \left (d x +c \right )^{n}}}{b d n \ln \left (F \right )}\)	\(28\)
norman	\(\frac {{\mathrm e}^{\left (a +b \,{\mathrm e}^{n \ln \left (d x +c \right )}\right ) \ln \left (F \right )}}{d b n \ln \left (F \right )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+n),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 27, normalized size = 1.00 \begin {gather*} \frac {F^{{\left (d x + c\right )}^{n} b + a}}{b d n \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+n),x, algorithm="maxima")

[Out]

F^((d*x + c)^n*b + a)/(b*d*n*log(F))

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Fricas [A]
time = 0.38, size = 31, normalized size = 1.15 \begin {gather*} \frac {e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b d n \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+n),x, algorithm="fricas")

[Out]

e^((d*x + c)^n*b*log(F) + a*log(F))/(b*d*n*log(F))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**n)*(d*x+c)**(-1+n),x)

[Out]

Timed out

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Giac [A]
time = 3.08, size = 27, normalized size = 1.00 \begin {gather*} \frac {F^{{\left (d x + c\right )}^{n} b + a}}{b d n \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+n),x, algorithm="giac")

[Out]

F^((d*x + c)^n*b + a)/(b*d*n*log(F))

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Mupad [B]
time = 3.68, size = 27, normalized size = 1.00 \begin {gather*} \frac {F^{a+b\,{\left (c+d\,x\right )}^n}}{b\,d\,n\,\ln \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^n)*(c + d*x)^(n - 1),x)

[Out]

F^(a + b*(c + d*x)^n)/(b*d*n*log(F))

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