Integrand size = 19, antiderivative size = 159 \[ \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {3 x^{7/2}}{b^4 \sqrt {a x+b x^3}}+\frac {9 \sqrt {x} \sqrt {a x+b x^3}}{2 b^5}-\frac {9 a \text {arctanh}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x+b x^3}}\right )}{2 b^{11/2}} \] Output:
-1/7*x^(25/2)/b/(b*x^3+a*x)^(7/2)-9/35*x^(19/2)/b^2/(b*x^3+a*x)^(5/2)-3/5* x^(13/2)/b^3/(b*x^3+a*x)^(3/2)-3*x^(7/2)/b^4/(b*x^3+a*x)^(1/2)+9/2*x^(1/2) *(b*x^3+a*x)^(1/2)/b^5-9/2*a*arctanh(b^(1/2)*x^(3/2)/(b*x^3+a*x)^(1/2))/b^ (11/2)
Time = 0.53 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.81 \[ \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {x^{9/2} \left (\sqrt {b} \left (a+b x^2\right ) \left (315 a^4 x+1050 a^3 b x^3+1218 a^2 b^2 x^5+528 a b^3 x^7+35 b^4 x^9\right )-630 a \left (a+b x^2\right )^{9/2} \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )\right )}{70 b^{11/2} \left (x \left (a+b x^2\right )\right )^{9/2}} \] Input:
Integrate[x^(29/2)/(a*x + b*x^3)^(9/2),x]
Output:
(x^(9/2)*(Sqrt[b]*(a + b*x^2)*(315*a^4*x + 1050*a^3*b*x^3 + 1218*a^2*b^2*x ^5 + 528*a*b^3*x^7 + 35*b^4*x^9) - 630*a*(a + b*x^2)^(9/2)*ArcTanh[(Sqrt[b ]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])]))/(70*b^(11/2)*(x*(a + b*x^2))^(9/2))
Time = 0.68 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1928, 1928, 1928, 1928, 1930, 1935, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1928 |
| \(\displaystyle \frac {9 \int \frac {x^{23/2}}{\left (b x^3+a x\right )^{7/2}}dx}{7 b}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}\) |
| \(\Big \downarrow \) 1928 |
| \(\displaystyle \frac {9 \left (\frac {7 \int \frac {x^{17/2}}{\left (b x^3+a x\right )^{5/2}}dx}{5 b}-\frac {x^{19/2}}{5 b \left (a x+b x^3\right )^{5/2}}\right )}{7 b}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}\) |
| \(\Big \downarrow \) 1928 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {x^{11/2}}{\left (b x^3+a x\right )^{3/2}}dx}{3 b}-\frac {x^{13/2}}{3 b \left (a x+b x^3\right )^{3/2}}\right )}{5 b}-\frac {x^{19/2}}{5 b \left (a x+b x^3\right )^{5/2}}\right )}{7 b}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}\) |
| \(\Big \downarrow \) 1928 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {x^{5/2}}{\sqrt {b x^3+a x}}dx}{b}-\frac {x^{7/2}}{b \sqrt {a x+b x^3}}\right )}{3 b}-\frac {x^{13/2}}{3 b \left (a x+b x^3\right )^{3/2}}\right )}{5 b}-\frac {x^{19/2}}{5 b \left (a x+b x^3\right )^{5/2}}\right )}{7 b}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {x} \sqrt {a x+b x^3}}{2 b}-\frac {a \int \frac {\sqrt {x}}{\sqrt {b x^3+a x}}dx}{2 b}\right )}{b}-\frac {x^{7/2}}{b \sqrt {a x+b x^3}}\right )}{3 b}-\frac {x^{13/2}}{3 b \left (a x+b x^3\right )^{3/2}}\right )}{5 b}-\frac {x^{19/2}}{5 b \left (a x+b x^3\right )^{5/2}}\right )}{7 b}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {x} \sqrt {a x+b x^3}}{2 b}-\frac {a \int \frac {1}{1-\frac {b x^3}{b x^3+a x}}d\frac {x^{3/2}}{\sqrt {b x^3+a x}}}{2 b}\right )}{b}-\frac {x^{7/2}}{b \sqrt {a x+b x^3}}\right )}{3 b}-\frac {x^{13/2}}{3 b \left (a x+b x^3\right )^{3/2}}\right )}{5 b}-\frac {x^{19/2}}{5 b \left (a x+b x^3\right )^{5/2}}\right )}{7 b}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {x} \sqrt {a x+b x^3}}{2 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x+b x^3}}\right )}{2 b^{3/2}}\right )}{b}-\frac {x^{7/2}}{b \sqrt {a x+b x^3}}\right )}{3 b}-\frac {x^{13/2}}{3 b \left (a x+b x^3\right )^{3/2}}\right )}{5 b}-\frac {x^{19/2}}{5 b \left (a x+b x^3\right )^{5/2}}\right )}{7 b}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}\) |
Input:
Int[x^(29/2)/(a*x + b*x^3)^(9/2),x]
Output:
