\(\int \frac {f^{a+b x^n}}{x^3} \, dx\) [181]

Optimal result

Integrand size = 13, antiderivative size = 39 \[ \int \frac {f^{a+b x^n}}{x^3} \, dx=-\frac {f^a \Gamma \left (-\frac {2}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{2/n}}{n x^2} \]

[Out]

-f^a*GAMMA(-2/n,-b*x^n*ln(f))*(-b*x^n*ln(f))^(2/n)/n/x^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2250} \[ \int \frac {f^{a+b x^n}}{x^3} \, dx=-\frac {f^a \left (-b \log (f) x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b x^n \log (f)\right )}{n x^2} \]

[In]

Int[f^(a + b*x^n)/x^3,x]

[Out]

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

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {f^a \Gamma \left (-\frac {2}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{2/n}}{n x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+b x^n}}{x^3} \, dx=-\frac {f^a \Gamma \left (-\frac {2}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{2/n}}{n x^2} \]

[In]

Integrate[f^(a + b*x^n)/x^3,x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 5.44

method	result	size
meijerg	\(\frac {f^{a} \ln \left (f \right )^{\frac {2}{n}} \left (-b \right )^{\frac {2}{n}} \left (-\frac {n \left (-b \right )^{-\frac {2}{n}} \ln \left (f \right )^{-\frac {2}{n}} \left (x^{n} \ln \left (f \right ) b n +n -2\right ) \Gamma \left (1+\frac {2}{n}\right ) \Gamma \left (\frac {n -2}{n}+1\right ) L_{\frac {2}{n}}^{\left (\frac {n -2}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right )}{2 x^{2} \left (n -2\right ) \Gamma \left (\frac {2}{n}+\frac {n -2}{n}+1\right )}+\frac {n^{2} x^{n -2} \left (-b \right )^{-\frac {2}{n}} \ln \left (f \right )^{1-\frac {2}{n}} b L_{\frac {2}{n}}^{\left (\frac {n -2}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (1+\frac {2}{n}\right ) \Gamma \left (\frac {n -2}{n}+1\right )}{2 \left (n -2\right ) \Gamma \left (\frac {2}{n}+\frac {n -2}{n}+1\right )}\right )}{n}\)	\(212\)

[In]

int(f^(a+b*x^n)/x^3,x,method=_RETURNVERBOSE)

[Out]

f^a*ln(f)^(2/n)*(-b)^(2/n)/n*(-1/2*n/x^2*(-b)^(-2/n)*ln(f)^(-2/n)*(x^n*ln(f)*b*n+n-2)/(n-2)/GAMMA(2/n+(n-2)/n+
1)*GAMMA(1+2/n)*GAMMA((n-2)/n+1)*LaguerreL(2/n,(n-2)/n,b*x^n*ln(f))+1/2*n^2*x^(n-2)*(-b)^(-2/n)*ln(f)^(1-2/n)*
b/(n-2)*LaguerreL(2/n,(n-2)/n+1,b*x^n*ln(f))*GAMMA(1+2/n)*GAMMA((n-2)/n+1)/GAMMA(2/n+(n-2)/n+1))

Fricas [F]

\[ \int \frac {f^{a+b x^n}}{x^3} \, dx=\int { \frac {f^{b x^{n} + a}}{x^{3}} \,d x } \]

[In]

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

[Out]

integral(f^(b*x^n + a)/x^3, x)

Sympy [F]

\[ \int \frac {f^{a+b x^n}}{x^3} \, dx=\int \frac {f^{a + b x^{n}}}{x^{3}}\, dx \]

[In]

integrate(f**(a+b*x**n)/x**3,x)

[Out]

Integral(f**(a + b*x**n)/x**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+b x^n}}{x^3} \, dx=-\frac {\left (-b x^{n} \log \left (f\right )\right )^{\frac {2}{n}} f^{a} \Gamma \left (-\frac {2}{n}, -b x^{n} \log \left (f\right )\right )}{n x^{2}} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {f^{a+b x^n}}{x^3} \, dx=\int { \frac {f^{b x^{n} + a}}{x^{3}} \,d x } \]

[In]

integrate(f^(a+b*x^n)/x^3,x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)/x^3, x)

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.23 \[ \int \frac {f^{a+b x^n}}{x^3} \, dx=-\frac {f^a\,{\mathrm {e}}^{\frac {b\,x^n\,\ln \left (f\right )}{2}}\,{\mathrm {M}}_{\frac {1}{n}+\frac {1}{2},-\frac {1}{n}}\left (b\,x^n\,\ln \left (f\right )\right )\,{\left (b\,x^n\,\ln \left (f\right )\right )}^{\frac {1}{n}-\frac {1}{2}}}{2\,x^2} \]

[In]

int(f^(a + b*x^n)/x^3,x)

[Out]

-(f^a*exp((b*x^n*log(f))/2)*whittakerM(1/n + 1/2, -1/n, b*x^n*log(f))*(b*x^n*log(f))^(1/n - 1/2))/(2*x^2)