Integrand size = 19, antiderivative size = 104 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\frac {3 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{4 b^{5/2} n \log ^{\frac {5}{2}}(f)}-\frac {3 f^{a+b x^n} x^{n/2}}{2 b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^{3 n/2}}{b n \log (f)} \] Output:
3/4*f^a*Pi^(1/2)*erfi(b^(1/2)*x^(1/2*n)*ln(f)^(1/2))/b^(5/2)/n/ln(f)^(5/2) -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)
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.38 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {f^a x^{5 n/2} \Gamma \left (\frac {5}{2},-b x^n \log (f)\right )}{n \left (-b x^n \log (f)\right )^{5/2}} \] Input:
Integrate[f^(a + b*x^n)*x^(-1 + (5*n)/2),x]
Output:
-((f^a*x^((5*n)/2)*Gamma[5/2, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(5/2) ))
Time = 0.59 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2642, 2642, 2640, 2633}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^{\frac {5 n}{2}-1} f^{a+b x^n} \, dx\) |
\(\Big \downarrow \) 2642 |
\(\displaystyle \frac {x^{3 n/2} f^{a+b x^n}}{b n \log (f)}-\frac {3 \int f^{b x^n+a} x^{\frac {3 n}{2}-1}dx}{2 b \log (f)}\) |
\(\Big \downarrow \) 2642 |
\(\displaystyle \frac {x^{3 n/2} f^{a+b x^n}}{b n \log (f)}-\frac {3 \left (\frac {x^{n/2} f^{a+b x^n}}{b n \log (f)}-\frac {\int f^{b x^n+a} x^{\frac {n-2}{2}}dx}{2 b \log (f)}\right )}{2 b \log (f)}\) |
\(\Big \downarrow \) 2640 |
\(\displaystyle \frac {x^{3 n/2} f^{a+b x^n}}{b n \log (f)}-\frac {3 \left (\frac {x^{n/2} f^{a+b x^n}}{b n \log (f)}-\frac {\int f^{b x^n+a}dx^{n/2}}{b n \log (f)}\right )}{2 b \log (f)}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {x^{3 n/2} f^{a+b x^n}}{b n \log (f)}-\frac {3 \left (\frac {x^{n/2} f^{a+b x^n}}{b n \log (f)}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{2 b^{3/2} n \log ^{\frac {3}{2}}(f)}\right )}{2 b \log (f)}\) |
Input:
Int[f^(a + b*x^n)*x^(-1 + (5*n)/2),x]
Output:
(f^(a + b*x^n)*x^((3*n)/2))/(b*n*Log[f]) - (3*(-1/2*(f^a*Sqrt[Pi]*Erfi[Sqr t[b]*x^(n/2)*Sqrt[Log[f]]])/(b^(3/2)*n*Log[f]^(3/2)) + (f^(a + b*x^n)*x^(n /2))/(b*n*Log[f])))/(2*b*Log[f])
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]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ .), x_Symbol] :> Simp[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)]
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*L og[F])), x] - Simp[(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] && Intege rQ[2*Simplify[(m + 1)/n]] && LtQ[0, Simplify[(m + 1)/n], 5] && !RationalQ[ m] && SumSimplerQ[m, -n]
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79
method | result | size |
meijerg | \(\frac {f^{a} \left (-\frac {x^{\frac {n}{2}} \left (-b \right )^{\frac {5}{2}} \sqrt {\ln \left (f \right )}\, \left (-10 b \,x^{n} \ln \left (f \right )+15\right ) {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{10 b^{2}}+\frac {3 \left (-b \right )^{\frac {5}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {b}\, x^{\frac {n}{2}} \sqrt {\ln \left (f \right )}\right )}{4 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {5}{2}} \ln \left (f \right )^{\frac {5}{2}} n}\) | \(82\) |
risch | \(\frac {f^{a} f^{b \,x^{n}} x^{\frac {3 n}{2}}}{n b \ln \left (f \right )}-\frac {3 f^{a} x^{\frac {n}{2}} f^{b \,x^{n}}}{2 n \ln \left (f \right )^{2} b^{2}}+\frac {3 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{4 n \ln \left (f \right )^{2} b^{2} \sqrt {-b \ln \left (f \right )}}\) | \(96\) |
Input:
int(f^(a+b*x^n)*x^(-1+5/2*n),x,method=_RETURNVERBOSE)
Output:
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( b^(1/2)*x^(1/2*n)*ln(f)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {3 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x^{\frac {1}{2} \, n}\right ) - 2 \, {\left (2 \, b^{2} x^{\frac {3}{2} \, n} \log \left (f\right )^{2} - 3 \, b x^{\frac {1}{2} \, n} \log \left (f\right )\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{4 \, b^{3} n \log \left (f\right )^{3}} \] Input:
integrate(f^(a+b*x^n)*x^(-1+5/2*n),x, algorithm="fricas")
Output:
-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)
\[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int f^{a + b x^{n}} x^{\frac {5 n}{2} - 1}\, dx \] Input:
integrate(f**(a+b*x**n)*x**(-1+5/2*n),x)
Output:
Integral(f**(a + b*x**n)*x**(5*n/2 - 1), x)
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.32 \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=-\frac {f^{a} x^{\frac {5}{2} \, n} \Gamma \left (\frac {5}{2}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac {5}{2}} n} \] Input:
integrate(f^(a+b*x^n)*x^(-1+5/2*n),x, algorithm="maxima")
Output:
-f^a*x^(5/2*n)*gamma(5/2, -b*x^n*log(f))/((-b*x^n*log(f))^(5/2)*n)
\[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int { f^{b x^{n} + a} x^{\frac {5}{2} \, n - 1} \,d x } \] Input:
integrate(f^(a+b*x^n)*x^(-1+5/2*n),x, algorithm="giac")
Output:
integrate(f^(b*x^n + a)*x^(5/2*n - 1), x)
Timed out. \[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=\int f^{a+b\,x^n}\,x^{\frac {5\,n}{2}-1} \,d x \] Input:
int(f^(a + b*x^n)*x^((5*n)/2 - 1),x)
Output:
int(f^(a + b*x^n)*x^((5*n)/2 - 1), x)
\[ \int f^{a+b x^n} x^{-1+\frac {5 n}{2}} \, dx=f^{a} \left (\int \frac {x^{\frac {5 n}{2}} f^{x^{n} b}}{x}d x \right ) \] Input:
int(f^(a+b*x^n)*x^(-1+5/2*n),x)
Output:
f**a*int((x**((5*n)/2)*f**(x**n*b))/x,x)