3.180 \(\int \frac{1}{x^9 (a+b x^2)^3} \, dx\)

Optimal. Leaf size=112 \[ \frac{5 b^4}{2 a^6 \left (a+b x^2\right )}+\frac{b^4}{4 a^5 \left (a+b x^2\right )^2}+\frac{5 b^3}{a^6 x^2}-\frac{3 b^2}{2 a^5 x^4}-\frac{15 b^4 \log \left (a+b x^2\right )}{2 a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{b}{2 a^4 x^6}-\frac{1}{8 a^3 x^8} \]

[Out]

-1/(8*a^3*x^8) + b/(2*a^4*x^6) - (3*b^2)/(2*a^5*x^4) + (5*b^3)/(a^6*x^2) + b^4/(4*a^5*(a + b*x^2)^2) + (5*b^4)
/(2*a^6*(a + b*x^2)) + (15*b^4*Log[x])/a^7 - (15*b^4*Log[a + b*x^2])/(2*a^7)

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Rubi [A]  time = 0.0798041, antiderivative size = 112, 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 = {266, 44} \[ \frac{5 b^4}{2 a^6 \left (a+b x^2\right )}+\frac{b^4}{4 a^5 \left (a+b x^2\right )^2}+\frac{5 b^3}{a^6 x^2}-\frac{3 b^2}{2 a^5 x^4}-\frac{15 b^4 \log \left (a+b x^2\right )}{2 a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{b}{2 a^4 x^6}-\frac{1}{8 a^3 x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^2)^3),x]

[Out]

-1/(8*a^3*x^8) + b/(2*a^4*x^6) - (3*b^2)/(2*a^5*x^4) + (5*b^3)/(a^6*x^2) + b^4/(4*a^5*(a + b*x^2)^2) + (5*b^4)
/(2*a^6*(a + b*x^2)) + (15*b^4*Log[x])/a^7 - (15*b^4*Log[a + b*x^2])/(2*a^7)

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]]

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])

Rubi steps

\begin{align*} \int \frac{1}{x^9 \left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^5}-\frac{3 b}{a^4 x^4}+\frac{6 b^2}{a^5 x^3}-\frac{10 b^3}{a^6 x^2}+\frac{15 b^4}{a^7 x}-\frac{b^5}{a^5 (a+b x)^3}-\frac{5 b^5}{a^6 (a+b x)^2}-\frac{15 b^5}{a^7 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{8 a^3 x^8}+\frac{b}{2 a^4 x^6}-\frac{3 b^2}{2 a^5 x^4}+\frac{5 b^3}{a^6 x^2}+\frac{b^4}{4 a^5 \left (a+b x^2\right )^2}+\frac{5 b^4}{2 a^6 \left (a+b x^2\right )}+\frac{15 b^4 \log (x)}{a^7}-\frac{15 b^4 \log \left (a+b x^2\right )}{2 a^7}\\ \end{align*}

Mathematica [A]  time = 0.0576169, size = 96, normalized size = 0.86 \[ \frac{\frac{a \left (20 a^2 b^3 x^6-5 a^3 b^2 x^4+2 a^4 b x^2-a^5+90 a b^4 x^8+60 b^5 x^{10}\right )}{x^8 \left (a+b x^2\right )^2}-60 b^4 \log \left (a+b x^2\right )+120 b^4 \log (x)}{8 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^2)^3),x]

[Out]

((a*(-a^5 + 2*a^4*b*x^2 - 5*a^3*b^2*x^4 + 20*a^2*b^3*x^6 + 90*a*b^4*x^8 + 60*b^5*x^10))/(x^8*(a + b*x^2)^2) +
120*b^4*Log[x] - 60*b^4*Log[a + b*x^2])/(8*a^7)

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Maple [A]  time = 0.014, size = 101, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,{a}^{3}{x}^{8}}}+{\frac{b}{2\,{a}^{4}{x}^{6}}}-{\frac{3\,{b}^{2}}{2\,{a}^{5}{x}^{4}}}+5\,{\frac{{b}^{3}}{{a}^{6}{x}^{2}}}+{\frac{{b}^{4}}{4\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{b}^{4}}{2\,{a}^{6} \left ( b{x}^{2}+a \right ) }}+15\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{7}}}-{\frac{15\,{b}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^2+a)^3,x)

[Out]

