\(\int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx\) [191]

Optimal result

Integrand size = 19, antiderivative size = 43 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2242, 2235} \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]

[In]

Int[f^(a + b*x^n)*x^(-1 + n/2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74

method	result	size
meijerg	\(\frac {f^{a} \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right ) \sqrt {\pi }}{n \sqrt {b}\, \sqrt {\ln \left (f \right )}}\)	\(32\)
risch	\(\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{n \sqrt {-b \ln \left (f \right )}}\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=-\frac {\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x x^{\frac {1}{2} \, n - 1}\right )}{b n \log \left (f\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=\frac {\sqrt {\pi } f^{a} x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {-b x^{n} \log \left (f\right )}\right ) - 1\right )}}{\sqrt {-b x^{n} \log \left (f\right )} n} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx=-\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (f\right )} \sqrt {x^{n}}\right )}{\sqrt {-b \log \left (f\right )} n} \]

[In]

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

[Out]

-sqrt(pi)*f^a*erf(-sqrt(-b*log(f))*sqrt(x^n))/(sqrt(-b*log(f))*n)

Mupad [F(-1)]

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

[In]

int(f^(a + b*x^n)*x^(n/2 - 1),x)

[Out]

int(f^(a + b*x^n)*x^(n/2 - 1), x)