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

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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [C]  time = 0.00, size = 24, normalized size = 0.34 \[ \frac {f^a \Gamma \left (3,-b x^n \log (f)\right )}{b^3 n \log ^3(f)} \]

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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fricas [A]  time = 0.41, size = 47, normalized size = 0.66 \[ \frac {{\left (b^{2} x^{2 \, n} \log \relax (f)^{2} - 2 \, b x^{n} \log \relax (f) + 2\right )} e^{\left (b x^{n} \log \relax (f) + a \log \relax (f)\right )}}{b^{3} n \log \relax (f)^{3}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{b x^{n} + a} x^{3 \, n - 1}\,{d x} \]

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)

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maple [A]  time = 0.02, size = 44, normalized size = 0.62 \[ \frac {\left (b^{2} x^{2 n} \ln \relax (f )^{2}-2 b \,x^{n} \ln \relax (f )+2\right ) f^{b \,x^{n}+a}}{b^{3} n \ln \relax (f )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^n+a)*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^(b*x^n+a)

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maxima [A]  time = 0.69, size = 51, normalized size = 0.72 \[ \frac {{\left (b^{2} f^{a} x^{2 \, n} \log \relax (f)^{2} - 2 \, b f^{a} x^{n} \log \relax (f) + 2 \, f^{a}\right )} f^{b x^{n}}}{b^{3} n \log \relax (f)^{3}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int f^{a+b\,x^n}\,x^{3\,n-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^n)*x^(3*n - 1),x)

[Out]

int(f^(a + b*x^n)*x^(3*n - 1), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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