3.166 \(\int \frac {1}{x^8 (a+b x^2)^2} \, dx\)

Optimal. Leaf size=94 \[ \frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}}+\frac {9 b^3}{2 a^5 x}-\frac {3 b^2}{2 a^4 x^3}+\frac {9 b}{10 a^3 x^5}-\frac {9}{14 a^2 x^7}+\frac {1}{2 a x^7 \left (a+b x^2\right )} \]

[Out]

-9/14/a^2/x^7+9/10*b/a^3/x^5-3/2*b^2/a^4/x^3+9/2*b^3/a^5/x+1/2/a/x^7/(b*x^2+a)+9/2*b^(7/2)*arctan(x*b^(1/2)/a^
(1/2))/a^(11/2)

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Rubi [A]  time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ -\frac {3 b^2}{2 a^4 x^3}+\frac {9 b^3}{2 a^5 x}+\frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}}+\frac {9 b}{10 a^3 x^5}-\frac {9}{14 a^2 x^7}+\frac {1}{2 a x^7 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^2)^2),x]

[Out]

-9/(14*a^2*x^7) + (9*b)/(10*a^3*x^5) - (3*b^2)/(2*a^4*x^3) + (9*b^3)/(2*a^5*x) + 1/(2*a*x^7*(a + b*x^2)) + (9*
b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(11/2))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {1}{x^8 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2 a x^7 \left (a+b x^2\right )}+\frac {9 \int \frac {1}{x^8 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac {9}{14 a^2 x^7}+\frac {1}{2 a x^7 \left (a+b x^2\right )}-\frac {(9 b) \int \frac {1}{x^6 \left (a+b x^2\right )} \, dx}{2 a^2}\\ &=-\frac {9}{14 a^2 x^7}+\frac {9 b}{10 a^3 x^5}+\frac {1}{2 a x^7 \left (a+b x^2\right )}+\frac {\left (9 b^2\right ) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{2 a^3}\\ &=-\frac {9}{14 a^2 x^7}+\frac {9 b}{10 a^3 x^5}-\frac {3 b^2}{2 a^4 x^3}+\frac {1}{2 a x^7 \left (a+b x^2\right )}-\frac {\left (9 b^3\right ) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{2 a^4}\\ &=-\frac {9}{14 a^2 x^7}+\frac {9 b}{10 a^3 x^5}-\frac {3 b^2}{2 a^4 x^3}+\frac {9 b^3}{2 a^5 x}+\frac {1}{2 a x^7 \left (a+b x^2\right )}+\frac {\left (9 b^4\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^5}\\ &=-\frac {9}{14 a^2 x^7}+\frac {9 b}{10 a^3 x^5}-\frac {3 b^2}{2 a^4 x^3}+\frac {9 b^3}{2 a^5 x}+\frac {1}{2 a x^7 \left (a+b x^2\right )}+\frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 91, normalized size = 0.97 \[ \frac {9 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}}+\frac {b^4 x}{2 a^5 \left (a+b x^2\right )}+\frac {4 b^3}{a^5 x}-\frac {b^2}{a^4 x^3}+\frac {2 b}{5 a^3 x^5}-\frac {1}{7 a^2 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^2)^2),x]

[Out]

-1/7*1/(a^2*x^7) + (2*b)/(5*a^3*x^5) - b^2/(a^4*x^3) + (4*b^3)/(a^5*x) + (b^4*x)/(2*a^5*(a + b*x^2)) + (9*b^(7
/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(11/2))

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fricas [A]  time = 1.01, size = 220, normalized size = 2.34 \[ \left [\frac {630 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} - 84 \, a^{2} b^{2} x^{4} + 36 \, a^{3} b x^{2} - 20 \, a^{4} + 315 \, {\left (b^{4} x^{9} + a b^{3} x^{7}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{140 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, \frac {315 \, b^{4} x^{8} + 210 \, a b^{3} x^{6} - 42 \, a^{2} b^{2} x^{4} + 18 \, a^{3} b x^{2} - 10 \, a^{4} + 315 \, {\left (b^{4} x^{9} + a b^{3} x^{7}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{70 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/140*(630*b^4*x^8 + 420*a*b^3*x^6 - 84*a^2*b^2*x^4 + 36*a^3*b*x^2 - 20*a^4 + 315*(b^4*x^9 + a*b^3*x^7)*sqrt(
-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^5*b*x^9 + a^6*x^7), 1/70*(315*b^4*x^8 + 210*a*b^3*x^
6 - 42*a^2*b^2*x^4 + 18*a^3*b*x^2 - 10*a^4 + 315*(b^4*x^9 + a*b^3*x^7)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^5*b*x
^9 + a^6*x^7)]

