3.137 \(\int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx\)

Optimal. Leaf size=62 \[ -\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)/x^7,x]

[Out]

-(f^(a + b/x^2)/(b^3*Log[f]^3)) + f^(a + b/x^2)/(b^2*x^2*Log[f]^2) - f^(a + b/x^2)/(2*b*x^4*Log[f])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2212

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)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin {align*} \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx &=-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)}-\frac {2 \int \frac {f^{a+\frac {b}{x^2}}}{x^5} \, dx}{b \log (f)}\\ &=\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)}+\frac {2 \int \frac {f^{a+\frac {b}{x^2}}}{x^3} \, dx}{b^2 \log ^2(f)}\\ &=-\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 45, normalized size = 0.73 \[ -\frac {f^{a+\frac {b}{x^2}} \left (b^2 \log ^2(f)-2 b x^2 \log (f)+2 x^4\right )}{2 b^3 x^4 \log ^3(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)/x^7,x]

[Out]

-1/2*(f^(a + b/x^2)*(2*x^4 - 2*b*x^2*Log[f] + b^2*Log[f]^2))/(b^3*x^4*Log[f]^3)

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 47, normalized size = 0.76 \[ -\frac {{\left (2 \, x^{4} - 2 \, b x^{2} \log \relax (f) + b^{2} \log \relax (f)^{2}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{3} x^{4} \log \relax (f)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^7,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^7,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)/x^7, x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 74, normalized size = 1.19 \[ \frac {-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b \ln \relax (f )}+\frac {x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}}{x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)/x^7,x)

[Out]

(1/b^2*x^4*exp((a+b/x^2)*ln(f))/ln(f)^2-1/b^3/ln(f)^3*x^6*exp((a+b/x^2)*ln(f))-1/2/b*x^2*exp((a+b/x^2)*ln(f))/
ln(f))/x^6

________________________________________________________________________________________

maxima [C]  time = 1.30, size = 22, normalized size = 0.35 \[ -\frac {f^{a} \Gamma \left (3, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, b^{3} \log \relax (f)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^7,x, algorithm="maxima")

[Out]

-1/2*f^a*gamma(3, -b*log(f)/x^2)/(b^3*log(f)^3)

________________________________________________________________________________________

mupad [B]  time = 3.55, size = 47, normalized size = 0.76 \[ -\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \relax (f)}-\frac {x^2}{b^2\,{\ln \relax (f)}^2}+\frac {x^4}{b^3\,{\ln \relax (f)}^3}\right )}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^2)/x^7,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 44, normalized size = 0.71 \[ \frac {f^{a + \frac {b}{x^{2}}} \left (- b^{2} \log {\relax (f )}^{2} + 2 b x^{2} \log {\relax (f )} - 2 x^{4}\right )}{2 b^{3} x^{4} \log {\relax (f )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)/x**7,x)

[Out]

f**(a + b/x**2)*(-b**2*log(f)**2 + 2*b*x**2*log(f) - 2*x**4)/(2*b**3*x**4*log(f)**3)

________________________________________________________________________________________