3.440 \(\int \frac {(a+b x^2)^{9/2}}{x^{24}} \, dx\)

Optimal. Leaf size=164 \[ -\frac {1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}} \]

[Out]

-1/23*(b*x^2+a)^(11/2)/a/x^23+4/161*b*(b*x^2+a)^(11/2)/a^2/x^21-40/3059*b^2*(b*x^2+a)^(11/2)/a^3/x^19+320/5200
3*b^3*(b*x^2+a)^(11/2)/a^4/x^17-128/52003*b^4*(b*x^2+a)^(11/2)/a^5/x^15+512/676039*b^5*(b*x^2+a)^(11/2)/a^6/x^
13-1024/7436429*b^6*(b*x^2+a)^(11/2)/a^7/x^11

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac {1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^24,x]

[Out]

-(a + b*x^2)^(11/2)/(23*a*x^23) + (4*b*(a + b*x^2)^(11/2))/(161*a^2*x^21) - (40*b^2*(a + b*x^2)^(11/2))/(3059*
a^3*x^19) + (320*b^3*(a + b*x^2)^(11/2))/(52003*a^4*x^17) - (128*b^4*(a + b*x^2)^(11/2))/(52003*a^5*x^15) + (5
12*b^5*(a + b*x^2)^(11/2))/(676039*a^6*x^13) - (1024*b^6*(a + b*x^2)^(11/2))/(7436429*a^7*x^11)

Rule 264

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 271

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}-\frac {(12 b) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx}{23 a}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}+\frac {\left (40 b^2\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx}{161 a^2}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}-\frac {\left (320 b^3\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx}{3059 a^3}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}+\frac {\left (1920 b^4\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{52003 a^4}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}-\frac {\left (512 b^5\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{52003 a^5}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}+\frac {\left (1024 b^6\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{676039 a^6}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac {1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 86, normalized size = 0.52 \[ -\frac {\left (a+b x^2\right )^{11/2} \left (323323 a^6-184756 a^5 b x^2+97240 a^4 b^2 x^4-45760 a^3 b^3 x^6+18304 a^2 b^4 x^8-5632 a b^5 x^{10}+1024 b^6 x^{12}\right )}{7436429 a^7 x^{23}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^24,x]

[Out]

-1/7436429*((a + b*x^2)^(11/2)*(323323*a^6 - 184756*a^5*b*x^2 + 97240*a^4*b^2*x^4 - 45760*a^3*b^3*x^6 + 18304*
a^2*b^4*x^8 - 5632*a*b^5*x^10 + 1024*b^6*x^12))/(a^7*x^23)

________________________________________________________________________________________

fricas [A]  time = 1.74, size = 137, normalized size = 0.84 \[ -\frac {{\left (1024 \, b^{11} x^{22} - 512 \, a b^{10} x^{20} + 384 \, a^{2} b^{9} x^{18} - 320 \, a^{3} b^{8} x^{16} + 280 \, a^{4} b^{7} x^{14} - 252 \, a^{5} b^{6} x^{12} + 231 \, a^{6} b^{5} x^{10} + 530959 \, a^{7} b^{4} x^{8} + 1826110 \, a^{8} b^{3} x^{6} + 2406690 \, a^{9} b^{2} x^{4} + 1431859 \, a^{10} b x^{2} + 323323 \, a^{11}\right )} \sqrt {b x^{2} + a}}{7436429 \, a^{7} x^{23}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^24,x, algorithm="fricas")

[Out]

-1/7436429*(1024*b^11*x^22 - 512*a*b^10*x^20 + 384*a^2*b^9*x^18 - 320*a^3*b^8*x^16 + 280*a^4*b^7*x^14 - 252*a^
5*b^6*x^12 + 231*a^6*b^5*x^10 + 530959*a^7*b^4*x^8 + 1826110*a^8*b^3*x^6 + 2406690*a^9*b^2*x^4 + 1431859*a^10*
b*x^2 + 323323*a^11)*sqrt(b*x^2 + a)/(a^7*x^23)