-1/7*x^(25/2)/(b*(a*x + b*x^3)^(7/2)) + (9*(-1/5*x^(19/2)/(b*(a*x + b*x^3) ^(5/2)) + (7*(-1/3*x^(13/2)/(b*(a*x + b*x^3)^(3/2)) + (5*(-(x^(7/2)/(b*Sqr t[a*x + b*x^3])) + (3*((Sqrt[x]*Sqrt[a*x + b*x^3])/(2*b) - (a*ArcTanh[(Sqr t[b]*x^(3/2))/Sqrt[a*x + b*x^3]])/(2*b^(3/2))))/b))/(3*b)))/(5*b)))/(7*b)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(n - j)*( p + 1))), x] - Simp[c^n*((m + j*p - n + j + 1)/(b*(n - j)*(p + 1))) Int[( c*x)^(m - n)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && !In tegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1] & & GtQ[m + j*p + 1, n - j]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))) I nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Time = 0.49 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(-\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (-35 x^{9} b^{\frac {9}{2}}+315 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a \,b^{3} x^{6} \sqrt {b \,x^{2}+a}-528 b^{\frac {7}{2}} a \,x^{7}+945 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{2} b^{2} x^{4} \sqrt {b \,x^{2}+a}-1218 b^{\frac {5}{2}} a^{2} x^{5}+945 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{3} b \,x^{2} \sqrt {b \,x^{2}+a}-1050 b^{\frac {3}{2}} a^{3} x^{3}+315 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{4} \sqrt {b \,x^{2}+a}-315 \sqrt {b}\, a^{4} x \right )}{70 b^{\frac {11}{2}} \sqrt {x}\, \left (b \,x^{2}+a \right )^{4}}\) | \(212\) |
| risch | \(\frac {x^{\frac {3}{2}} \left (b \,x^{2}+a \right )}{2 b^{5} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {\left (-\frac {53 a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{560 b^{7} \left (x -\frac {\sqrt {-a b}}{b}\right )^{3}}-\frac {571 a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 b^{6} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {97 a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{35 b^{6} \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {9 a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {11}{2}}}+\frac {571 a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 b^{6} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {97 a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{35 b^{6} \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {a^{3} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{112 b^{7} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{4}}-\frac {a^{3} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{112 b^{7} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{4}}-\frac {53 a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{560 b^{7} \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x}}{\sqrt {x \left (b \,x^{2}+a \right )}}\) | \(595\) |
Input:
int(x^(29/2)/(b*x^3+a*x)^(9/2),x,method=_RETURNVERBOSE)
Output:
-1/70*(x*(b*x^2+a))^(1/2)/b^(11/2)*(-35*x^9*b^(9/2)+315*ln(b^(1/2)*x+(b*x^ 2+a)^(1/2))*a*b^3*x^6*(b*x^2+a)^(1/2)-528*b^(7/2)*a*x^7+945*ln(b^(1/2)*x+( b*x^2+a)^(1/2))*a^2*b^2*x^4*(b*x^2+a)^(1/2)-1218*b^(5/2)*a^2*x^5+945*ln(b^ (1/2)*x+(b*x^2+a)^(1/2))*a^3*b*x^2*(b*x^2+a)^(1/2)-1050*b^(3/2)*a^3*x^3+31 5*ln(b^(1/2)*x+(b*x^2+a)^(1/2))*a^4*(b*x^2+a)^(1/2)-315*b^(1/2)*a^4*x)/x^( 1/2)/(b*x^2+a)^4
Time = 0.11 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.40 \[ \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\left [\frac {315 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {b} \log \left (2 \, b x^{2} - 2 \, \sqrt {b x^{3} + a x} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (35 \, b^{5} x^{8} + 528 \, a b^{4} x^{6} + 1218 \, a^{2} b^{3} x^{4} + 1050 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{140 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}}, \frac {315 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{3} + a x} \sqrt {-b} \sqrt {x}}{b x^{2} + a}\right ) + {\left (35 \, b^{5} x^{8} + 528 \, a b^{4} x^{6} + 1218 \, a^{2} b^{3} x^{4} + 1050 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{70 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}}\right ] \] Input:
