\(\int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx\) [193]

Optimal result

Integrand size = 19, antiderivative size = 96 \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=-\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {4 b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right ) \log ^{\frac {3}{2}}(f)}{3 n} \]

[Out]

-2/3*f^(a+b*x^n)/n/(x^(3/2*n))-4/3*b*f^(a+b*x^n)*ln(f)/n/(x^(1/2*n))+4/3*b^(3/2)*f^a*erfi(x^(1/2*n)*b^(1/2)*ln
(f)^(1/2))*ln(f)^(3/2)*Pi^(1/2)/n

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2246, 2242, 2235} \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\frac {4 \sqrt {\pi } b^{3/2} f^a \log ^{\frac {3}{2}}(f) \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{3 n}-\frac {2 x^{-3 n/2} f^{a+b x^n}}{3 n}-\frac {4 b \log (f) x^{-n/2} f^{a+b x^n}}{3 n} \]

[In]

Int[f^(a + b*x^n)*x^(-1 - (3*n)/2),x]

[Out]

(-2*f^(a + b*x^n))/(3*n*x^((3*n)/2)) - (4*b*f^(a + b*x^n)*Log[f])/(3*n*x^(n/2)) + (4*b^(3/2)*f^a*Sqrt[Pi]*Erfi
[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]]*Log[f]^(3/2))/(3*n)

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2242

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rule 2246

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), 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]] && LtQ[-4, Simpl
ify[(m + 1)/n], 5] &&  !RationalQ[m] && SumSimplerQ[m, n]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}+\frac {1}{3} (2 b \log (f)) \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx \\ & = -\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {1}{3} \left (4 b^2 \log ^2(f)\right ) \int f^{a+b x^n} x^{\frac {1}{2} (-2+n)} \, dx \\ & = -\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {\left (8 b^2 \log ^2(f)\right ) \text {Subst}\left (\int f^{a+b x^2} \, dx,x,x^{1+\frac {1}{2} (-2+n)}\right )}{3 n} \\ & = -\frac {2 f^{a+b x^n} x^{-3 n/2}}{3 n}-\frac {4 b f^{a+b x^n} x^{-n/2} \log (f)}{3 n}+\frac {4 b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right ) \log ^{\frac {3}{2}}(f)}{3 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.41 \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=-\frac {f^a x^{-3 n/2} \Gamma \left (-\frac {3}{2},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{3/2}}{n} \]

[In]

Integrate[f^(a + b*x^n)*x^(-1 - (3*n)/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82

method	result	size
meijerg	\(\frac {f^{a} \left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}} \left (-\frac {2 x^{-\frac {3 n}{2}} \left (2 b \,x^{n} \ln \left (f \right )+1\right ) {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{3 \left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}}}+\frac {4 b^{\frac {3}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{3 \left (-b \right )^{\frac {3}{2}}}\right )}{n}\)	\(79\)
risch	\(-\frac {2 f^{a} x^{-\frac {3 n}{2}} f^{b \,x^{n}}}{3 n}-\frac {4 f^{a} \ln \left (f \right ) b \,x^{-\frac {n}{2}} f^{b \,x^{n}}}{3 n}+\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{3 n \sqrt {-b \ln \left (f \right )}}\)	\(88\)

[In]

int(f^(a+b*x^n)*x^(-1-3/2*n),x,method=_RETURNVERBOSE)

[Out]

f^a*(-b)^(3/2)*ln(f)^(3/2)/n*(-2/3*x^(-3/2*n)/(-b)^(3/2)/ln(f)^(3/2)*(2*b*x^n*ln(f)+1)*exp(b*x^n*ln(f))+4/3/(-
b)^(3/2)*b^(3/2)*Pi^(1/2)*erfi(x^(1/2*n)*b^(1/2)*ln(f)^(1/2)))

Fricas [F]

\[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\int { f^{b x^{n} + a} x^{-\frac {3}{2} \, n - 1} \,d x } \]

[In]

integrate(f^(a+b*x^n)*x^(-1-3/2*n),x, algorithm="fricas")

[Out]

integral(f^(b*x^n + a)*x^(-3/2*n - 1), x)

Sympy [F]

\[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\int f^{a + b x^{n}} x^{- \frac {3 n}{2} - 1}\, dx \]

[In]

integrate(f**(a+b*x**n)*x**(-1-3/2*n),x)

[Out]

Integral(f**(a + b*x**n)*x**(-3*n/2 - 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=-\frac {\left (-b x^{n} \log \left (f\right )\right )^{\frac {3}{2}} f^{a} \Gamma \left (-\frac {3}{2}, -b x^{n} \log \left (f\right )\right )}{n x^{\frac {3}{2} \, n}} \]

[In]

integrate(f^(a+b*x^n)*x^(-1-3/2*n),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\int { f^{b x^{n} + a} x^{-\frac {3}{2} \, n - 1} \,d x } \]

[In]

integrate(f^(a+b*x^n)*x^(-1-3/2*n),x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)*x^(-3/2*n - 1), x)

Mupad [F(-1)]

Timed out. \[ \int f^{a+b x^n} x^{-1-\frac {3 n}{2}} \, dx=\int \frac {f^{a+b\,x^n}}{x^{\frac {3\,n}{2}+1}} \,d x \]

[In]

int(f^(a + b*x^n)/x^((3*n)/2 + 1),x)

[Out]

int(f^(a + b*x^n)/x^((3*n)/2 + 1), x)