3.189 \(\int f^{a+b x^n} x^{-1+\frac{5 n}{2}} \, dx\)

Optimal. Leaf size=104 \[ \frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{4 b^{5/2} n \log ^{\frac{5}{2}}(f)}-\frac{3 x^{n/2} f^{a+b x^n}}{2 b^2 n \log ^2(f)}+\frac{x^{3 n/2} f^{a+b x^n}}{b n \log (f)} \]

[Out]

(3*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]])/(4*b^(5/2)*n*Log[f]^(5/2)) - (3*f^(a + b*x^n)*x^(n/2))/(2*
b^2*n*Log[f]^2) + (f^(a + b*x^n)*x^((3*n)/2))/(b*n*Log[f])

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Rubi [A]  time = 0.110245, antiderivative size = 104, 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 = {2213, 2211, 2204} \[ \frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (f)} x^{n/2}\right )}{4 b^{5/2} n \log ^{\frac{5}{2}}(f)}-\frac{3 x^{n/2} f^{a+b x^n}}{2 b^2 n \log ^2(f)}+\frac{x^{3 n/2} f^{a+b x^n}}{b n \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 + (5*n)/2),x]

[Out]

(3*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]])/(4*b^(5/2)*n*Log[f]^(5/2)) - (3*f^(a + b*x^n)*x^(n/2))/(2*
b^2*n*Log[f]^2) + (f^(a + b*x^n)*x^((3*n)/2))/(b*n*Log[f])

Rule 2213

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]

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{5 n}{2}} \, dx &=\frac{f^{a+b x^n} x^{3 n/2}}{b n \log (f)}-\frac{3 \int f^{a+b x^n} x^{-1+\frac{3 n}{2}} \, dx}{2 b \log (f)}\\ &=-\frac{3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac{f^{a+b x^n} x^{3 n/2}}{b n \log (f)}+\frac{3 \int f^{a+b x^n} x^{\frac{1}{2} (-2+n)} \, dx}{4 b^2 \log ^2(f)}\\ &=-\frac{3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac{f^{a+b x^n} x^{3 n/2}}{b n \log (f)}+\frac{3 \operatorname{Subst}\left (\int f^{a+b x^2} \, dx,x,x^{1+\frac{1}{2} (-2+n)}\right )}{2 b^2 n \log ^2(f)}\\ &=\frac{3 f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x^{n/2} \sqrt{\log (f)}\right )}{4 b^{5/2} n \log ^{\frac{5}{2}}(f)}-\frac{3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac{f^{a+b x^n} x^{3 n/2}}{b n \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0105505, size = 39, normalized size = 0.38 \[ -\frac{f^a x^{5 n/2} \text{Gamma}\left (\frac{5}{2},-b \log (f) x^n\right )}{n \left (-b \log (f) x^n\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x^(-1 + (5*n)/2),x]

[Out]

-((f^a*x^((5*n)/2)*Gamma[5/2, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(5/2)))

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Maple [A]  time = 0.084, size = 96, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}{f}^{b{x}^{n}}}{\ln \left ( f \right ) bn} \left ({x}^{{\frac{n}{2}}} \right ) ^{3}}-{\frac{3\,{f}^{a}{f}^{b{x}^{n}}}{2\,n \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{x}^{{\frac{n}{2}}}}+{\frac{3\,{f}^{a}\sqrt{\pi }}{4\,n \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{\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+5/2*n),x)

[Out]

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

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Maxima [A]  time = 1.20442, size = 45, normalized size = 0.43 \begin{align*} -\frac{f^{a} x^{\frac{5}{2} \, n} \Gamma \left (\frac{5}{2}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac{5}{2}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^(-1+5/2*n),x, algorithm="maxima")

[Out]

-f^a*x^(5/2*n)*gamma(5/2, -b*x^n*log(f))/((-b*x^n*log(f))^(5/2)*n)

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Fricas [A]  time = 1.64528, size = 228, normalized size = 2.19 \begin{align*} -\frac{3 \, \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 \,{\left (2 \, b^{2} x^{\frac{3}{2} \, n} \log \left (f\right )^{2} - 3 \, b x^{\frac{1}{2} \, n} \log \left (f\right )\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{4 \, b^{3} n \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^(-1+5/2*n),x, algorithm="fricas")

[Out]

-1/4*(3*sqrt(pi)*sqrt(-b*log(f))*f^a*erf(sqrt(-b*log(f))*x^(1/2*n)) - 2*(2*b^2*x^(3/2*n)*log(f)^2 - 3*b*x^(1/2
*n)*log(f))*e^(b*x^n*log(f) + a*log(f)))/(b^3*n*log(f)^3)

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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+5/2*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{5}{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+5/2*n),x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)*x^(5/2*n - 1), x)