________________________________________________________________________________________

giac [B]  time = 1.23, size = 462, normalized size = 2.82 \[ \frac {2048 \, {\left (4249388 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{32} b^{\frac {23}{2}} + 28683369 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{30} a b^{\frac {23}{2}} + 100922965 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{28} a^{2} b^{\frac {23}{2}} + 215656441 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} a^{3} b^{\frac {23}{2}} + 313006057 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a^{4} b^{\frac {23}{2}} + 311653979 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{5} b^{\frac {23}{2}} + 216800507 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{6} b^{\frac {23}{2}} + 100105775 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{7} b^{\frac {23}{2}} + 29173683 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{8} b^{\frac {23}{2}} + 4004231 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{9} b^{\frac {23}{2}} + 100947 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{10} b^{\frac {23}{2}} - 33649 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{11} b^{\frac {23}{2}} + 8855 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{12} b^{\frac {23}{2}} - 1771 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{13} b^{\frac {23}{2}} + 253 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{14} b^{\frac {23}{2}} - 23 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{15} b^{\frac {23}{2}} + a^{16} b^{\frac {23}{2}}\right )}}{7436429 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{23}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^24,x, algorithm="giac")

[Out]

2048/7436429*(4249388*(sqrt(b)*x - sqrt(b*x^2 + a))^32*b^(23/2) + 28683369*(sqrt(b)*x - sqrt(b*x^2 + a))^30*a*
b^(23/2) + 100922965*(sqrt(b)*x - sqrt(b*x^2 + a))^28*a^2*b^(23/2) + 215656441*(sqrt(b)*x - sqrt(b*x^2 + a))^2
6*a^3*b^(23/2) + 313006057*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a^4*b^(23/2) + 311653979*(sqrt(b)*x - sqrt(b*x^2 +
a))^22*a^5*b^(23/2) + 216800507*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a^6*b^(23/2) + 100105775*(sqrt(b)*x - sqrt(b
*x^2 + a))^18*a^7*b^(23/2) + 29173683*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^8*b^(23/2) + 4004231*(sqrt(b)*x - sqr
t(b*x^2 + a))^14*a^9*b^(23/2) + 100947*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^10*b^(23/2) - 33649*(sqrt(b)*x - sqr
t(b*x^2 + a))^10*a^11*b^(23/2) + 8855*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^12*b^(23/2) - 1771*(sqrt(b)*x - sqrt(b
*x^2 + a))^6*a^13*b^(23/2) + 253*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^14*b^(23/2) - 23*(sqrt(b)*x - sqrt(b*x^2 +
a))^2*a^15*b^(23/2) + a^16*b^(23/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^23

________________________________________________________________________________________

maple [A]  time = 0.01, size = 83, normalized size = 0.51 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (1024 x^{12} b^{6}-5632 a \,x^{10} b^{5}+18304 a^{2} x^{8} b^{4}-45760 a^{3} x^{6} b^{3}+97240 a^{4} x^{4} b^{2}-184756 a^{5} x^{2} b +323323 a^{6}\right )}{7436429 a^{7} x^{23}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^24,x)

[Out]

-1/7436429*(b*x^2+a)^(11/2)*(1024*b^6*x^12-5632*a*b^5*x^10+18304*a^2*b^4*x^8-45760*a^3*b^3*x^6+97240*a^4*b^2*x
^4-184756*a^5*b*x^2+323323*a^6)/x^23/a^7

________________________________________________________________________________________

maxima [A]  time = 1.59, size = 136, normalized size = 0.83 \[ -\frac {1024 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{6}}{7436429 \, a^{7} x^{11}} + \frac {512 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{5}}{676039 \, a^{6} x^{13}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{52003 \, a^{5} x^{15}} + \frac {320 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{52003 \, a^{4} x^{17}} - \frac {40 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{3059 \, a^{3} x^{19}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{161 \, a^{2} x^{21}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{23 \, a x^{23}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^24,x, algorithm="maxima")

[Out]

-1024/7436429*(b*x^2 + a)^(11/2)*b^6/(a^7*x^11) + 512/676039*(b*x^2 + a)^(11/2)*b^5/(a^6*x^13) - 128/52003*(b*
x^2 + a)^(11/2)*b^4/(a^5*x^15) + 320/52003*(b*x^2 + a)^(11/2)*b^3/(a^4*x^17) - 40/3059*(b*x^2 + a)^(11/2)*b^2/
(a^3*x^19) + 4/161*(b*x^2 + a)^(11/2)*b/(a^2*x^21) - 1/23*(b*x^2 + a)^(11/2)/(a*x^23)

