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

Optimal. Leaf size=96 \[ \frac{4 \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{3 n}-\frac{2 x^{-3 n/2} f^{a+b x^n}}{3 n}-\frac{4 b \log (f) x^{-n/2} f^{a+b x^n}}{3 n} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0955575, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2215, 2211, 2204} \[ \frac{4 \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{3 n}-\frac{2 x^{-3 n/2} f^{a+b x^n}}{3 n}-\frac{4 b \log (f) x^{-n/2} f^{a+b x^n}}{3 n} \]

Antiderivative was successfully verified.

[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 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{3 n}{2}} \, dx &=-\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 ) \operatorname{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]  time = 0.007734, size = 39, normalized size = 0.41 \[ -\frac{f^a x^{-3 n/2} \left (-b \log (f) x^n\right )^{3/2} \text{Gamma}\left (-\frac{3}{2},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[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]  time = 0.054, size = 88, normalized size = 0.9 \begin{align*} -{\frac{2\,{f}^{a}{f}^{b{x}^{n}}}{3\,n} \left ({x}^{{\frac{n}{2}}} \right ) ^{-3}}-{\frac{4\,{f}^{a}\ln \left ( f \right ) b{f}^{b{x}^{n}}}{3\,n} \left ({x}^{{\frac{n}{2}}} \right ) ^{-1}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}\sqrt{\pi }}{3\,n}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }{x}^{{\frac{n}{2}}} \right ){\frac{1}{\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^(-1-3/2*n),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.33736, size = 47, normalized size = 0.49 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (f^{b x^{n} + a} x^{-\frac{3}{2} \, n - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(-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**(-1-3/2*n),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{b x^{n} + a} x^{-\frac{3}{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^(-1-3/2*n),x, algorithm="giac")

[Out]

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