3.224 \(\int \frac{1}{x^6 (a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=233 \[ -\frac{7436429 b^2}{65536 a^{12} x}-\frac{7436429 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{25/2}}+\frac{7436429 b}{196608 a^{11} x^3}+\frac{1062347}{65536 a^9 x^5 \left (a+b x^2\right )}+\frac{1062347}{294912 a^8 x^5 \left (a+b x^2\right )^2}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}-\frac{7436429}{327680 a^{10} x^5}+\frac{1}{18 a x^5 \left (a+b x^2\right )^9} \]

[Out]

-7436429/(327680*a^10*x^5) + (7436429*b)/(196608*a^11*x^3) - (7436429*b^2)/(65536*a^12*x) + 1/(18*a*x^5*(a + b
*x^2)^9) + 23/(288*a^2*x^5*(a + b*x^2)^8) + 23/(192*a^3*x^5*(a + b*x^2)^7) + 437/(2304*a^4*x^5*(a + b*x^2)^6)
+ 7429/(23040*a^5*x^5*(a + b*x^2)^5) + 7429/(12288*a^6*x^5*(a + b*x^2)^4) + 96577/(73728*a^7*x^5*(a + b*x^2)^3
) + 1062347/(294912*a^8*x^5*(a + b*x^2)^2) + 1062347/(65536*a^9*x^5*(a + b*x^2)) - (7436429*b^(5/2)*ArcTan[(Sq
rt[b]*x)/Sqrt[a]])/(65536*a^(25/2))

________________________________________________________________________________________

Rubi [A]  time = 0.157845, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ -\frac{7436429 b^2}{65536 a^{12} x}-\frac{7436429 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{25/2}}+\frac{7436429 b}{196608 a^{11} x^3}+\frac{1062347}{65536 a^9 x^5 \left (a+b x^2\right )}+\frac{1062347}{294912 a^8 x^5 \left (a+b x^2\right )^2}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}-\frac{7436429}{327680 a^{10} x^5}+\frac{1}{18 a x^5 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^2)^10),x]

[Out]

-7436429/(327680*a^10*x^5) + (7436429*b)/(196608*a^11*x^3) - (7436429*b^2)/(65536*a^12*x) + 1/(18*a*x^5*(a + b
*x^2)^9) + 23/(288*a^2*x^5*(a + b*x^2)^8) + 23/(192*a^3*x^5*(a + b*x^2)^7) + 437/(2304*a^4*x^5*(a + b*x^2)^6)
+ 7429/(23040*a^5*x^5*(a + b*x^2)^5) + 7429/(12288*a^6*x^5*(a + b*x^2)^4) + 96577/(73728*a^7*x^5*(a + b*x^2)^3
) + 1062347/(294912*a^8*x^5*(a + b*x^2)^2) + 1062347/(65536*a^9*x^5*(a + b*x^2)) - (7436429*b^(5/2)*ArcTan[(Sq
rt[b]*x)/Sqrt[a]])/(65536*a^(25/2))

