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

Optimal. Leaf size=209 \[ \frac{230945}{196608 a^9 x \left (a+b x^2\right )}+\frac{46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac{46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac{46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}-\frac{230945 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{21/2}}-\frac{230945}{65536 a^{10} x}+\frac{1}{18 a x \left (a+b x^2\right )^9} \]

[Out]

-230945/(65536*a^10*x) + 1/(18*a*x*(a + b*x^2)^9) + 19/(288*a^2*x*(a + b*x^2)^8) + 323/(4032*a^3*x*(a + b*x^2)
^7) + 1615/(16128*a^4*x*(a + b*x^2)^6) + 4199/(32256*a^5*x*(a + b*x^2)^5) + 46189/(258048*a^6*x*(a + b*x^2)^4)
+ 46189/(172032*a^7*x*(a + b*x^2)^3) + 46189/(98304*a^8*x*(a + b*x^2)^2) + 230945/(196608*a^9*x*(a + b*x^2))
- (230945*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(21/2))

________________________________________________________________________________________

Rubi [A]  time = 0.130814, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ \frac{230945}{196608 a^9 x \left (a+b x^2\right )}+\frac{46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac{46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac{46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}-\frac{230945 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{21/2}}-\frac{230945}{65536 a^{10} x}+\frac{1}{18 a x \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^2)^10),x]

[Out]

-230945/(65536*a^10*x) + 1/(18*a*x*(a + b*x^2)^9) + 19/(288*a^2*x*(a + b*x^2)^8) + 323/(4032*a^3*x*(a + b*x^2)
^7) + 1615/(16128*a^4*x*(a + b*x^2)^6) + 4199/(32256*a^5*x*(a + b*x^2)^5) + 46189/(258048*a^6*x*(a + b*x^2)^4)
+ 46189/(172032*a^7*x*(a + b*x^2)^3) + 46189/(98304*a^8*x*(a + b*x^2)^2) + 230945/(196608*a^9*x*(a + b*x^2))
- (230945*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(21/2))

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]

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]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b x^2\right )^{10}} \, dx &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19 \int \frac{1}{x^2 \left (a+b x^2\right )^9} \, dx}{18 a}\\ &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323 \int \frac{1}{x^2 \left (a+b x^2\right )^8} \, dx}{288 a^2}\\ &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615 \int \frac{1}{x^2 \left (a+b x^2\right )^7} \, dx}{1344 a^3}\\ &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{20995 \int \frac{1}{x^2 \left (a+b x^2\right )^6} \, dx}{16128 a^4}\\ &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{46189 \int \frac{1}{x^2 \left (a+b x^2\right )^5} \, dx}{32256 a^5}\\ &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac{46189 \int \frac{1}{x^2 \left (a+b x^2\right )^4} \, dx}{28672 a^6}\\ &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac{46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac{46189 \int \frac{1}{x^2 \left (a+b x^2\right )^3} \, dx}{24576 a^7}\\ &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac{46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac{46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac{230945 \int \frac{1}{x^2 \left (a+b x^2\right )^2} \, dx}{98304 a^8}\\ &=\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac{46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac{46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac{230945}{196608 a^9 x \left (a+b x^2\right )}+\frac{230945 \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{65536 a^9}\\ &=-\frac{230945}{65536 a^{10} x}+\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac{46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac{46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac{230945}{196608 a^9 x \left (a+b x^2\right )}-\frac{(230945 b) \int \frac{1}{a+b x^2} \, dx}{65536 a^{10}}\\ &=-\frac{230945}{65536 a^{10} x}+\frac{1}{18 a x \left (a+b x^2\right )^9}+\frac{19}{288 a^2 x \left (a+b x^2\right )^8}+\frac{323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac{1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac{4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac{46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac{46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac{46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac{230945}{196608 a^9 x \left (a+b x^2\right )}-\frac{230945 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{21/2}}\\ \end{align*}

