3.372 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1+2 n} \, dx\)

Optimal. Leaf size=63 \[ \frac{(c+d x)^n F^{a+b (c+d x)^n}}{b d n \log (F)}-\frac{F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)} \]

[Out]

-(F^(a + b*(c + d*x)^n)/(b^2*d*n*Log[F]^2)) + (F^(a + b*(c + d*x)^n)*(c + d*x)^n)/(b*d*n*Log[F])

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Rubi [A]  time = 0.074351, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2213, 2209} \[ \frac{(c+d x)^n F^{a+b (c+d x)^n}}{b d n \log (F)}-\frac{F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 2*n),x]

[Out]

-(F^(a + b*(c + d*x)^n)/(b^2*d*n*Log[F]^2)) + (F^(a + b*(c + d*x)^n)*(c + d*x)^n)/(b*d*n*Log[F])

Rule 2213

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), 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]] && Lt
Q[0, Simplify[(m + 1)/n], 5] &&  !RationalQ[m] && SumSimplerQ[m, -n]

Rule 2209

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+2 n} \, dx &=\frac{F^{a+b (c+d x)^n} (c+d x)^n}{b d n \log (F)}-\frac{\int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx}{b \log (F)}\\ &=-\frac{F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac{F^{a+b (c+d x)^n} (c+d x)^n}{b d n \log (F)}\\ \end{align*}

Mathematica [C]  time = 0.0063835, size = 32, normalized size = 0.51 \[ -\frac{F^a \text{Gamma}\left (2,-b \log (F) (c+d x)^n\right )}{b^2 d n \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 2*n),x]

[Out]

-((F^a*Gamma[2, -(b*(c + d*x)^n*Log[F])])/(b^2*d*n*Log[F]^2))

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Maple [A]  time = 0.056, size = 74, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{n\ln \left ( dx+c \right ) }}{{\rm e}^{ \left ( a+b{{\rm e}^{n\ln \left ( dx+c \right ) }} \right ) \ln \left ( F \right ) }}}{\ln \left ( F \right ) bdn}}-{\frac{{{\rm e}^{ \left ( a+b{{\rm e}^{n\ln \left ( dx+c \right ) }} \right ) \ln \left ( F \right ) }}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}dn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+2*n),x)

[Out]

1/d/b/n/ln(F)*exp(n*ln(d*x+c))*exp((a+b*exp(n*ln(d*x+c)))*ln(F))-1/b^2/n/d/ln(F)^2*exp((a+b*exp(n*ln(d*x+c)))*
ln(F))

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Maxima [A]  time = 1.0147, size = 61, normalized size = 0.97 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{n} F^{a} b \log \left (F\right ) - F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{2} d n \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+2*n),x, algorithm="maxima")

[Out]

((d*x + c)^n*F^a*b*log(F) - F^a)*F^((d*x + c)^n*b)/(b^2*d*n*log(F)^2)

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Fricas [A]  time = 1.57775, size = 112, normalized size = 1.78 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) - 1\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{2} d n \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+2*n),x, algorithm="fricas")

[Out]

((d*x + c)^n*b*log(F) - 1)*e^((d*x + c)^n*b*log(F) + a*log(F))/(b^2*d*n*log(F)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**n)*(d*x+c)**(-1+2*n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+2*n),x, algorithm="giac")

[Out]

integrate((d*x + c)^(2*n - 1)*F^((d*x + c)^n*b + a), x)