3.184 \(\int f^{a+b x^n} x^{-1+2 n} \, dx\)

Optimal. Leaf size=45 \[ \frac{x^n f^{a+b x^n}}{b n \log (f)}-\frac{f^{a+b x^n}}{b^2 n \log ^2(f)} \]

[Out]

-(f^(a + b*x^n)/(b^2*n*Log[f]^2)) + (f^(a + b*x^n)*x^n)/(b*n*Log[f])

________________________________________________________________________________________

Rubi [A]  time = 0.048403, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2213, 2209} \[ \frac{x^n f^{a+b x^n}}{b n \log (f)}-\frac{f^{a+b x^n}}{b^2 n \log ^2(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 + 2*n),x]

[Out]

-(f^(a + b*x^n)/(b^2*n*Log[f]^2)) + (f^(a + b*x^n)*x^n)/(b*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 x^n} x^{-1+2 n} \, dx &=\frac{f^{a+b x^n} x^n}{b n \log (f)}-\frac{\int f^{a+b x^n} x^{-1+n} \, dx}{b \log (f)}\\ &=-\frac{f^{a+b x^n}}{b^2 n \log ^2(f)}+\frac{f^{a+b x^n} x^n}{b n \log (f)}\\ \end{align*}

Mathematica [C]  time = 0.0042438, size = 25, normalized size = 0.56 \[ -\frac{f^a \text{Gamma}\left (2,-b \log (f) x^n\right )}{b^2 n \log ^2(f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((f^a*Gamma[2, -(b*x^n*Log[f])])/(b^2*n*Log[f]^2))

________________________________________________________________________________________

Maple [A]  time = 0.031, size = 56, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{{\rm e}^{ \left ( b{{\rm e}^{n\ln \left ( x \right ) }}+a \right ) \ln \left ( f \right ) }}}{\ln \left ( f \right ) bn}}-{\frac{{{\rm e}^{ \left ( b{{\rm e}^{n\ln \left ( x \right ) }}+a \right ) \ln \left ( f \right ) }}}{{b}^{2}n \left ( \ln \left ( f \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b*x^n)*x^(-1+2*n),x)

[Out]

1/ln(f)/b/n*exp(n*ln(x))*exp((b*exp(n*ln(x))+a)*ln(f))-1/ln(f)^2/b^2/n*exp((b*exp(n*ln(x))+a)*ln(f))

________________________________________________________________________________________

Maxima [A]  time = 1.14659, size = 46, normalized size = 1.02 \begin{align*} \frac{{\left (b f^{a} x^{n} \log \left (f\right ) - f^{a}\right )} f^{b x^{n}}}{b^{2} n \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*f^a*x^n*log(f) - f^a)*f^(b*x^n)/(b^2*n*log(f)^2)

________________________________________________________________________________________

Fricas [A]  time = 1.55595, size = 88, normalized size = 1.96 \begin{align*} \frac{{\left (b x^{n} \log \left (f\right ) - 1\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{b^{2} n \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^n*log(f) - 1)*e^(b*x^n*log(f) + a*log(f))/(b^2*n*log(f)^2)

________________________________________________________________________________________

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*x**n)*x**(-1+2*n),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^n + a)*x^(2*n - 1), x)