3.1886 \(\int \frac {1}{(a+\frac {b}{x^2})^3 x^7} \, dx\)

Optimal. Leaf size=54 \[ -\frac {\log \left (a x^2+b\right )}{2 b^3}+\frac {1}{2 b^2 \left (a x^2+b\right )}+\frac {1}{4 b \left (a x^2+b\right )^2}+\frac {\log (x)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 54, 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} \[ \frac {1}{2 b^2 \left (a x^2+b\right )}-\frac {\log \left (a x^2+b\right )}{2 b^3}+\frac {1}{4 b \left (a x^2+b\right )^2}+\frac {\log (x)}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^3*x^7),x]

[Out]

1/(4*b*(b + a*x^2)^2) + 1/(2*b^2*(b + a*x^2)) + Log[x]/b^3 - Log[b + a*x^2]/(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 )^3 x^7} \, dx &=\int \frac {1}{x \left (b+a x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (b+a x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b^3 x}-\frac {a}{b (b+a x)^3}-\frac {a}{b^2 (b+a x)^2}-\frac {a}{b^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4 b \left (b+a x^2\right )^2}+\frac {1}{2 b^2 \left (b+a x^2\right )}+\frac {\log (x)}{b^3}-\frac {\log \left (b+a x^2\right )}{2 b^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 43, normalized size = 0.80 \[ \frac {\frac {b \left (2 a x^2+3 b\right )}{\left (a x^2+b\right )^2}-2 \log \left (a x^2+b\right )+4 \log (x)}{4 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^3*x^7),x]

[Out]

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

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fricas [A]  time = 0.79, size = 90, normalized size = 1.67 \[ \frac {2 \, a b x^{2} + 3 \, b^{2} - 2 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right ) + 4 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \relax (x)}{4 \, {\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^7,x, algorithm="fricas")

[Out]

1/4*(2*a*b*x^2 + 3*b^2 - 2*(a^2*x^4 + 2*a*b*x^2 + b^2)*log(a*x^2 + b) + 4*(a^2*x^4 + 2*a*b*x^2 + b^2)*log(x))/
(a^2*b^3*x^4 + 2*a*b^4*x^2 + b^5)

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giac [A]  time = 0.15, size = 59, normalized size = 1.09 \[ \frac {\log \left (x^{2}\right )}{2 \, b^{3}} - \frac {\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{3}} + \frac {3 \, a^{2} x^{4} + 8 \, a b x^{2} + 6 \, b^{2}}{4 \, {\left (a x^{2} + b\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^7,x, algorithm="giac")

[Out]

1/2*log(x^2)/b^3 - 1/2*log(abs(a*x^2 + b))/b^3 + 1/4*(3*a^2*x^4 + 8*a*b*x^2 + 6*b^2)/((a*x^2 + b)^2*b^3)

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maple [A]  time = 0.01, size = 49, normalized size = 0.91 \[ \frac {1}{4 \left (a \,x^{2}+b \right )^{2} b}+\frac {1}{2 \left (a \,x^{2}+b \right ) b^{2}}+\frac {\ln \relax (x )}{b^{3}}-\frac {\ln \left (a \,x^{2}+b \right )}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^2)^3/x^7,x)

[Out]

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

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maxima [A]  time = 0.89, size = 60, normalized size = 1.11 \[ \frac {2 \, a x^{2} + 3 \, b}{4 \, {\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )}} - \frac {\log \left (a x^{2} + b\right )}{2 \, b^{3}} + \frac {\log \left (x^{2}\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^7,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.14, size = 56, normalized size = 1.04 \[ \frac {\ln \relax (x)}{b^3}+\frac {\frac {3}{4\,b}+\frac {a\,x^2}{2\,b^2}}{a^2\,x^4+2\,a\,b\,x^2+b^2}-\frac {\ln \left (a\,x^2+b\right )}{2\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b/x^2)^3),x)

[Out]

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

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sympy [A]  time = 0.43, size = 56, normalized size = 1.04 \[ \frac {2 a x^{2} + 3 b}{4 a^{2} b^{2} x^{4} + 8 a b^{3} x^{2} + 4 b^{4}} + \frac {\log {\relax (x )}}{b^{3}} - \frac {\log {\left (x^{2} + \frac {b}{a} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**3/x**7,x)

[Out]

(2*a*x**2 + 3*b)/(4*a**2*b**2*x**4 + 8*a*b**3*x**2 + 4*b**4) + log(x)/b**3 - log(x**2 + b/a)/(2*b**3)

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