3.2.67 \(\int \frac {1}{x^9 (a+b x^2)^2} \, dx\) [167]

Optimal. Leaf size=93 \[ -\frac {1}{8 a^2 x^8}+\frac {b}{3 a^3 x^6}-\frac {3 b^2}{4 a^4 x^4}+\frac {2 b^3}{a^5 x^2}+\frac {b^4}{2 a^5 \left (a+b x^2\right )}+\frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log \left (a+b x^2\right )}{2 a^6} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 93, 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 = {272, 46} \begin {gather*} -\frac {5 b^4 \log \left (a+b x^2\right )}{2 a^6}+\frac {5 b^4 \log (x)}{a^6}+\frac {b^4}{2 a^5 \left (a+b x^2\right )}+\frac {2 b^3}{a^5 x^2}-\frac {3 b^2}{4 a^4 x^4}+\frac {b}{3 a^3 x^6}-\frac {1}{8 a^2 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^2)^2),x]

[Out]

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

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

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Mathematica [A]
time = 0.06, size = 79, normalized size = 0.85 \begin {gather*} \frac {a \left (-\frac {3 a^3}{x^8}+\frac {8 a^2 b}{x^6}-\frac {18 a b^2}{x^4}+12 b^3 \left (\frac {4}{x^2}+\frac {b}{a+b x^2}\right )\right )+120 b^4 \log (x)-60 b^4 \log \left (a+b x^2\right )}{24 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^2)^2),x]

[Out]

(a*((-3*a^3)/x^8 + (8*a^2*b)/x^6 - (18*a*b^2)/x^4 + 12*b^3*(4/x^2 + b/(a + b*x^2))) + 120*b^4*Log[x] - 60*b^4*
Log[a + b*x^2])/(24*a^6)

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Maple [A]
time = 0.05, size = 88, normalized size = 0.95

method	result	size
default	\(-\frac {b^{5} \left (-\frac {a}{b \left (b \,x^{2}+a \right )}+\frac {5 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{6}}-\frac {1}{8 a^{2} x^{8}}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}+\frac {2 b^{3}}{a^{5} x^{2}}-\frac {3 b^{2}}{4 a^{4} x^{4}}+\frac {b}{3 a^{3} x^{6}}\)	\(88\)
norman	\(\frac {-\frac {1}{8 a}+\frac {5 b \,x^{2}}{24 a^{2}}-\frac {5 b^{2} x^{4}}{12 a^{3}}+\frac {5 b^{3} x^{6}}{4 a^{4}}-\frac {5 b^{5} x^{10}}{2 a^{6}}}{x^{8} \left (b \,x^{2}+a \right )}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{6}}\)	\(89\)
risch	\(\frac {\frac {5 b^{4} x^{8}}{2 a^{5}}+\frac {5 b^{3} x^{6}}{4 a^{4}}-\frac {5 b^{2} x^{4}}{12 a^{3}}+\frac {5 b \,x^{2}}{24 a^{2}}-\frac {1}{8 a}}{x^{8} \left (b \,x^{2}+a \right )}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{6}}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 92, normalized size = 0.99 \begin {gather*} \frac {60 \, b^{4} x^{8} + 30 \, a b^{3} x^{6} - 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} - 3 \, a^{4}}{24 \, {\left (a^{5} b x^{10} + a^{6} x^{8}\right )}} - \frac {5 \, b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{6}} + \frac {5 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(60*b^4*x^8 + 30*a*b^3*x^6 - 10*a^2*b^2*x^4 + 5*a^3*b*x^2 - 3*a^4)/(a^5*b*x^10 + a^6*x^8) - 5/2*b^4*log(b
*x^2 + a)/a^6 + 5/2*b^4*log(x^2)/a^6

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Fricas [A]
time = 1.19, size = 112, normalized size = 1.20 \begin {gather*} \frac {60 \, a b^{4} x^{8} + 30 \, a^{2} b^{3} x^{6} - 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} - 3 \, a^{5} - 60 \, {\left (b^{5} x^{10} + a b^{4} x^{8}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{5} x^{10} + a b^{4} x^{8}\right )} \log \left (x\right )}{24 \, {\left (a^{6} b x^{10} + a^{7} x^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(60*a*b^4*x^8 + 30*a^2*b^3*x^6 - 10*a^3*b^2*x^4 + 5*a^4*b*x^2 - 3*a^5 - 60*(b^5*x^10 + a*b^4*x^8)*log(b*x
^2 + a) + 120*(b^5*x^10 + a*b^4*x^8)*log(x))/(a^6*b*x^10 + a^7*x^8)

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Sympy [A]
time = 0.24, size = 94, normalized size = 1.01 \begin {gather*} \frac {- 3 a^{4} + 5 a^{3} b x^{2} - 10 a^{2} b^{2} x^{4} + 30 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{6} x^{8} + 24 a^{5} b x^{10}} + \frac {5 b^{4} \log {\left (x \right )}}{a^{6}} - \frac {5 b^{4} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a**4 + 5*a**3*b*x**2 - 10*a**2*b**2*x**4 + 30*a*b**3*x**6 + 60*b**4*x**8)/(24*a**6*x**8 + 24*a**5*b*x**10)
+ 5*b**4*log(x)/a**6 - 5*b**4*log(a/b + x**2)/(2*a**6)

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Giac [A]
time = 1.79, size = 110, normalized size = 1.18 \begin {gather*} \frac {5 \, b^{4} \log \left (x^{2}\right )}{2 \, a^{6}} - \frac {5 \, b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{6}} + \frac {5 \, b^{5} x^{2} + 6 \, a b^{4}}{2 \, {\left (b x^{2} + a\right )} a^{6}} - \frac {125 \, b^{4} x^{8} - 48 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} - 8 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{6} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*b^4*log(x^2)/a^6 - 5/2*b^4*log(abs(b*x^2 + a))/a^6 + 1/2*(5*b^5*x^2 + 6*a*b^4)/((b*x^2 + a)*a^6) - 1/24*(1
25*b^4*x^8 - 48*a*b^3*x^6 + 18*a^2*b^2*x^4 - 8*a^3*b*x^2 + 3*a^4)/(a^6*x^8)

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Mupad [B]
time = 4.73, size = 89, normalized size = 0.96 \begin {gather*} \frac {\frac {5\,b\,x^2}{24\,a^2}-\frac {1}{8\,a}-\frac {5\,b^2\,x^4}{12\,a^3}+\frac {5\,b^3\,x^6}{4\,a^4}+\frac {5\,b^4\,x^8}{2\,a^5}}{b\,x^{10}+a\,x^8}-\frac {5\,b^4\,\ln \left (b\,x^2+a\right )}{2\,a^6}+\frac {5\,b^4\,\ln \left (x\right )}{a^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b*x^2)^2),x)

[Out]

((5*b*x^2)/(24*a^2) - 1/(8*a) - (5*b^2*x^4)/(12*a^3) + (5*b^3*x^6)/(4*a^4) + (5*b^4*x^8)/(2*a^5))/(a*x^8 + b*x
^10) - (5*b^4*log(a + b*x^2))/(2*a^6) + (5*b^4*log(x))/a^6

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