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giac [A]  time = 0.62, size = 81, normalized size = 0.86 \[ \frac {9 \, b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} + \frac {b^{4} x}{2 \, {\left (b x^{2} + a\right )} a^{5}} + \frac {140 \, b^{3} x^{6} - 35 \, a b^{2} x^{4} + 14 \, a^{2} b x^{2} - 5 \, a^{3}}{35 \, a^{5} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^2+a)^2,x, algorithm="giac")

[Out]

9/2*b^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/2*b^4*x/((b*x^2 + a)*a^5) + 1/35*(140*b^3*x^6 - 35*a*b^2*x^4
+ 14*a^2*b*x^2 - 5*a^3)/(a^5*x^7)

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maple [A]  time = 0.01, size = 81, normalized size = 0.86 \[ \frac {b^{4} x}{2 \left (b \,x^{2}+a \right ) a^{5}}+\frac {9 b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{5}}+\frac {4 b^{3}}{a^{5} x}-\frac {b^{2}}{a^{4} x^{3}}+\frac {2 b}{5 a^{3} x^{5}}-\frac {1}{7 a^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^2+a)^2,x)

[Out]

-1/7/a^2/x^7+4*b^3/a^5/x-b^2/a^4/x^3+2/5*b/a^3/x^5+1/2*b^4/a^5*x/(b*x^2+a)+9/2*b^4/a^5/(a*b)^(1/2)*arctan(1/(a
*b)^(1/2)*b*x)

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maxima [A]  time = 2.97, size = 86, normalized size = 0.91 \[ \frac {315 \, b^{4} x^{8} + 210 \, a b^{3} x^{6} - 42 \, a^{2} b^{2} x^{4} + 18 \, a^{3} b x^{2} - 10 \, a^{4}}{70 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} + \frac {9 \, b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/70*(315*b^4*x^8 + 210*a*b^3*x^6 - 42*a^2*b^2*x^4 + 18*a^3*b*x^2 - 10*a^4)/(a^5*b*x^9 + a^6*x^7) + 9/2*b^4*ar
ctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5)

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mupad [B]  time = 4.56, size = 80, normalized size = 0.85 \[ \frac {\frac {9\,b\,x^2}{35\,a^2}-\frac {1}{7\,a}-\frac {3\,b^2\,x^4}{5\,a^3}+\frac {3\,b^3\,x^6}{a^4}+\frac {9\,b^4\,x^8}{2\,a^5}}{b\,x^9+a\,x^7}+\frac {9\,b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{11/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a + b*x^2)^2),x)

[Out]

((9*b*x^2)/(35*a^2) - 1/(7*a) - (3*b^2*x^4)/(5*a^3) + (3*b^3*x^6)/a^4 + (9*b^4*x^8)/(2*a^5))/(a*x^7 + b*x^9) +
(9*b^(7/2)*atan((b^(1/2)*x)/a^(1/2)))/(2*a^(11/2))

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sympy [A]  time = 0.45, size = 138, normalized size = 1.47 \[ - \frac {9 \sqrt {- \frac {b^{7}}{a^{11}}} \log {\left (- \frac {a^{6} \sqrt {- \frac {b^{7}}{a^{11}}}}{b^{4}} + x \right )}}{4} + \frac {9 \sqrt {- \frac {b^{7}}{a^{11}}} \log {\left (\frac {a^{6} \sqrt {- \frac {b^{7}}{a^{11}}}}{b^{4}} + x \right )}}{4} + \frac {- 10 a^{4} + 18 a^{3} b x^{2} - 42 a^{2} b^{2} x^{4} + 210 a b^{3} x^{6} + 315 b^{4} x^{8}}{70 a^{6} x^{7} + 70 a^{5} b x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(b*x**2+a)**2,x)

[Out]

-9*sqrt(-b**7/a**11)*log(-a**6*sqrt(-b**7/a**11)/b**4 + x)/4 + 9*sqrt(-b**7/a**11)*log(a**6*sqrt(-b**7/a**11)/
b**4 + x)/4 + (-10*a**4 + 18*a**3*b*x**2 - 42*a**2*b**2*x**4 + 210*a*b**3*x**6 + 315*b**4*x**8)/(70*a**6*x**7
+ 70*a**5*b*x**9)

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