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

Optimal. Leaf size=66 \[ \frac{2 \sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{n}-\frac{2 x^{-n/2} f^{a+b x^n}}{n} \]

[Out]

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

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Rubi [A]  time = 0.0652999, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2215, 2211, 2204} \[ \frac{2 \sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{n}-\frac{2 x^{-n/2} f^{a+b x^n}}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 - n/2),x]

[Out]

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

Rule 2215

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]

Rule 2211

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 2204

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]

Rubi steps

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

Mathematica [A]  time = 0.00714, size = 39, normalized size = 0.59 \[ -\frac{f^a x^{-n/2} \sqrt{-b \log (f) x^n} \text{Gamma}\left (-\frac{1}{2},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((f^a*Gamma[-1/2, -(b*x^n*Log[f])]*Sqrt[-(b*x^n*Log[f])])/(n*x^(n/2)))

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Maple [A]  time = 0.061, size = 59, normalized size = 0.9 \begin{align*} -2\,{\frac{{f}^{a}{f}^{b{x}^{n}}}{n{x}^{n/2}}}+2\,{\frac{{f}^{a}\ln \left ( f \right ) b\sqrt{\pi }{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }{x}^{n/2} \right ) }{n\sqrt{-b\ln \left ( f \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.20819, size = 47, normalized size = 0.71 \begin{align*} -\frac{\sqrt{-b x^{n} \log \left (f\right )} f^{a} \Gamma \left (-\frac{1}{2}, -b x^{n} \log \left (f\right )\right )}{n x^{\frac{1}{2} \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*x^n*log(f))*f^a*gamma(-1/2, -b*x^n*log(f))/(n*x^(1/2*n))

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Fricas [A]  time = 1.60989, size = 208, normalized size = 3.15 \begin{align*} -\frac{2 \,{\left (\sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x x^{-\frac{1}{2} \, n - 1}}\right ) + x x^{-\frac{1}{2} \, n - 1} e^{\left (\frac{a x^{2} x^{-n - 2} \log \left (f\right ) + b \log \left (f\right )}{x^{2} x^{-n - 2}}\right )}\right )}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{b x^{n} + a} x^{-\frac{1}{2} \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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