3.183 \(\int f^{a+b x^n} x^{-1+3 n} \, dx\)

Optimal. Leaf size=71 \[ -\frac{2 x^n f^{a+b x^n}}{b^2 n \log ^2(f)}+\frac{2 f^{a+b x^n}}{b^3 n \log ^3(f)}+\frac{x^{2 n} f^{a+b x^n}}{b n \log (f)} \]

[Out]

(2*f^(a + b*x^n))/(b^3*n*Log[f]^3) - (2*f^(a + b*x^n)*x^n)/(b^2*n*Log[f]^2) + (f^(a + b*x^n)*x^(2*n))/(b*n*Log
[f])

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Rubi [A]  time = 0.075849, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2213, 2209} \[ -\frac{2 x^n f^{a+b x^n}}{b^2 n \log ^2(f)}+\frac{2 f^{a+b x^n}}{b^3 n \log ^3(f)}+\frac{x^{2 n} f^{a+b x^n}}{b n \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 + 3*n),x]

[Out]

(2*f^(a + b*x^n))/(b^3*n*Log[f]^3) - (2*f^(a + b*x^n)*x^n)/(b^2*n*Log[f]^2) + (f^(a + b*x^n)*x^(2*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+3 n} \, dx &=\frac{f^{a+b x^n} x^{2 n}}{b n \log (f)}-\frac{2 \int f^{a+b x^n} x^{-1+2 n} \, dx}{b \log (f)}\\ &=-\frac{2 f^{a+b x^n} x^n}{b^2 n \log ^2(f)}+\frac{f^{a+b x^n} x^{2 n}}{b n \log (f)}+\frac{2 \int f^{a+b x^n} x^{-1+n} \, dx}{b^2 \log ^2(f)}\\ &=\frac{2 f^{a+b x^n}}{b^3 n \log ^3(f)}-\frac{2 f^{a+b x^n} x^n}{b^2 n \log ^2(f)}+\frac{f^{a+b x^n} x^{2 n}}{b n \log (f)}\\ \end{align*}

Mathematica [C]  time = 0.0040479, size = 24, normalized size = 0.34 \[ \frac{f^a \text{Gamma}\left (3,-b \log (f) x^n\right )}{b^3 n \log ^3(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x^(-1 + 3*n),x]

[Out]

(f^a*Gamma[3, -(b*x^n*Log[f])])/(b^3*n*Log[f]^3)

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Maple [A]  time = 0.02, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( \left ({x}^{n} \right ) ^{2}{b}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}-2\,b{x}^{n}\ln \left ( f \right ) +2 \right ){f}^{a+b{x}^{n}}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b*x^n)*x^(-1+3*n),x)

[Out]

((x^n)^2*b^2*ln(f)^2-2*b*x^n*ln(f)+2)/b^3/ln(f)^3/n*f^(a+b*x^n)

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Maxima [A]  time = 1.1279, size = 69, normalized size = 0.97 \begin{align*} \frac{{\left (b^{2} f^{a} x^{2 \, n} \log \left (f\right )^{2} - 2 \, b f^{a} x^{n} \log \left (f\right ) + 2 \, f^{a}\right )} f^{b x^{n}}}{b^{3} n \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^(-1+3*n),x, algorithm="maxima")

[Out]

(b^2*f^a*x^(2*n)*log(f)^2 - 2*b*f^a*x^n*log(f) + 2*f^a)*f^(b*x^n)/(b^3*n*log(f)^3)

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Fricas [A]  time = 1.53393, size = 122, normalized size = 1.72 \begin{align*} \frac{{\left (b^{2} x^{2 \, n} \log \left (f\right )^{2} - 2 \, b x^{n} \log \left (f\right ) + 2\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{b^{3} n \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^(-1+3*n),x, algorithm="fricas")

[Out]

(b^2*x^(2*n)*log(f)^2 - 2*b*x^n*log(f) + 2)*e^(b*x^n*log(f) + a*log(f))/(b^3*n*log(f)^3)

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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*x**n)*x**(-1+3*n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{b x^{n} + a} x^{3 \, 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+3*n),x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)*x^(3*n - 1), x)