3.205 \(\int \frac {1}{x (a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=166 \[ -\frac {\log \left (a+b x^2\right )}{2 a^{10}}+\frac {\log (x)}{a^{10}}+\frac {1}{2 a^9 \left (a+b x^2\right )}+\frac {1}{4 a^8 \left (a+b x^2\right )^2}+\frac {1}{6 a^7 \left (a+b x^2\right )^3}+\frac {1}{8 a^6 \left (a+b x^2\right )^4}+\frac {1}{10 a^5 \left (a+b x^2\right )^5}+\frac {1}{12 a^4 \left (a+b x^2\right )^6}+\frac {1}{14 a^3 \left (a+b x^2\right )^7}+\frac {1}{16 a^2 \left (a+b x^2\right )^8}+\frac {1}{18 a \left (a+b x^2\right )^9} \]

[Out]

1/18/a/(b*x^2+a)^9+1/16/a^2/(b*x^2+a)^8+1/14/a^3/(b*x^2+a)^7+1/12/a^4/(b*x^2+a)^6+1/10/a^5/(b*x^2+a)^5+1/8/a^6
/(b*x^2+a)^4+1/6/a^7/(b*x^2+a)^3+1/4/a^8/(b*x^2+a)^2+1/2/a^9/(b*x^2+a)+ln(x)/a^10-1/2*ln(b*x^2+a)/a^10

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Rubi [A]  time = 0.13, antiderivative size = 166, 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} \[ \frac {1}{2 a^9 \left (a+b x^2\right )}+\frac {1}{4 a^8 \left (a+b x^2\right )^2}+\frac {1}{6 a^7 \left (a+b x^2\right )^3}+\frac {1}{8 a^6 \left (a+b x^2\right )^4}+\frac {1}{10 a^5 \left (a+b x^2\right )^5}+\frac {1}{12 a^4 \left (a+b x^2\right )^6}+\frac {1}{14 a^3 \left (a+b x^2\right )^7}+\frac {1}{16 a^2 \left (a+b x^2\right )^8}-\frac {\log \left (a+b x^2\right )}{2 a^{10}}+\frac {\log (x)}{a^{10}}+\frac {1}{18 a \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^2)^10),x]

[Out]

1/(18*a*(a + b*x^2)^9) + 1/(16*a^2*(a + b*x^2)^8) + 1/(14*a^3*(a + b*x^2)^7) + 1/(12*a^4*(a + b*x^2)^6) + 1/(1
0*a^5*(a + b*x^2)^5) + 1/(8*a^6*(a + b*x^2)^4) + 1/(6*a^7*(a + b*x^2)^3) + 1/(4*a^8*(a + b*x^2)^2) + 1/(2*a^9*
(a + b*x^2)) + Log[x]/a^10 - Log[a + b*x^2]/(2*a^10)

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 \left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^{10} x}-\frac {b}{a (a+b x)^{10}}-\frac {b}{a^2 (a+b x)^9}-\frac {b}{a^3 (a+b x)^8}-\frac {b}{a^4 (a+b x)^7}-\frac {b}{a^5 (a+b x)^6}-\frac {b}{a^6 (a+b x)^5}-\frac {b}{a^7 (a+b x)^4}-\frac {b}{a^8 (a+b x)^3}-\frac {b}{a^9 (a+b x)^2}-\frac {b}{a^{10} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{18 a \left (a+b x^2\right )^9}+\frac {1}{16 a^2 \left (a+b x^2\right )^8}+\frac {1}{14 a^3 \left (a+b x^2\right )^7}+\frac {1}{12 a^4 \left (a+b x^2\right )^6}+\frac {1}{10 a^5 \left (a+b x^2\right )^5}+\frac {1}{8 a^6 \left (a+b x^2\right )^4}+\frac {1}{6 a^7 \left (a+b x^2\right )^3}+\frac {1}{4 a^8 \left (a+b x^2\right )^2}+\frac {1}{2 a^9 \left (a+b x^2\right )}+\frac {\log (x)}{a^{10}}-\frac {\log \left (a+b x^2\right )}{2 a^{10}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 120, normalized size = 0.72 \[ \frac {\frac {a \left (7129 a^8+41481 a^7 b x^2+120564 a^6 b^2 x^4+210756 a^5 b^3 x^6+236754 a^4 b^4 x^8+173250 a^3 b^5 x^{10}+80220 a^2 b^6 x^{12}+21420 a b^7 x^{14}+2520 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}-2520 \log \left (a+b x^2\right )+5040 \log (x)}{5040 a^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^2)^10),x]