Mathematica [A]  time = 0.0944645, size = 147, normalized size = 0.7 \[ \frac{-\frac{\sqrt{a} \left (483044562 a^2 b^7 x^{14}[phone] a^3 b^6 x^{12}[phone] a^4 b^5 x^{10}[phone] a^5 b^4 x^8+850547502 a^6 b^3 x^6+318434718 a^7 b^2 x^4+63897057 a^8 b x^2+4128768 a^9+126095970 a b^8 x^{16}[phone] b^9 x^{18}\right )}{x \left (a+b x^2\right )^9}-14549535 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{4128768 a^{21/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^2)^10),x]

[Out]

(-((Sqrt[a]*(4128768*a^9 + 63897057*a^8*b*x^2 + 318434718*a^7*b^2*x^4 + 850547502*a^6*b^3*x^6 + 1404993798*a^5
*b^4*x^8 + 1513521152*a^4*b^5*x^10 + 1071677178*a^3*b^6*x^12 + 483044562*a^2*b^7*x^14 + 126095970*a*b^8*x^16 +
14549535*b^9*x^18))/(x*(a + b*x^2)^9)) - 14549535*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(21/2))

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 206, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{10}x}}-{\frac{424415\,bx}{65536\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{4042835\,{b}^{2}{x}^{3}}{98304\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{3997865\,{b}^{3}{x}^{5}}{32768\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{49153835\,{b}^{4}{x}^{7}}{229376\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{30313\,{b}^{5}{x}^{9}}{126\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{40270037\,{b}^{6}{x}^{11}}{229376\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{2654039\,{b}^{7}{x}^{13}}{32768\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{2117549\,{b}^{8}{x}^{15}}{98304\,{a}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{165409\,{b}^{9}{x}^{17}}{65536\,{a}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{230945\,b}{65536\,{a}^{10}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^10/x-424415/65536/a^2*b/(b*x^2+a)^9*x-4042835/98304/a^3*b^2/(b*x^2+a)^9*x^3-3997865/32768/a^4*b^3/(b*x^2+
a)^9*x^5-49153835/229376/a^5*b^4/(b*x^2+a)^9*x^7-30313/126/a^6*b^5/(b*x^2+a)^9*x^9-40270037/229376/a^7*b^6/(b*
x^2+a)^9*x^11-2654039/32768/a^8*b^7/(b*x^2+a)^9*x^13-2117549/98304/a^9*b^8/(b*x^2+a)^9*x^15-165409/65536/a^10*
b^9/(b*x^2+a)^9*x^17-230945/65536/a^10*b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54778, size = 1686, normalized size = 8.07 \begin{align*} \left [-\frac{29099070 \, b^{9} x^{18} + 252191940 \, a b^{8} x^{16} + 966089124 \, a^{2} b^{7} x^{14} + 2143354356 \, a^{3} b^{6} x^{12} + 3027042304 \, a^{4} b^{5} x^{10} + 2809987596 \, a^{5} b^{4} x^{8} + 1701095004 \, a^{6} b^{3} x^{6} + 636869436 \, a^{7} b^{2} x^{4} + 127794114 \, a^{8} b x^{2} + 8257536 \, a^{9} - 14549535 \,{\left (b^{9} x^{19} + 9 \, a b^{8} x^{17} + 36 \, a^{2} b^{7} x^{15} + 84 \, a^{3} b^{6} x^{13} + 