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

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

[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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Rubi [A]  time = 0.08, antiderivative size = 112, normalized size of antiderivative = 1.00, 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 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 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}{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*}

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Mathematica [A]  time = 0.06, size = 96, normalized size = 0.86 \[ \frac {\frac {a \left (-a^5+2 a^4 b x^2-5 a^3 b^2 x^4+20 a^2 b^3 x^6+90 a b^4 x^8+60 b^5 x^{10}\right )}{x^8 \left (a+b x^2\right )^2}-60 b^4 \log \left (a+b x^2\right )+120 b^4 \log (x)}{8 a^7} \]

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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fricas [A]  time = 0.95, size = 156, normalized size = 1.39 \[ \frac {60 \, a b^{5} x^{10} + 90 \, a^{2} b^{4} x^{8} + 20 \, a^{3} b^{3} x^{6} - 5 \, a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{2} - a^{6} - 60 \, {\left (b^{6} x^{12} + 2 \, a b^{5} x^{10} + a^{2} b^{4} x^{8}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{6} x^{12} + 2 \, a b^{5} x^{10} + a^{2} b^{4} x^{8}\right )} \log \relax (x)}{8 \, {\left (a^{7} b^{2} x^{12} + 2 \, a^{8} b x^{10} + a^{9} x^{8}\right )}} \]

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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giac [A]  time = 0.63, size = 119, normalized size = 1.06 \[ \frac {15 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{7}} - \frac {15 \, b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{7}} + \frac {45 \, b^{6} x^{4} + 100 \, a b^{5} x^{2} + 56 \, a^{2} b^{4}}{4 \, {\left (b x^{2} + a\right )}^{2} a^{7}} - \frac {125 \, b^{4} x^{8} - 40 \, a b^{3} x^{6} + 12 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}}{8 \, a^{7} x^{8}} \]

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)

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maple [A]  time = 0.01, size = 101, normalized size = 0.90 \[ \frac {b^{4}}{4 \left (b \,x^{2}+a \right )^{2} a^{5}}+\frac {5 b^{4}}{2 \left (b \,x^{2}+a \right ) a^{6}}+\frac {15 b^{4} \ln \relax (x )}{a^{7}}-\frac {15 b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{7}}+\frac {5 b^{3}}{a^{6} x^{2}}-\frac {3 b^{2}}{2 a^{5} x^{4}}+\frac {b}{2 a^{4} x^{6}}-\frac {1}{8 a^{3} x^{8}} \]

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 = 1.34, size = 114, normalized size = 1.02 \[ \frac {60 \, b^{5} x^{10} + 90 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} - 5 \, a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} - a^{5}}{8 \, {\left (a^{6} b^{2} x^{12} + 2 \, a^{7} b x^{10} + a^{8} x^{8}\right )}} - \frac {15 \, b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{7}} + \frac {15 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{7}} \]

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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mupad [B]  time = 4.86, size = 111, normalized size = 0.99 \[ \frac {\frac {b\,x^2}{4\,a^2}-\frac {1}{8\,a}-\frac {5\,b^2\,x^4}{8\,a^3}+\frac {5\,b^3\,x^6}{2\,a^4}+\frac {45\,b^4\,x^8}{4\,a^5}+\frac {15\,b^5\,x^{10}}{2\,a^6}}{a^2\,x^8+2\,a\,b\,x^{10}+b^2\,x^{12}}-\frac {15\,b^4\,\ln \left (b\,x^2+a\right )}{2\,a^7}+\frac {15\,b^4\,\ln \relax (x)}{a^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b*x^2)^3),x)

[Out]

((b*x^2)/(4*a^2) - 1/(8*a) - (5*b^2*x^4)/(8*a^3) + (5*b^3*x^6)/(2*a^4) + (45*b^4*x^8)/(4*a^5) + (15*b^5*x^10)/
(2*a^6))/(a^2*x^8 + b^2*x^12 + 2*a*b*x^10) - (15*b^4*log(a + b*x^2))/(2*a^7) + (15*b^4*log(x))/a^7

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sympy [A]  time = 0.65, size = 116, normalized size = 1.04 \[ \frac {- a^{5} + 2 a^{4} b x^{2} - 5 a^{3} b^{2} x^{4} + 20 a^{2} b^{3} x^{6} + 90 a b^{4} x^{8} + 60 b^{5} x^{10}}{8 a^{8} x^{8} + 16 a^{7} b x^{10} + 8 a^{6} b^{2} x^{12}} + \frac {15 b^{4} \log {\relax (x )}}{a^{7}} - \frac {15 b^{4} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{7}} \]

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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