[Out]

((a*(7129*a^8 + 41481*a^7*b*x^2 + 120564*a^6*b^2*x^4 + 210756*a^5*b^3*x^6 + 236754*a^4*b^4*x^8 + 173250*a^3*b^
5*x^10 + 80220*a^2*b^6*x^12 + 21420*a*b^7*x^14 + 2520*b^8*x^16))/(a + b*x^2)^9 + 5040*Log[x] - 2520*Log[a + b*
x^2])/(5040*a^10)

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fricas [B]  time = 0.91, size = 398, normalized size = 2.40 \[ \frac {2520 \, a b^{8} x^{16} + 21420 \, a^{2} b^{7} x^{14} + 80220 \, a^{3} b^{6} x^{12} + 173250 \, a^{4} b^{5} x^{10} + 236754 \, a^{5} b^{4} x^{8} + 210756 \, a^{6} b^{3} x^{6} + 120564 \, a^{7} b^{2} x^{4} + 41481 \, a^{8} b x^{2} + 7129 \, a^{9} - 2520 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (b x^{2} + a\right ) + 5040 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \relax (x)}{5040 \, {\left (a^{10} b^{9} x^{18} + 9 \, a^{11} b^{8} x^{16} + 36 \, a^{12} b^{7} x^{14} + 84 \, a^{13} b^{6} x^{12} + 126 \, a^{14} b^{5} x^{10} + 126 \, a^{15} b^{4} x^{8} + 84 \, a^{16} b^{3} x^{6} + 36 \, a^{17} b^{2} x^{4} + 9 \, a^{18} b x^{2} + a^{19}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

1/5040*(2520*a*b^8*x^16 + 21420*a^2*b^7*x^14 + 80220*a^3*b^6*x^12 + 173250*a^4*b^5*x^10 + 236754*a^5*b^4*x^8 +
210756*a^6*b^3*x^6 + 120564*a^7*b^2*x^4 + 41481*a^8*b*x^2 + 7129*a^9 - 2520*(b^9*x^18 + 9*a*b^8*x^16 + 36*a^2
*b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 36*a^7*b^2*x^4 + 9*a^8*b*x
^2 + a^9)*log(b*x^2 + a) + 5040*(b^9*x^18 + 9*a*b^8*x^16 + 36*a^2*b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x^1
0 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 36*a^7*b^2*x^4 + 9*a^8*b*x^2 + a^9)*log(x))/(a^10*b^9*x^18 + 9*a^11*b^8
*x^16 + 36*a^12*b^7*x^14 + 84*a^13*b^6*x^12 + 126*a^14*b^5*x^10 + 126*a^15*b^4*x^8 + 84*a^16*b^3*x^6 + 36*a^17
*b^2*x^4 + 9*a^18*b*x^2 + a^19)

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giac [A]  time = 0.63, size = 136, normalized size = 0.82 \[ \frac {\log \left (x^{2}\right )}{2 \, a^{10}} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{10}} + \frac {7129 \, b^{9} x^{18} + 66681 \, a b^{8} x^{16} + 278064 \, a^{2} b^{7} x^{14} + 679056 \, a^{3} b^{6} x^{12} + 1071504 \, a^{4} b^{5} x^{10} + 1135008 \, a^{5} b^{4} x^{8} + 809592 \, a^{6} b^{3} x^{6} + 377208 \, a^{7} b^{2} x^{4} + 105642 \, a^{8} b x^{2} + 14258 \, a^{9}}{5040 \, {\left (b x^{2} + a\right )}^{9} a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^2+a)^10,x, algorithm="giac")

