\(\int f^{a+b x^n} x^3 \, dx\) [175]

Optimal result

Integrand size = 13, antiderivative size = 39 \[ \int f^{a+b x^n} x^3 \, dx=-\frac {f^a x^4 \Gamma \left (\frac {4}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-4/n}}{n} \]

[Out]

-f^a*x^4*GAMMA(4/n,-b*x^n*ln(f))/n/((-b*x^n*ln(f))^(4/n))

Rubi [A] (verified)

Time = 0.02 (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} \[ \int f^{a+b x^n} x^3 \, dx=-\frac {x^4 f^a \left (-b \log (f) x^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-b x^n \log (f)\right )}{n} \]

[In]

Int[f^(a + b*x^n)*x^3,x]

[Out]

-((f^a*x^4*Gamma[4/n, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(4/n)))

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 x^4 \Gamma \left (\frac {4}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-4/n}}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^n} x^3 \, dx=-\frac {f^a x^4 \Gamma \left (\frac {4}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-4/n}}{n} \]

[In]

Integrate[f^(a + b*x^n)*x^3,x]

[Out]

-((f^a*x^4*Gamma[4/n, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(4/n)))

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	\(\frac {f^{a} \left (-b \right )^{-\frac {4}{n}} \ln \left (f \right )^{-\frac {4}{n}} \left (\frac {n \,x^{4} \left (-b \right )^{\frac {4}{n}} \ln \left (f \right )^{\frac {4}{n}} \left (x^{n} \ln \left (f \right ) b n +n +4\right ) \Gamma \left (1-\frac {4}{n}\right ) \Gamma \left (\frac {n +4}{n}+1\right ) L_{-\frac {4}{n}}^{\left (\frac {n +4}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right )}{4 \left (n +4\right ) \Gamma \left (-\frac {4}{n}+\frac {n +4}{n}+1\right )}-\frac {n^{2} x^{n +4} \left (-b \right )^{\frac {4}{n}} \ln \left (f \right )^{1+\frac {4}{n}} b L_{-\frac {4}{n}}^{\left (\frac {n +4}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (1-\frac {4}{n}\right ) \Gamma \left (\frac {n +4}{n}+1\right )}{4 \left (n +4\right ) \Gamma \left (-\frac {4}{n}+\frac {n +4}{n}+1\right )}\right )}{n}\)	\(212\)

[In]

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

[Out]

f^a*(-b)^(-4/n)*ln(f)^(-4/n)/n*(1/4*n*x^4*(-b)^(4/n)*ln(f)^(4/n)*(x^n*ln(f)*b*n+n+4)/(n+4)/GAMMA(-4/n+(n+4)/n+
1)*GAMMA(1-4/n)*GAMMA((n+4)/n+1)*LaguerreL(-4/n,(n+4)/n,b*x^n*ln(f))-1/4*n^2*x^(n+4)*(-b)^(4/n)*ln(f)^(1+4/n)*
b/(n+4)*LaguerreL(-4/n,(n+4)/n+1,b*x^n*ln(f))*GAMMA(1-4/n)*GAMMA((n+4)/n+1)/GAMMA(-4/n+(n+4)/n+1))

Fricas [F]

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

[In]

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

[Out]

integral(f^(b*x^n + a)*x^3, x)

Sympy [F]

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

[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 = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int f^{a+b x^n} x^3 \, dx=-\frac {f^{a} x^{4} \Gamma \left (\frac {4}{n}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac {4}{n}} n} \]

[In]

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

[Out]

-f^a*x^4*gamma(4/n, -b*x^n*log(f))/((-b*x^n*log(f))^(4/n)*n)

Giac [F]

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

[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 = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38 \[ \int f^{a+b x^n} x^3 \, dx=\frac {f^a\,x^4\,{\mathrm {e}}^{\frac {b\,x^n\,\ln \left (f\right )}{2}}\,{\mathrm {M}}_{\frac {1}{2}-\frac {2}{n},\frac {2}{n}}\left (b\,x^n\,\ln \left (f\right )\right )}{4\,{\left (b\,x^n\,\ln \left (f\right )\right )}^{\frac {2}{n}+\frac {1}{2}}} \]

[In]

int(f^(a + b*x^n)*x^3,x)

[Out]

(f^a*x^4*exp((b*x^n*log(f))/2)*whittakerM(1/2 - 2/n, 2/n, b*x^n*log(f)))/(4*(b*x^n*log(f))^(2/n + 1/2))