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^6 \left (a+b x^2\right )^{10}} \, dx &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23 \int \frac{1}{x^6 \left (a+b x^2\right )^9} \, dx}{18 a}\\ &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{161 \int \frac{1}{x^6 \left (a+b x^2\right )^8} \, dx}{96 a^2}\\ &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437 \int \frac{1}{x^6 \left (a+b x^2\right )^7} \, dx}{192 a^3}\\ &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429 \int \frac{1}{x^6 \left (a+b x^2\right )^6} \, dx}{2304 a^4}\\ &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429 \int \frac{1}{x^6 \left (a+b x^2\right )^5} \, dx}{1536 a^5}\\ &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{96577 \int \frac{1}{x^6 \left (a+b x^2\right )^4} \, dx}{12288 a^6}\\ &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{1062347 \int \frac{1}{x^6 \left (a+b x^2\right )^3} \, dx}{73728 a^7}\\ &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{1062347}{294912 a^8 x^5 \left (a+b x^2\right )^2}+\frac{1062347 \int \frac{1}{x^6 \left (a+b x^2\right )^2} \, dx}{32768 a^8}\\ &=\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{1062347}{294912 a^8 x^5 \left (a+b x^2\right )^2}+\frac{1062347}{65536 a^9 x^5 \left (a+b x^2\right )}+\frac{7436429 \int \frac{1}{x^6 \left (a+b x^2\right )} \, dx}{65536 a^9}\\ &=-\frac{7436429}{327680 a^{10} x^5}+\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{1062347}{294912 a^8 x^5 \left (a+b x^2\right )^2}+\frac{1062347}{65536 a^9 x^5 \left (a+b x^2\right )}-\frac{(7436429 b) \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{65536 a^{10}}\\ &=-\frac{7436429}{327680 a^{10} x^5}+\frac{7436429 b}{196608 a^{11} x^3}+\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{1062347}{294912 a^8 x^5 \left (a+b x^2\right )^2}+\frac{1062347}{65536 a^9 x^5 \left (a+b x^2\right )}+\frac{\left (7436429 b^2\right ) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{65536 a^{11}}\\ &=-\frac{7436429}{327680 a^{10} x^5}+\frac{7436429 b}{196608 a^{11} x^3}-\frac{7436429 b^2}{65536 a^{12} x}+\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{1062347}{294912 a^8 x^5 \left (a+b x^2\right )^2}+\frac{1062347}{65536 a^9 x^5 \left (a+b x^2\right )}-\frac{\left (7436429 b^3\right ) \int \frac{1}{a+b x^2} \, dx}{65536 a^{12}}\\ &=-\frac{7436429}{327680 a^{10} x^5}+\frac{7436429 b}{196608 a^{11} x^3}-\frac{7436429 b^2}{65536 a^{12} x}+\frac{1}{18 a x^5 \left (a+b x^2\right )^9}+\frac{23}{288 a^2 x^5 \left (a+b x^2\right )^8}+\frac{23}{192 a^3 x^5 \left (a+b x^2\right )^7}+\frac{437}{2304 a^4 x^5 \left (a+b x^2\right )^6}+\frac{7429}{23040 a^5 x^5 \left (a+b x^2\right )^5}+\frac{7429}{12288 a^6 x^5 \left (a+b x^2\right )^4}+\frac{96577}{73728 a^7 x^5 \left (a+b x^2\right )^3}+\frac{1062347}{294912 a^8 x^5 \left (a+b x^2\right )^2}+\frac{1062347}{65536 a^9 x^5 \left (a+b x^2\right )}-\frac{7436429 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{25/2}}\\ \end{align*}

Mathematica [A]  time = 0.0875366, size = 169, normalized size = 0.73 \[ \frac{-\frac{\sqrt{a} \left (11110024926 a^2 b^9 x^{18}[phone] a^3 b^8 x^{16}[phone] a^4 b^7 x^{14}[phone] a^5 b^6 x^{12}[phone] a^6 b^5 x^{10}[phone] a^7 b^4 x^8+1469632311 a^8 b^3 x^6+94961664 a^9 b^2 x^4-4521984 a^{10} b x^2+589824 a^{11}[phone] a b^{10} x^{20}[phone] b^{11} x^{22}\right )}{x^5 \left (a+b x^2\right )^9}-334639305 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2949120 a^{25/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^2)^10),x]

[Out]

(-((Sqrt[a]*(589824*a^11 - 4521984*a^10*b*x^2 + 94961664*a^9*b^2*x^4 + 1469632311*a^8*b^3*x^6 + 7323998514*a^7
*b^4*x^8 + 19562592546*a^6*b^5*x^10 + 32314857354*a^5*b^6*x^12 + 34810986496*a^4*b^7*x^14 + 24648575094*a^3*b^
8*x^16 + 11110024926*a^2*b^9*x^18 + 2900207310*a*b^10*x^20 + 334639305*b^11*x^22))/(x^5*(a + b*x^2)^9)) - 3346
39305*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2949120*a^(25/2))

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Maple [A]  time = 0.022, size = 230, normalized size = 1. \begin{align*} -{\frac{1}{5\,{a}^{10}{x}^{5}}}-55\,{\frac{{b}^{2}}{{a}^{12}x}}+{\frac{10\,b}{3\,{a}^{11}{x}^{3}}}-{\frac{6981491\,{b}^{3}x}{65536\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{74539223\,{b}^{4}{x}^{3}}{98304\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{394553929\,{b}^{5}{x}^{5}}{163840\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{725918941\,{b}^{6}{x}^{7}}{163840\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{463199\,{b}^{7}{x}^{9}}{90\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{631790371\,{b}^{8}{x}^{11}}{163840\,{a}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{297702839\,{b}^{9}{x}^{13}}{163840\,{a}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{48340777\,{b}^{10}{x}^{15}}{98304\,{a}^{11} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{3831949\,{b}^{11}{x}^{17}}{65536\,{a}^{12} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{7436429\,{b}^{3}}{65536\,{a}^{12}}\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^6/(b*x^2+a)^10,x)

[Out]