[Out]

1/2*log(x^2)/a^10 - 1/2*log(abs(b*x^2 + a))/a^10 + 1/5040*(7129*b^9*x^18 + 66681*a*b^8*x^16 + 278064*a^2*b^7*x
^14 + 679056*a^3*b^6*x^12 + 1071504*a^4*b^5*x^10 + 1135008*a^5*b^4*x^8 + 809592*a^6*b^3*x^6 + 377208*a^7*b^2*x
^4 + 105642*a^8*b*x^2 + 14258*a^9)/((b*x^2 + a)^9*a^10)

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maple [A]  time = 0.02, size = 147, normalized size = 0.89 \[ \frac {1}{18 \left (b \,x^{2}+a \right )^{9} a}+\frac {1}{16 \left (b \,x^{2}+a \right )^{8} a^{2}}+\frac {1}{14 \left (b \,x^{2}+a \right )^{7} a^{3}}+\frac {1}{12 \left (b \,x^{2}+a \right )^{6} a^{4}}+\frac {1}{10 \left (b \,x^{2}+a \right )^{5} a^{5}}+\frac {1}{8 \left (b \,x^{2}+a \right )^{4} a^{6}}+\frac {1}{6 \left (b \,x^{2}+a \right )^{3} a^{7}}+\frac {1}{4 \left (b \,x^{2}+a \right )^{2} a^{8}}+\frac {1}{2 \left (b \,x^{2}+a \right ) a^{9}}+\frac {\ln \relax (x )}{a^{10}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^2+a)^10,x)

[Out]

1/18/a/(b*x^2+a)^9+1/16/a^2/(b*x^2+a)^8+1/14/a^3/(b*x^2+a)^7+1/12/a^4/(b*x^2+a)^6+1/10/a^5/(b*x^2+a)^5+1/8/a^6
/(b*x^2+a)^4+1/6/a^7/(b*x^2+a)^3+1/4/a^8/(b*x^2+a)^2+1/2/a^9/(b*x^2+a)+ln(x)/a^10-1/2*ln(b*x^2+a)/a^10

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maxima [A]  time = 1.65, size = 214, normalized size = 1.29 \[ \frac {2520 \, b^{8} x^{16} + 21420 \, a b^{7} x^{14} + 80220 \, a^{2} b^{6} x^{12} + 173250 \, a^{3} b^{5} x^{10} + 236754 \, a^{4} b^{4} x^{8} + 210756 \, a^{5} b^{3} x^{6} + 120564 \, a^{6} b^{2} x^{4} + 41481 \, a^{7} b x^{2} + 7129 \, a^{8}}{5040 \, {\left (a^{9} b^{9} x^{18} + 9 \, a^{10} b^{8} x^{16} + 36 \, a^{11} b^{7} x^{14} + 84 \, a^{12} b^{6} x^{12} + 126 \, a^{13} b^{5} x^{10} + 126 \, a^{14} b^{4} x^{8} + 84 \, a^{15} b^{3} x^{6} + 36 \, a^{16} b^{2} x^{4} + 9 \, a^{17} b x^{2} + a^{18}\right )}} - \frac {\log \left (b x^{2} + a\right )}{2 \, a^{10}} + \frac {\log \left (x^{2}\right )}{2 \, a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

