3.1871 \(\int \frac {1}{(a+\frac {b}{x^2})^2 x^7} \, dx\)

Optimal. Leaf size=49 \[ \frac {a \log \left (a x^2+b\right )}{b^3}-\frac {2 a \log (x)}{b^3}-\frac {a}{2 b^2 \left (a x^2+b\right )}-\frac {1}{2 b^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 44} \[ -\frac {a}{2 b^2 \left (a x^2+b\right )}+\frac {a \log \left (a x^2+b\right )}{b^3}-\frac {2 a \log (x)}{b^3}-\frac {1}{2 b^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^2*x^7),x]

[Out]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^7} \, dx &=\int \frac {1}{x^3 \left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b^2 x^2}-\frac {2 a}{b^3 x}+\frac {a^2}{b^2 (b+a x)^2}+\frac {2 a^2}{b^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 b^2 x^2}-\frac {a}{2 b^2 \left (b+a x^2\right )}-\frac {2 a \log (x)}{b^3}+\frac {a \log \left (b+a x^2\right )}{b^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 41, normalized size = 0.84 \[ -\frac {b \left (\frac {a}{a x^2+b}+\frac {1}{x^2}\right )-2 a \log \left (a x^2+b\right )+4 a \log (x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^2*x^7),x]

[Out]

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

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fricas [A]  time = 0.94, size = 73, normalized size = 1.49 \[ -\frac {2 \, a b x^{2} + b^{2} - 2 \, {\left (a^{2} x^{4} + a b x^{2}\right )} \log \left (a x^{2} + b\right ) + 4 \, {\left (a^{2} x^{4} + a b x^{2}\right )} \log \relax (x)}{2 \, {\left (a b^{3} x^{4} + b^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^7,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 51, normalized size = 1.04 \[ -\frac {a \log \left (x^{2}\right )}{b^{3}} + \frac {a \log \left ({\left | a x^{2} + b \right |}\right )}{b^{3}} - \frac {2 \, a x^{2} + b}{2 \, {\left (a x^{4} + b x^{2}\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^7,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 46, normalized size = 0.94 \[ -\frac {a}{2 \left (a \,x^{2}+b \right ) b^{2}}-\frac {2 a \ln \relax (x )}{b^{3}}+\frac {a \ln \left (a \,x^{2}+b \right )}{b^{3}}-\frac {1}{2 b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^2)^2/x^7,x)

[Out]

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

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maxima [A]  time = 0.91, size = 52, normalized size = 1.06 \[ -\frac {2 \, a x^{2} + b}{2 \, {\left (a b^{2} x^{4} + b^{3} x^{2}\right )}} + \frac {a \log \left (a x^{2} + b\right )}{b^{3}} - \frac {a \log \left (x^{2}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^7,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.12, size = 51, normalized size = 1.04 \[ \frac {a\,\ln \left (a\,x^2+b\right )}{b^3}-\frac {\frac {1}{2\,b}+\frac {a\,x^2}{b^2}}{a\,x^4+b\,x^2}-\frac {2\,a\,\ln \relax (x)}{b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b/x^2)^2),x)

[Out]

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

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sympy [A]  time = 0.39, size = 51, normalized size = 1.04 \[ - \frac {2 a \log {\relax (x )}}{b^{3}} + \frac {a \log {\left (x^{2} + \frac {b}{a} \right )}}{b^{3}} + \frac {- 2 a x^{2} - b}{2 a b^{2} x^{4} + 2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**2/x**7,x)

[Out]

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

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