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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.0693236, antiderivative size = 62, normalized size of antiderivative = 1., 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 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])

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]

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 = 0.0089231, 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]

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

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 74, normalized size = 1.2 \begin{align*}{\frac{1}{{x}^{6}} \left ({\frac{{x}^{4}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}}-{\frac{{x}^{6}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}}-{\frac{{x}^{2}}{2\,b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [C]  time = 1.16573, size = 30, normalized size = 0.48 \begin{align*} -\frac{f^{a} \Gamma \left (3, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, b^{3} \log \left (f\right )^{3}} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.65512, size = 115, normalized size = 1.85 \begin{align*} -\frac{{\left (2 \, x^{4} - 2 \, b x^{2} \log \left (f\right ) + b^{2} \log \left (f\right )^{2}\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{2 \, b^{3} x^{4} \log \left (f\right )^{3}} \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.13622, size = 44, normalized size = 0.71 \begin{align*} \frac{f^{a + \frac{b}{x^{2}}} \left (- b^{2} \log{\left (f \right )}^{2} + 2 b x^{2} \log{\left (f \right )} - 2 x^{4}\right )}{2 b^{3} x^{4} \log{\left (f \right )}^{3}} \end{align*}

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{7}}\,{d x} \end{align*}

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)