________________________________________________________________________________________

mupad [B]  time = 10.47, size = 231, normalized size = 1.41 \[ \frac {36\,b^6\,\sqrt {b\,x^2+a}}{1062347\,a^2\,x^{11}}-\frac {3713\,b^4\,\sqrt {b\,x^2+a}}{52003\,x^{15}}-\frac {12770\,a\,b^3\,\sqrt {b\,x^2+a}}{52003\,x^{17}}-\frac {31\,a^3\,b\,\sqrt {b\,x^2+a}}{161\,x^{21}}-\frac {3\,b^5\,\sqrt {b\,x^2+a}}{96577\,a\,x^{13}}-\frac {a^4\,\sqrt {b\,x^2+a}}{23\,x^{23}}-\frac {40\,b^7\,\sqrt {b\,x^2+a}}{1062347\,a^3\,x^9}+\frac {320\,b^8\,\sqrt {b\,x^2+a}}{7436429\,a^4\,x^7}-\frac {384\,b^9\,\sqrt {b\,x^2+a}}{7436429\,a^5\,x^5}+\frac {512\,b^{10}\,\sqrt {b\,x^2+a}}{7436429\,a^6\,x^3}-\frac {1024\,b^{11}\,\sqrt {b\,x^2+a}}{7436429\,a^7\,x}-\frac {990\,a^2\,b^2\,\sqrt {b\,x^2+a}}{3059\,x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^24,x)

[Out]

(36*b^6*(a + b*x^2)^(1/2))/(1062347*a^2*x^11) - (3713*b^4*(a + b*x^2)^(1/2))/(52003*x^15) - (12770*a*b^3*(a +
b*x^2)^(1/2))/(52003*x^17) - (31*a^3*b*(a + b*x^2)^(1/2))/(161*x^21) - (3*b^5*(a + b*x^2)^(1/2))/(96577*a*x^13
) - (a^4*(a + b*x^2)^(1/2))/(23*x^23) - (40*b^7*(a + b*x^2)^(1/2))/(1062347*a^3*x^9) + (320*b^8*(a + b*x^2)^(1
/2))/(7436429*a^4*x^7) - (384*b^9*(a + b*x^2)^(1/2))/(7436429*a^5*x^5) + (512*b^10*(a + b*x^2)^(1/2))/(7436429
*a^6*x^3) - (1024*b^11*(a + b*x^2)^(1/2))/(7436429*a^7*x) - (990*a^2*b^2*(a + b*x^2)^(1/2))/(3059*x^19)

________________________________________________________________________________________

sympy [B]  time = 9.07, size = 1950, normalized size = 11.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**24,x)

[Out]

-323323*a**17*b**(73/2)*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 1115464
35*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 +
7436429*a**7*b**42*x**34) - 3371797*a**16*b**(75/2)*x**2*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 446
18574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**
30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 15847689*a**15*b**(77/2)*x**4*sqrt(a/(b*x**2) + 1
)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**3
9*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 44210595*a**14*
b**(79/2)*x**6*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*
b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a
**7*b**42*x**34) - 81074994*a**13*b**(81/2)*x**8*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a*
*12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 446
18574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 102129258*a**12*b**(83/2)*x**10*sqrt(a/(b*x**2) + 1)/(743
6429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**2
8 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 89502546*a**11*b**(85
/2)*x**12*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38
*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b
**42*x**34) - 53885062*a**10*b**(87/2)*x**14*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*
b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 4461857
4*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 21329935*a**9*b**(89/2)*x**16*sqrt(a/(b*x**2) + 1)/(7436429*a
**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 11
1546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 5012953*a**8*b**(91/2)*x**1
8*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 +
148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**
34) - 531157*a**7*b**(93/2)*x**20*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24
+ 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**4
1*x**32 + 7436429*a**7*b**42*x**34) - 231*a**6*b**(95/2)*x**22*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22
+ 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**
40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 2772*a**5*b**(97/2)*x**24*sqrt(a/(b*x**2) +
1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b*
*39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 9240*a**4*b**
(99/2)*x**26*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b*
*38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**
7*b**42*x**34) - 14784*a**3*b**(101/2)*x**28*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*
b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 4461857
4*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 12672*a**2*b**(103/2)*x**30*sqrt(a/(b*x**2) + 1)/(7436429*a**
13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 1115
46435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 5632*a*b**(105/2)*x**32*sqrt(
a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728
580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 1
024*b**(107/2)*x**34*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*
a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 743
6429*a**7*b**42*x**34)

________________________________________________________________________________________