-1/8/a^3/x^8+1/2*b/a^4/x^6-3/2*b^2/a^5/x^4+5*b^3/a^6/x^2+1/4*b^4/a^5/(b*x^2+a)^2+5/2*b^4/a^6/(b*x^2+a)+15*b^4*
ln(x)/a^7-15/2*b^4*ln(b*x^2+a)/a^7

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Maxima [A]  time = 2.71982, size = 154, normalized size = 1.38 \begin{align*} \frac{60 \, b^{5} x^{10} + 90 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} - 5 \, a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} - a^{5}}{8 \,{\left (a^{6} b^{2} x^{12} + 2 \, a^{7} b x^{10} + a^{8} x^{8}\right )}} - \frac{15 \, b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{7}} + \frac{15 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/8*(60*b^5*x^10 + 90*a*b^4*x^8 + 20*a^2*b^3*x^6 - 5*a^3*b^2*x^4 + 2*a^4*b*x^2 - a^5)/(a^6*b^2*x^12 + 2*a^7*b*
x^10 + a^8*x^8) - 15/2*b^4*log(b*x^2 + a)/a^7 + 15/2*b^4*log(x^2)/a^7

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Fricas [A]  time = 1.24437, size = 329, normalized size = 2.94 \begin{align*} \frac{60 \, a b^{5} x^{10} + 90 \, a^{2} b^{4} x^{8} + 20 \, a^{3} b^{3} x^{6} - 5 \, a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{2} - a^{6} - 60 \,{\left (b^{6} x^{12} + 2 \, a b^{5} x^{10} + a^{2} b^{4} x^{8}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{6} x^{12} + 2 \, a b^{5} x^{10} + a^{2} b^{4} x^{8}\right )} \log \left (x\right )}{8 \,{\left (a^{7} b^{2} x^{12} + 2 \, a^{8} b x^{10} + a^{9} x^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/8*(60*a*b^5*x^10 + 90*a^2*b^4*x^8 + 20*a^3*b^3*x^6 - 5*a^4*b^2*x^4 + 2*a^5*b*x^2 - a^6 - 60*(b^6*x^12 + 2*a*
b^5*x^10 + a^2*b^4*x^8)*log(b*x^2 + a) + 120*(b^6*x^12 + 2*a*b^5*x^10 + a^2*b^4*x^8)*log(x))/(a^7*b^2*x^12 + 2
*a^8*b*x^10 + a^9*x^8)

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Sympy [A]  time = 3.1706, size = 116, normalized size = 1.04 \begin{align*} \frac{- a^{5} + 2 a^{4} b x^{2} - 5 a^{3} b^{2} x^{4} + 20 a^{2} b^{3} x^{6} + 90 a b^{4} x^{8} + 60 b^{5} x^{10}}{8 a^{8} x^{8} + 16 a^{7} b x^{10} + 8 a^{6} b^{2} x^{12}} + \frac{15 b^{4} \log{\left (x \right )}}{a^{7}} - \frac{15 b^{4} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**2+a)**3,x)

[Out]

(-a**5 + 2*a**4*b*x**2 - 5*a**3*b**2*x**4 + 20*a**2*b**3*x**6 + 90*a*b**4*x**8 + 60*b**5*x**10)/(8*a**8*x**8 +
16*a**7*b*x**10 + 8*a**6*b**2*x**12) + 15*b**4*log(x)/a**7 - 15*b**4*log(a/b + x**2)/(2*a**7)

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Giac [A]  time = 1.60655, size = 161, normalized size = 1.44 \begin{align*} \frac{15 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{7}} - \frac{15 \, b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{7}} + \frac{45 \, b^{6} x^{4} + 100 \, a b^{5} x^{2} + 56 \, a^{2} b^{4}}{4 \,{\left (b x^{2} + a\right )}^{2} a^{7}} - \frac{125 \, b^{4} x^{8} - 40 \, a b^{3} x^{6} + 12 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}}{8 \, a^{7} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^2+a)^3,x, algorithm="giac")

[Out]

15/2*b^4*log(x^2)/a^7 - 15/2*b^4*log(abs(b*x^2 + a))/a^7 + 1/4*(45*b^6*x^4 + 100*a*b^5*x^2 + 56*a^2*b^4)/((b*x
^2 + a)^2*a^7) - 1/8*(125*b^4*x^8 - 40*a*b^3*x^6 + 12*a^2*b^2*x^4 - 4*a^3*b*x^2 + a^4)/(a^7*x^8)