\(\int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx\) [190]

Optimal result

Integrand size = 19, antiderivative size = 74 \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{2 b^{3/2} n \log ^{\frac {3}{2}}(f)}+\frac {f^{a+b x^n} x^{n/2}}{b n \log (f)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2244, 2242, 2235} \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=\frac {x^{n/2} f^{a+b x^n}}{b n \log (f)}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{2 b^{3/2} n \log ^{\frac {3}{2}}(f)} \]

[In]

Int[f^(a + b*x^n)*x^(-1 + (3*n)/2),x]

[Out]

-1/2*(f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]])/(b^(3/2)*n*Log[f]^(3/2)) + (f^(a + b*x^n)*x^(n/2))/(b*n
*Log[f])

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 2244

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.53 \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=-\frac {f^a x^{3 n/2} \Gamma \left (\frac {3}{2},-b x^n \log (f)\right )}{n \left (-b x^n \log (f)\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91

method	result	size
risch	\(\frac {f^{a} x^{\frac {n}{2}} f^{b \,x^{n}}}{n \ln \left (f \right ) b}-\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{2 n \ln \left (f \right ) b \sqrt {-b \ln \left (f \right )}}\)	\(67\)
meijerg	\(\frac {f^{a} \left (\frac {x^{\frac {n}{2}} \left (-b \right )^{\frac {3}{2}} \sqrt {\ln \left (f \right )}\, {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{b}-\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{\left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}} n}\)	\(71\)

[In]

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

[Out]

1/n*f^a/ln(f)/b*x^(1/2*n)*f^(b*x^n)-1/2/n*f^a/ln(f)/b*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)*x^(1/2*n)
)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86 \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=\frac {2 \, b x^{\frac {1}{2} \, n} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )} \log \left (f\right ) + \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x^{\frac {1}{2} \, n}\right )}{2 \, b^{2} n \log \left (f\right )^{2}} \]

[In]

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

[Out]

1/2*(2*b*x^(1/2*n)*e^(b*x^n*log(f) + a*log(f))*log(f) + sqrt(pi)*sqrt(-b*log(f))*f^a*erf(sqrt(-b*log(f))*x^(1/
2*n)))/(b^2*n*log(f)^2)

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 = 33, normalized size of antiderivative = 0.45 \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=-\frac {f^{a} x^{\frac {3}{2} \, n} \Gamma \left (\frac {3}{2}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac {3}{2}} n} \]

[In]

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

[Out]

-f^a*x^(3/2*n)*gamma(3/2, -b*x^n*log(f))/((-b*x^n*log(f))^(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. \[ \int f^{a+b x^n} x^{-1+\frac {3 n}{2}} \, dx=\int f^{a+b\,x^n}\,x^{\frac {3\,n}{2}-1} \,d x \]

[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)