integrate(x^(29/2)/(b*x^3+a*x)^(9/2),x, algorithm="fricas")
Output:
[1/140*(315*(a*b^4*x^8 + 4*a^2*b^3*x^6 + 6*a^3*b^2*x^4 + 4*a^4*b*x^2 + a^5 )*sqrt(b)*log(2*b*x^2 - 2*sqrt(b*x^3 + a*x)*sqrt(b)*sqrt(x) + a) + 2*(35*b ^5*x^8 + 528*a*b^4*x^6 + 1218*a^2*b^3*x^4 + 1050*a^3*b^2*x^2 + 315*a^4*b)* sqrt(b*x^3 + a*x)*sqrt(x))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3 *b^7*x^2 + a^4*b^6), 1/70*(315*(a*b^4*x^8 + 4*a^2*b^3*x^6 + 6*a^3*b^2*x^4 + 4*a^4*b*x^2 + a^5)*sqrt(-b)*arctan(sqrt(b*x^3 + a*x)*sqrt(-b)*sqrt(x)/(b *x^2 + a)) + (35*b^5*x^8 + 528*a*b^4*x^6 + 1218*a^2*b^3*x^4 + 1050*a^3*b^2 *x^2 + 315*a^4*b)*sqrt(b*x^3 + a*x)*sqrt(x))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a ^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4*b^6)]
Timed out. \[ \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\text {Timed out} \] Input:
integrate(x**(29/2)/(b*x**3+a*x)**(9/2),x)
Output:
Timed out
\[ \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int { \frac {x^{\frac {29}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate(x^(29/2)/(b*x^3+a*x)^(9/2),x, algorithm="maxima")
Output:
integrate(x^(29/2)/(b*x^3 + a*x)^(9/2), x)
Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.57 \[ \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left (x^{2} {\left (\frac {35 \, x^{2}}{b} + \frac {528 \, a}{b^{2}}\right )} + \frac {1218 \, a^{2}}{b^{3}}\right )} x^{2} + \frac {1050 \, a^{3}}{b^{4}}\right )} x^{2} + \frac {315 \, a^{4}}{b^{5}}\right )} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {9 \, a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {11}{2}}} \] Input:
integrate(x^(29/2)/(b*x^3+a*x)^(9/2),x, algorithm="giac")
Output:
1/70*(((x^2*(35*x^2/b + 528*a/b^2) + 1218*a^2/b^3)*x^2 + 1050*a^3/b^4)*x^2 + 315*a^4/b^5)*x/(b*x^2 + a)^(7/2) + 9/2*a*log(abs(-sqrt(b)*x + sqrt(b*x^ 2 + a)))/b^(11/2)
Timed out. \[ \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {x^{29/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \] Input:
int(x^(29/2)/(a*x + b*x^3)^(9/2),x)
Output:
int(x^(29/2)/(a*x + b*x^3)^(9/2), x)
Time = 0.23 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.13 \[ \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {315 \sqrt {b \,x^{2}+a}\, a^{4} b x +1050 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{3}+1218 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{5}+528 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{7}+35 \sqrt {b \,x^{2}+a}\, b^{5} x^{9}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5}-1260 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,x^{2}-1890 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} x^{4}-1260 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} x^{6}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{8}-213 \sqrt {b}\, a^{5}-852 \sqrt {b}\, a^{4} b \,x^{2}-1278 \sqrt {b}\, a^{3} b^{2} x^{4}-852 \sqrt {b}\, a^{2} b^{3} x^{6}-213 \sqrt {b}\, a \,b^{4} x^{8}}{70 b^{6} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(x^(29/2)/(b*x^3+a*x)^(9/2),x)
Output:
(315*sqrt(a + b*x**2)*a**4*b*x + 1050*sqrt(a + b*x**2)*a**3*b**2*x**3 + 12 18*sqrt(a + b*x**2)*a**2*b**3*x**5 + 528*sqrt(a + b*x**2)*a*b**4*x**7 + 35 *sqrt(a + b*x**2)*b**5*x**9 - 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)* x)/sqrt(a))*a**5 - 1260*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a) )*a**4*b*x**2 - 1890*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a **3*b**2*x**4 - 1260*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a **2*b**3*x**6 - 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a* b**4*x**8 - 213*sqrt(b)*a**5 - 852*sqrt(b)*a**4*b*x**2 - 1278*sqrt(b)*a**3 *b**2*x**4 - 852*sqrt(b)*a**2*b**3*x**6 - 213*sqrt(b)*a*b**4*x**8)/(70*b** 6*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x**8))