\(\int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx\) [189]

Optimal result

Integrand size = 19, antiderivative size = 104 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\frac {3 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{4 b^{5/2} n \log ^{\frac {5}{2}}(f)}-\frac {3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^{3 n/2}}{b n \log (f)} \]

[Out]

-3/2*f^(a+b*x^n)*x^(1/2*n)/b^2/n/ln(f)^2+f^(a+b*x^n)*x^(3/2*n)/b/n/ln(f)+3/4*f^a*erfi(x^(1/2*n)*b^(1/2)*ln(f)^
(1/2))*Pi^(1/2)/b^(5/2)/n/ln(f)^(5/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2244, 2242, 2235} \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\frac {3 \sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{4 b^{5/2} n \log ^{\frac {5}{2}}(f)}-\frac {3 x^{n/2} f^{a+b x^n}}{2 b^2 n \log ^2(f)}+\frac {x^{3 n/2} f^{a+b x^n}}{b n \log (f)} \]

[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 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^{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 \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.38 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {f^a x^{5 n/2} \Gamma \left (\frac {5}{2},-b x^n \log (f)\right )}{n \left (-b x^n \log (f)\right )^{5/2}} \]

[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)))

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79

method	result	size
meijerg	\(\frac {f^{a} \left (-\frac {x^{\frac {n}{2}} \left (-b \right )^{\frac {5}{2}} \sqrt {\ln \left (f \right )}\, \left (-10 b \,x^{n} \ln \left (f \right )+15\right ) {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{10 b^{2}}+\frac {3 \left (-b \right )^{\frac {5}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{4 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {5}{2}} \ln \left (f \right )^{\frac {5}{2}} n}\)	\(82\)
risch	\(\frac {f^{a} f^{b \,x^{n}} x^{\frac {3 n}{2}}}{n b \ln \left (f \right )}-\frac {3 f^{a} x^{\frac {n}{2}} f^{b \,x^{n}}}{2 n \ln \left (f \right )^{2} b^{2}}+\frac {3 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{4 n \ln \left (f \right )^{2} b^{2} \sqrt {-b \ln \left (f \right )}}\)	\(96\)

[In]

int(f^(a+b*x^n)*x^(-1+5/2*n),x,method=_RETURNVERBOSE)

[Out]

f^a/(-b)^(5/2)/ln(f)^(5/2)/n*(-1/10*x^(1/2*n)*(-b)^(5/2)*ln(f)^(1/2)*(-10*b*x^n*ln(f)+15)/b^2*exp(b*x^n*ln(f))
+3/4*(-b)^(5/2)/b^(5/2)*Pi^(1/2)*erfi(x^(1/2*n)*b^(1/2)*ln(f)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {3 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x^{\frac {1}{2} \, n}\right ) - 2 \, {\left (2 \, b^{2} x^{\frac {3}{2} \, n} \log \left (f\right )^{2} - 3 \, b x^{\frac {1}{2} \, n} \log \left (f\right )\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{4 \, b^{3} n \log \left (f\right )^{3}} \]

[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)

Sympy [F]

\[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int f^{a + b x^{n}} x^{\frac {5 n}{2} - 1}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.32 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {f^{a} x^{\frac {5}{2} \, n} \Gamma \left (\frac {5}{2}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac {5}{2}} n} \]

[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)

Giac [F]

\[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int { f^{b x^{n} + a} x^{\frac {5}{2} \, n - 1} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int f^{a+b\,x^n}\,x^{\frac {5\,n}{2}-1} \,d x \]

[In]

int(f^(a + b*x^n)*x^((5*n)/2 - 1),x)

[Out]

int(f^(a + b*x^n)*x^((5*n)/2 - 1), x)