126 \, a^{4} b^{5} x^{11} + 126 \, a^{5} b^{4} x^{9} + 84 \, a^{6} b^{3} x^{7} + 36 \, a^{7} b^{2} x^{5} + 9 \, a^{8} b x^{3} + a^{9} x\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{8257536 \,{\left (a^{10} b^{9} x^{19} + 9 \, a^{11} b^{8} x^{17} + 36 \, a^{12} b^{7} x^{15} + 84 \, a^{13} b^{6} x^{13} + 126 \, a^{14} b^{5} x^{11} + 126 \, a^{15} b^{4} x^{9} + 84 \, a^{16} b^{3} x^{7} + 36 \, a^{17} b^{2} x^{5} + 9 \, a^{18} b x^{3} + a^{19} x\right )}}, -\frac{14549535 \, b^{9} x^{18} + 126095970 \, a b^{8} x^{16} + 483044562 \, a^{2} b^{7} x^{14} + 1071677178 \, a^{3} b^{6} x^{12} + 1513521152 \, a^{4} b^{5} x^{10} + 1404993798 \, a^{5} b^{4} x^{8} + 850547502 \, a^{6} b^{3} x^{6} + 318434718 \, a^{7} b^{2} x^{4} + 63897057 \, a^{8} b x^{2} + 4128768 \, a^{9} + 14549535 \,{\left (b^{9} x^{19} + 9 \, a b^{8} x^{17} + 36 \, a^{2} b^{7} x^{15} + 84 \, a^{3} b^{6} x^{13} + 126 \, a^{4} b^{5} x^{11} + 126 \, a^{5} b^{4} x^{9} + 84 \, a^{6} b^{3} x^{7} + 36 \, a^{7} b^{2} x^{5} + 9 \, a^{8} b x^{3} + a^{9} x\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{4128768 \,{\left (a^{10} b^{9} x^{19} + 9 \, a^{11} b^{8} x^{17} + 36 \, a^{12} b^{7} x^{15} + 84 \, a^{13} b^{6} x^{13} + 126 \, a^{14} b^{5} x^{11} + 126 \, a^{15} b^{4} x^{9} + 84 \, a^{16} b^{3} x^{7} + 36 \, a^{17} b^{2} x^{5} + 9 \, a^{18} b x^{3} + a^{19} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8257536*(29099070*b^9*x^18 + 252191940*a*b^8*x^16 + 966089124*a^2*b^7*x^14 + 2143354356*a^3*b^6*x^12 + 302
7042304*a^4*b^5*x^10 + 2809987596*a^5*b^4*x^8 + 1701095004*a^6*b^3*x^6 + 636869436*a^7*b^2*x^4 + 127794114*a^8
*b*x^2 + 8257536*a^9 - 14549535*(b^9*x^19 + 9*a*b^8*x^17 + 36*a^2*b^7*x^15 + 84*a^3*b^6*x^13 + 126*a^4*b^5*x^1
1 + 126*a^5*b^4*x^9 + 84*a^6*b^3*x^7 + 36*a^7*b^2*x^5 + 9*a^8*b*x^3 + a^9*x)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqr
t(-b/a) - a)/(b*x^2 + a)))/(a^10*b^9*x^19 + 9*a^11*b^8*x^17 + 36*a^12*b^7*x^15 + 84*a^13*b^6*x^13 + 126*a^14*b
^5*x^11 + 126*a^15*b^4*x^9 + 84*a^16*b^3*x^7 + 36*a^17*b^2*x^5 + 9*a^18*b*x^3 + a^19*x), -1/4128768*(14549535*
b^9*x^18 + 126095970*a*b^8*x^16 + 483044562*a^2*b^7*x^14 + 1071677178*a^3*b^6*x^12 + 1513521152*a^4*b^5*x^10 +
1404993798*a^5*b^4*x^8 + 850547502*a^6*b^3*x^6 + 318434718*a^7*b^2*x^4 + 63897057*a^8*b*x^2 + 4128768*a^9 + 1
4549535*(b^9*x^19 + 9*a*b^8*x^17 + 36*a^2*b^7*x^15 + 84*a^3*b^6*x^13 + 126*a^4*b^5*x^11 + 126*a^5*b^4*x^9 + 84
*a^6*b^3*x^7 + 36*a^7*b^2*x^5 + 9*a^8*b*x^3 + a^9*x)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^10*b^9*x^19 + 9*a^11*b^
8*x^17 + 36*a^12*b^7*x^15 + 84*a^13*b^6*x^13 + 126*a^14*b^5*x^11 + 126*a^15*b^4*x^9 + 84*a^16*b^3*x^7 + 36*a^1
7*b^2*x^5 + 9*a^18*b*x^3 + a^19*x)]

