Integrand size = 13, antiderivative size = 73 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {1}{3} f^{a+\frac {b}{x^2}} x^3+\frac {2}{3} b f^{a+\frac {b}{x^2}} x \log (f)-\frac {2}{3} b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {3}{2}}(f) \] Output:
1/3*f^(a+b/x^2)*x^3+2/3*b*f^(a+b/x^2)*x*ln(f)-2/3*b^(3/2)*f^a*Pi^(1/2)*erf i(b^(1/2)*ln(f)^(1/2)/x)*ln(f)^(3/2)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {1}{3} f^a \left (-2 b^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {3}{2}}(f)+f^{\frac {b}{x^2}} x \left (x^2+2 b \log (f)\right )\right ) \] Input:
Integrate[f^(a + b/x^2)*x^2,x]
Output:
(f^a*(-2*b^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(3/2) + f^ (b/x^2)*x*(x^2 + 2*b*Log[f])))/3
Time = 0.45 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2643, 2635, 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.
\(\displaystyle \int x^2 f^{a+\frac {b}{x^2}} \, dx\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle \frac {2}{3} b \log (f) \int f^{a+\frac {b}{x^2}}dx+\frac {1}{3} x^3 f^{a+\frac {b}{x^2}}\) |
\(\Big \downarrow \) 2635 |
\(\displaystyle \frac {2}{3} b \log (f) \left (2 b \log (f) \int \frac {f^{a+\frac {b}{x^2}}}{x^2}dx+x f^{a+\frac {b}{x^2}}\right )+\frac {1}{3} x^3 f^{a+\frac {b}{x^2}}\) |
\(\Big \downarrow \) 2640 |
\(\displaystyle \frac {2}{3} b \log (f) \left (x f^{a+\frac {b}{x^2}}-2 b \log (f) \int f^{a+\frac {b}{x^2}}d\frac {1}{x}\right )+\frac {1}{3} x^3 f^{a+\frac {b}{x^2}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {2}{3} b \log (f) \left (x f^{a+\frac {b}{x^2}}-\sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )\right )+\frac {1}{3} x^3 f^{a+\frac {b}{x^2}}\) |
Input:
Int[f^(a + b/x^2)*x^2,x]
Output:
(f^(a + b/x^2)*x^3)/3 + (2*b*(f^(a + b/x^2)*x - Sqrt[b]*f^a*Sqrt[Pi]*Erfi[ (Sqrt[b]*Sqrt[Log[f]])/x]*Sqrt[Log[f]])*Log[f])/3
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_)), x_Symbol] :> Simp[(c + d*x)*(F^(a + b*(c + d*x)^n)/d), x] - Simp[b*n*Log[F] Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] && ILtQ[n, 0]
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 + 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))) , x] - Simp[b*n*(Log[F]/(m + 1)) Int[(c + d*x)^(m + n)*F^(a + b*(c + d*x) ^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[ -4, (m + 1)/n, 5] && IntegerQ[n] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {f^{a} f^{\frac {b}{x^{2}}} x^{3}}{3}+\frac {2 f^{a} \ln \left (f \right ) b x \,f^{\frac {b}{x^{2}}}}{3}-\frac {2 f^{a} \ln \left (f \right )^{2} b^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{3 \sqrt {-b \ln \left (f \right )}}\) | \(67\) |
meijerg | \(\frac {f^{a} b \ln \left (f \right )^{\frac {3}{2}} \sqrt {-b}\, \left (-\frac {2 x^{3} \left (\frac {2 b \ln \left (f \right )}{x^{2}}+1\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{3 \left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}}}+\frac {4 b^{\frac {3}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{3 \left (-b \right )^{\frac {3}{2}}}\right )}{2}\) | \(74\) |
Input:
int(f^(a+b/x^2)*x^2,x,method=_RETURNVERBOSE)
Output:
1/3*f^a*f^(b/x^2)*x^3+2/3*f^a*ln(f)*b*x*f^(b/x^2)-2/3*f^a*ln(f)^2*b^2*Pi^( 1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)/x)
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {2}{3} \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} b f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) \log \left (f\right ) + \frac {1}{3} \, {\left (x^{3} + 2 \, b x \log \left (f\right )\right )} f^{\frac {a x^{2} + b}{x^{2}}} \] Input:
integrate(f^(a+b/x^2)*x^2,x, algorithm="fricas")
Output:
2/3*sqrt(pi)*sqrt(-b*log(f))*b*f^a*erf(sqrt(-b*log(f))/x)*log(f) + 1/3*(x^ 3 + 2*b*x*log(f))*f^((a*x^2 + b)/x^2)
\[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\int f^{a + \frac {b}{x^{2}}} x^{2}\, dx \] Input:
integrate(f**(a+b/x**2)*x**2,x)
Output:
Integral(f**(a + b/x**2)*x**2, x)
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.38 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {1}{2} \, f^{a} x^{3} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {3}{2}} \Gamma \left (-\frac {3}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right ) \] Input:
integrate(f^(a+b/x^2)*x^2,x, algorithm="maxima")
Output:
1/2*f^a*x^3*(-b*log(f)/x^2)^(3/2)*gamma(-3/2, -b*log(f)/x^2)
\[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{2} \,d x } \] Input:
integrate(f^(a+b/x^2)*x^2,x, algorithm="giac")
Output:
integrate(f^(a + b/x^2)*x^2, x)
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=x^3\,\left (\frac {f^a\,f^{\frac {b}{x^2}}}{3}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,\ln \left (f\right )}{3\,x^2}\right )-\frac {2\,b^2\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )\,{\ln \left (f\right )}^2}{3\,\sqrt {b\,\ln \left (f\right )}} \] Input:
int(f^(a + b/x^2)*x^2,x)
Output:
x^3*((f^a*f^(b/x^2))/3 + (2*b*f^a*f^(b/x^2)*log(f))/(3*x^2)) - (2*b^2*f^a* pi^(1/2)*erfi((b*log(f))/(x*(b*log(f))^(1/2)))*log(f)^2)/(3*(b*log(f))^(1/ 2))
\[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {-4 f^{\frac {a \,x^{2}+b}{x^{2}}} \mathrm {log}\left (f \right )^{2} b^{2}+2 f^{\frac {a \,x^{2}+b}{x^{2}}} \mathrm {log}\left (f \right ) b \,x^{2}+f^{\frac {a \,x^{2}+b}{x^{2}}} x^{4}-8 \left (\int \frac {f^{\frac {a \,x^{2}+b}{x^{2}}}}{x^{4}}d x \right ) \mathrm {log}\left (f \right )^{3} b^{3} x}{3 x} \] Input:
int(f^(a+b/x^2)*x^2,x)
Output:
( - 4*f**((a*x**2 + b)/x**2)*log(f)**2*b**2 + 2*f**((a*x**2 + b)/x**2)*log (f)*b*x**2 + f**((a*x**2 + b)/x**2)*x**4 - 8*int(f**((a*x**2 + b)/x**2)/x* *4,x)*log(f)**3*b**3*x)/(3*x)