\(\int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx\) [375]

Optimal result

Integrand size = 25, antiderivative size = 56 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx=-\frac {F^{a+b (c+d x)^n} (c+d x)^{-n}}{d n}+\frac {b F^a \operatorname {ExpIntegralEi}\left (b (c+d x)^n \log (F)\right ) \log (F)}{d n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2246, 2241} \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx=\frac {b F^a \log (F) \operatorname {ExpIntegralEi}\left (b (c+d x)^n \log (F)\right )}{d n}-\frac {(c+d x)^{-n} F^{a+b (c+d x)^n}}{d n} \]

[In]

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

[Out]

-(F^(a + b*(c + d*x)^n)/(d*n*(c + d*x)^n)) + (b*F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F])/(d*n)

Rule 2241

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

Rule 2246

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), Int[(c + d*x)^Simplify[m + n]*F^(a +
b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[-4, Simpl
ify[(m + 1)/n], 5] &&  !RationalQ[m] && SumSimplerQ[m, n]

Rubi steps \begin{align*} \text {integral}& = -\frac {F^{a+b (c+d x)^n} (c+d x)^{-n}}{d n}+(b \log (F)) \int \frac {F^{a+b (c+d x)^n}}{c+d x} \, dx \\ & = -\frac {F^{a+b (c+d x)^n} (c+d x)^{-n}}{d n}+\frac {b F^a \text {Ei}\left (b (c+d x)^n \log (F)\right ) \log (F)}{d n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.48 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx=\frac {b F^a \Gamma \left (-1,-b (c+d x)^n \log (F)\right ) \log (F)}{d n} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09

method	result	size
risch	\(-\frac {F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-n}}{n d}-\frac {\ln \left (F \right ) b \,F^{a} \operatorname {Ei}_{1}\left (-b \left (d x +c \right )^{n} \ln \left (F \right )\right )}{n d}\)	\(61\)

[In]

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

[Out]

-1/n/d*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)-1/n/d*ln(F)*b*F^a*Ei(1,-b*(d*x+c)^n*ln(F))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.11 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx=\frac {{\left (d x + c\right )}^{n} F^{a} b {\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right ) - e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{{\left (d x + c\right )}^{n} d n} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx=\int \frac {F^{a+b\,{\left (c+d\,x\right )}^n}}{{\left (c+d\,x\right )}^{n+1}} \,d x \]

[In]

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

[Out]

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