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Sympy [A]  time = 118.924, size = 280, normalized size = 1.34 \begin{align*} \frac{230945 \sqrt{- \frac{b}{a^{21}}} \log{\left (- \frac{a^{11} \sqrt{- \frac{b}{a^{21}}}}{b} + x \right )}}{131072} - \frac{230945 \sqrt{- \frac{b}{a^{21}}} \log{\left (\frac{a^{11} \sqrt{- \frac{b}{a^{21}}}}{b} + x \right )}}{131072} - \frac{4128768 a^{9} + 63897057 a^{8} b x^{2} + 318434718 a^{7} b^{2} x^{4} + 850547502 a^{6} b^{3} x^{6} + 1404993798 a^{5} b^{4} x^{8} + 1513521152 a^{4} b^{5} x^{10} + 1071677178 a^{3} b^{6} x^{12} + 483044562 a^{2} b^{7} x^{14} + 126095970 a b^{8} x^{16} + 14549535 b^{9} x^{18}}{4128768 a^{19} x + 37158912 a^{18} b x^{3} + 148635648 a^{17} b^{2} x^{5} + 346816512 a^{16} b^{3} x^{7} + 520224768 a^{15} b^{4} x^{9} + 520224768 a^{14} b^{5} x^{11} + 346816512 a^{13} b^{6} x^{13} + 148635648 a^{12} b^{7} x^{15} + 37158912 a^{11} b^{8} x^{17} + 4128768 a^{10} b^{9} x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

230945*sqrt(-b/a**21)*log(-a**11*sqrt(-b/a**21)/b + x)/131072 - 230945*sqrt(-b/a**21)*log(a**11*sqrt(-b/a**21)
/b + x)/131072 - (4128768*a**9 + 63897057*a**8*b*x**2 + 318434718*a**7*b**2*x**4 + 850547502*a**6*b**3*x**6 +
1404993798*a**5*b**4*x**8 + 1513521152*a**4*b**5*x**10 + 1071677178*a**3*b**6*x**12 + 483044562*a**2*b**7*x**1
4 + 126095970*a*b**8*x**16 + 14549535*b**9*x**18)/(4128768*a**19*x + 37158912*a**18*b*x**3 + 148635648*a**17*b
**2*x**5 + 346816512*a**16*b**3*x**7 + 520224768*a**15*b**4*x**9 + 520224768*a**14*b**5*x**11 + 346816512*a**1
3*b**6*x**13 + 148635648*a**12*b**7*x**15 + 37158912*a**11*b**8*x**17 + 4128768*a**10*b**9*x**19)

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Giac [A]  time = 2.36857, size = 181, normalized size = 0.87 \begin{align*} -\frac{230945 \, b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{10}} - \frac{1}{a^{10} x} - \frac{10420767 \, b^{9} x^{17} + 88937058 \, a b^{8} x^{15} + 334408914 \, a^{2} b^{7} x^{13} + 724860666 \, a^{3} b^{6} x^{11} + 993296384 \, a^{4} b^{5} x^{9} + 884769030 \, a^{5} b^{4} x^{7} + 503730990 \, a^{6} b^{3} x^{5} + 169799070 \, a^{7} b^{2} x^{3} + 26738145 \, a^{8} b x}{4128768 \,{\left (b x^{2} + a\right )}^{9} a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-230945/65536*b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^10) - 1/(a^10*x) - 1/4128768*(10420767*b^9*x^17 + 88937058*
a*b^8*x^15 + 334408914*a^2*b^7*x^13 + 724860666*a^3*b^6*x^11 + 993296384*a^4*b^5*x^9 + 884769030*a^5*b^4*x^7 +
503730990*a^6*b^3*x^5 + 169799070*a^7*b^2*x^3 + 26738145*a^8*b*x)/((b*x^2 + a)^9*a^10)