1/5040*(2520*b^8*x^16 + 21420*a*b^7*x^14 + 80220*a^2*b^6*x^12 + 173250*a^3*b^5*x^10 + 236754*a^4*b^4*x^8 + 210
756*a^5*b^3*x^6 + 120564*a^6*b^2*x^4 + 41481*a^7*b*x^2 + 7129*a^8)/(a^9*b^9*x^18 + 9*a^10*b^8*x^16 + 36*a^11*b
^7*x^14 + 84*a^12*b^6*x^12 + 126*a^13*b^5*x^10 + 126*a^14*b^4*x^8 + 84*a^15*b^3*x^6 + 36*a^16*b^2*x^4 + 9*a^17
*b*x^2 + a^18) - 1/2*log(b*x^2 + a)/a^10 + 1/2*log(x^2)/a^10

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mupad [B]  time = 5.46, size = 210, normalized size = 1.27 \[ \frac {\ln \relax (x)}{a^{10}}+\frac {\frac {7129}{5040\,a}+\frac {4609\,b\,x^2}{560\,a^2}+\frac {3349\,b^2\,x^4}{140\,a^3}+\frac {2509\,b^3\,x^6}{60\,a^4}+\frac {1879\,b^4\,x^8}{40\,a^5}+\frac {275\,b^5\,x^{10}}{8\,a^6}+\frac {191\,b^6\,x^{12}}{12\,a^7}+\frac {17\,b^7\,x^{14}}{4\,a^8}+\frac {b^8\,x^{16}}{2\,a^9}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}-\frac {\ln \left (b\,x^2+a\right )}{2\,a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^2)^10),x)

[Out]

log(x)/a^10 + (7129/(5040*a) + (4609*b*x^2)/(560*a^2) + (3349*b^2*x^4)/(140*a^3) + (2509*b^3*x^6)/(60*a^4) + (
1879*b^4*x^8)/(40*a^5) + (275*b^5*x^10)/(8*a^6) + (191*b^6*x^12)/(12*a^7) + (17*b^7*x^14)/(4*a^8) + (b^8*x^16)
/(2*a^9))/(a^9 + b^9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 36*a^7*b^2*x^4 + 84*a^6*b^3*x^6 + 126*a^5*b^4*x^8 + 1
26*a^4*b^5*x^10 + 84*a^3*b^6*x^12 + 36*a^2*b^7*x^14) - log(a + b*x^2)/(2*a^10)

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sympy [A]  time = 1.31, size = 223, normalized size = 1.34 \[ \frac {7129 a^{8} + 41481 a^{7} b x^{2} + 120564 a^{6} b^{2} x^{4} + 210756 a^{5} b^{3} x^{6} + 236754 a^{4} b^{4} x^{8} + 173250 a^{3} b^{5} x^{10} + 80220 a^{2} b^{6} x^{12} + 21420 a b^{7} x^{14} + 2520 b^{8} x^{16}}{5040 a^{18} + 45360 a^{17} b x^{2} + 181440 a^{16} b^{2} x^{4} + 423360 a^{15} b^{3} x^{6} + 635040 a^{14} b^{4} x^{8} + 635040 a^{13} b^{5} x^{10} + 423360 a^{12} b^{6} x^{12} + 181440 a^{11} b^{7} x^{14} + 45360 a^{10} b^{8} x^{16} + 5040 a^{9} b^{9} x^{18}} + \frac {\log {\relax (x )}}{a^{10}} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**2+a)**10,x)

[Out]

(7129*a**8 + 41481*a**7*b*x**2 + 120564*a**6*b**2*x**4 + 210756*a**5*b**3*x**6 + 236754*a**4*b**4*x**8 + 17325
0*a**3*b**5*x**10 + 80220*a**2*b**6*x**12 + 21420*a*b**7*x**14 + 2520*b**8*x**16)/(5040*a**18 + 45360*a**17*b*
x**2 + 181440*a**16*b**2*x**4 + 423360*a**15*b**3*x**6 + 635040*a**14*b**4*x**8 + 635040*a**13*b**5*x**10 + 42
3360*a**12*b**6*x**12 + 181440*a**11*b**7*x**14 + 45360*a**10*b**8*x**16 + 5040*a**9*b**9*x**18) + log(x)/a**1
0 - log(a/b + x**2)/(2*a**10)

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