3.19.73 \(\int \frac {1}{(a+\frac {b}{x^2})^2 x^9} \, dx\) [1873]

Optimal. Leaf size=66 \[ -\frac {1}{4 b^2 x^4}+\frac {a}{b^3 x^2}+\frac {a^2}{2 b^3 \left (b+a x^2\right )}+\frac {3 a^2 \log (x)}{b^4}-\frac {3 a^2 \log \left (b+a x^2\right )}{2 b^4} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 66, 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 = {269, 272, 46} \begin {gather*} -\frac {3 a^2 \log \left (a x^2+b\right )}{2 b^4}+\frac {3 a^2 \log (x)}{b^4}+\frac {a^2}{2 b^3 \left (a x^2+b\right )}+\frac {a}{b^3 x^2}-\frac {1}{4 b^2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^2*x^9),x]

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 269

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 272

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^9} \, dx &=\int \frac {1}{x^5 \left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b^2 x^3}-\frac {2 a}{b^3 x^2}+\frac {3 a^2}{b^4 x}-\frac {a^3}{b^3 (b+a x)^2}-\frac {3 a^3}{b^4 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 b^2 x^4}+\frac {a}{b^3 x^2}+\frac {a^2}{2 b^3 \left (b+a x^2\right )}+\frac {3 a^2 \log (x)}{b^4}-\frac {3 a^2 \log \left (b+a x^2\right )}{2 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 57, normalized size = 0.86 \begin {gather*} \frac {b \left (-\frac {b}{x^4}+\frac {4 a}{x^2}+\frac {2 a^2}{b+a x^2}\right )+12 a^2 \log (x)-6 a^2 \log \left (b+a x^2\right )}{4 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^2*x^9),x]

[Out]

(b*(-(b/x^4) + (4*a)/x^2 + (2*a^2)/(b + a*x^2)) + 12*a^2*Log[x] - 6*a^2*Log[b + a*x^2])/(4*b^4)

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Maple [A]
time = 0.04, size = 65, normalized size = 0.98

method	result	size
default	\(-\frac {a^{3} \left (-\frac {b}{a \left (a \,x^{2}+b \right )}+\frac {3 \ln \left (a \,x^{2}+b \right )}{a}\right )}{2 b^{4}}-\frac {1}{4 b^{2} x^{4}}+\frac {a}{b^{3} x^{2}}+\frac {3 a^{2} \ln \left (x \right )}{b^{4}}\)	\(65\)
risch	\(\frac {\frac {3 a^{2} x^{4}}{2 b^{3}}+\frac {3 a \,x^{2}}{4 b^{2}}-\frac {1}{4 b}}{\left (a \,x^{2}+b \right ) x^{4}}+\frac {3 a^{2} \ln \left (x \right )}{b^{4}}-\frac {3 a^{2} \ln \left (a \,x^{2}+b \right )}{2 b^{4}}\)	\(67\)
norman	\(\frac {-\frac {x^{4}}{4 b}+\frac {3 a \,x^{6}}{4 b^{2}}-\frac {3 a^{3} x^{10}}{2 b^{4}}}{\left (a \,x^{2}+b \right ) x^{8}}+\frac {3 a^{2} \ln \left (x \right )}{b^{4}}-\frac {3 a^{2} \ln \left (a \,x^{2}+b \right )}{2 b^{4}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x^2+a)^2/x^9,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 70, normalized size = 1.06 \begin {gather*} \frac {6 \, a^{2} x^{4} + 3 \, a b x^{2} - b^{2}}{4 \, {\left (a b^{3} x^{6} + b^{4} x^{4}\right )}} - \frac {3 \, a^{2} \log \left (a x^{2} + b\right )}{2 \, b^{4}} + \frac {3 \, a^{2} \log \left (x^{2}\right )}{2 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^9,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 90, normalized size = 1.36 \begin {gather*} \frac {6 \, a^{2} b x^{4} + 3 \, a b^{2} x^{2} - b^{3} - 6 \, {\left (a^{3} x^{6} + a^{2} b x^{4}\right )} \log \left (a x^{2} + b\right ) + 12 \, {\left (a^{3} x^{6} + a^{2} b x^{4}\right )} \log \left (x\right )}{4 \, {\left (a b^{4} x^{6} + b^{5} x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^9,x, algorithm="fricas")

[Out]

1/4*(6*a^2*b*x^4 + 3*a*b^2*x^2 - b^3 - 6*(a^3*x^6 + a^2*b*x^4)*log(a*x^2 + b) + 12*(a^3*x^6 + a^2*b*x^4)*log(x
))/(a*b^4*x^6 + b^5*x^4)

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Sympy [A]
time = 0.20, size = 68, normalized size = 1.03 \begin {gather*} \frac {3 a^{2} \log {\left (x \right )}}{b^{4}} - \frac {3 a^{2} \log {\left (x^{2} + \frac {b}{a} \right )}}{2 b^{4}} + \frac {6 a^{2} x^{4} + 3 a b x^{2} - b^{2}}{4 a b^{3} x^{6} + 4 b^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**2/x**9,x)

[Out]

3*a**2*log(x)/b**4 - 3*a**2*log(x**2 + b/a)/(2*b**4) + (6*a**2*x**4 + 3*a*b*x**2 - b**2)/(4*a*b**3*x**6 + 4*b*
*4*x**4)

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Giac [A]
time = 1.87, size = 86, normalized size = 1.30 \begin {gather*} \frac {3 \, a^{2} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac {3 \, a^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{4}} + \frac {3 \, a^{3} x^{2} + 4 \, a^{2} b}{2 \, {\left (a x^{2} + b\right )} b^{4}} - \frac {9 \, a^{2} x^{4} - 4 \, a b x^{2} + b^{2}}{4 \, b^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^9,x, algorithm="giac")

[Out]

3/2*a^2*log(x^2)/b^4 - 3/2*a^2*log(abs(a*x^2 + b))/b^4 + 1/2*(3*a^3*x^2 + 4*a^2*b)/((a*x^2 + b)*b^4) - 1/4*(9*
a^2*x^4 - 4*a*b*x^2 + b^2)/(b^4*x^4)

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Mupad [B]
time = 0.08, size = 67, normalized size = 1.02 \begin {gather*} \frac {\frac {3\,a\,x^2}{4\,b^2}-\frac {1}{4\,b}+\frac {3\,a^2\,x^4}{2\,b^3}}{a\,x^6+b\,x^4}-\frac {3\,a^2\,\ln \left (a\,x^2+b\right )}{2\,b^4}+\frac {3\,a^2\,\ln \left (x\right )}{b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b/x^2)^2),x)

[Out]

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

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