-1/5/a^10/x^5-55*b^2/a^12/x+10/3*b/a^11/x^3-6981491/65536/a^4*b^3/(b*x^2+a)^9*x-74539223/98304/a^5*b^4/(b*x^2+
a)^9*x^3-394553929/163840/a^6*b^5/(b*x^2+a)^9*x^5-725918941/163840/a^7*b^6/(b*x^2+a)^9*x^7-463199/90/a^8*b^7/(
b*x^2+a)^9*x^9-631790371/163840/a^9*b^8/(b*x^2+a)^9*x^11-297702839/163840/a^10*b^9/(b*x^2+a)^9*x^13-48340777/9
8304/a^11*b^10/(b*x^2+a)^9*x^15-3831949/65536/a^12*b^11/(b*x^2+a)^9*x^17-7436429/65536/a^12*b^3/(a*b)^(1/2)*ar
ctan(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^6/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43982, size = 1901, normalized size = 8.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[-1/5898240*(669278610*b^11*x^22 + 5800414620*a*b^10*x^20 + 22220049852*a^2*b^9*x^18 + 49297150188*a^3*b^8*x^1
6 + 69621972992*a^4*b^7*x^14 + 64629714708*a^5*b^6*x^12 + 39125185092*a^6*b^5*x^10 + 14647997028*a^7*b^4*x^8 +
2939264622*a^8*b^3*x^6 + 189923328*a^9*b^2*x^4 - 9043968*a^10*b*x^2 + 1179648*a^11 - 334639305*(b^11*x^23 + 9
*a*b^10*x^21 + 36*a^2*b^9*x^19 + 84*a^3*b^8*x^17 + 126*a^4*b^7*x^15 + 126*a^5*b^6*x^13 + 84*a^6*b^5*x^11 + 36*
a^7*b^4*x^9 + 9*a^8*b^3*x^7 + a^9*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^12*b
^9*x^23 + 9*a^13*b^8*x^21 + 36*a^14*b^7*x^19 + 84*a^15*b^6*x^17 + 126*a^16*b^5*x^15 + 126*a^17*b^4*x^13 + 84*a
^18*b^3*x^11 + 36*a^19*b^2*x^9 + 9*a^20*b*x^7 + a^21*x^5), -1/2949120*(334639305*b^11*x^22 + 2900207310*a*b^10
*x^20 + 11110024926*a^2*b^9*x^18 + 24648575094*a^3*b^8*x^16 + 34810986496*a^4*b^7*x^14 + 32314857354*a^5*b^6*x
^12 + 19562592546*a^6*b^5*x^10 + 7323998514*a^7*b^4*x^8 + 1469632311*a^8*b^3*x^6 + 94961664*a^9*b^2*x^4 - 4521
984*a^10*b*x^2 + 589824*a^11 + 334639305*(b^11*x^23 + 9*a*b^10*x^21 + 36*a^2*b^9*x^19 + 84*a^3*b^8*x^17 + 126*
a^4*b^7*x^15 + 126*a^5*b^6*x^13 + 84*a^6*b^5*x^11 + 36*a^7*b^4*x^9 + 9*a^8*b^3*x^7 + a^9*b^2*x^5)*sqrt(b/a)*ar
ctan(x*sqrt(b/a)))/(a^12*b^9*x^23 + 9*a^13*b^8*x^21 + 36*a^14*b^7*x^19 + 84*a^15*b^6*x^17 + 126*a^16*b^5*x^15
+ 126*a^17*b^4*x^13 + 84*a^18*b^3*x^11 + 36*a^19*b^2*x^9 + 9*a^20*b*x^7 + a^21*x^5)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**2+a)**10,x)

[Out]

Timed out

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Giac [A]  time = 2.09083, size = 215, normalized size = 0.92 \begin{align*} -\frac{7436429 \, b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{12}} - \frac{825 \, b^{2} x^{4} - 50 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{12} x^{5}} - \frac{172437705 \, b^{11} x^{17} + 1450223310 \, a b^{10} x^{15} + 5358651102 \, a^{2} b^{9} x^{13} + 11372226678 \, a^{3} b^{8} x^{11} + 15178104832 \, a^{4} b^{7} x^{9} + 13066540938 \, a^{5} b^{6} x^{7} + 7101970722 \, a^{6} b^{5} x^{5} + 2236176690 \, a^{7} b^{4} x^{3} + 314167095 \, a^{8} b^{3} x}{2949120 \,{\left (b x^{2} + a\right )}^{9} a^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-7436429/65536*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^12) - 1/15*(825*b^2*x^4 - 50*a*b*x^2 + 3*a^2)/(a^12*x^5)
- 1/2949120*(172437705*b^11*x^17 + 1450223310*a*b^10*x^15 + 5358651102*a^2*b^9*x^13 + 11372226678*a^3*b^8*x^1
1 + 15178104832*a^4*b^7*x^9 + 13066540938*a^5*b^6*x^7 + 7101970722*a^6*b^5*x^5 + 2236176690*a^7*b^4*x^3 + 3141
67095*a^8*b^3*x)/((b*x^2 + a)^9*a^12)