\(\int \frac {1}{x^3 (c+d x) (a+b x^2)^{3/2}} \, dx\) [1236]

Optimal result

Integrand size = 22, antiderivative size = 187 \[ \int \frac {1}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {b^2 (c-d x)}{a^2 \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {\sqrt {a+b x^2}}{2 a^2 c x^2}+\frac {d \sqrt {a+b x^2}}{a^2 c^2 x}+\frac {d^5 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^3 \left (b c^2+a d^2\right )^{3/2}}+\frac {\left (3 b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2} c^3} \] Output:

-b^2*(-d*x+c)/a^2/(a*d^2+b*c^2)/(b*x^2+a)^(1/2)-1/2*(b*x^2+a)^(1/2)/a^2/c/ 
x^2+d*(b*x^2+a)^(1/2)/a^2/c^2/x+d^5*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/ 
2)/(b*x^2+a)^(1/2))/c^3/(a*d^2+b*c^2)^(3/2)+1/2*(-2*a*d^2+3*b*c^2)*arctanh 
((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)/c^3
 

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {c \left (b^2 c^2 x^2 (3 c-4 d x)+a^2 d^2 (c-2 d x)+a b \left (c^3-2 c^2 d x+c d^2 x^2-2 d^3 x^3\right )\right )}{a^2 \left (b c^2+a d^2\right ) x^2 \sqrt {a+b x^2}}+\frac {4 d^5 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {2 \left (3 b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}}{2 c^3} \] Input:

Integrate[1/(x^3*(c + d*x)*(a + b*x^2)^(3/2)),x]
 

Output:

-1/2*((c*(b^2*c^2*x^2*(3*c - 4*d*x) + a^2*d^2*(c - 2*d*x) + a*b*(c^3 - 2*c 
^2*d*x + c*d^2*x^2 - 2*d^3*x^3)))/(a^2*(b*c^2 + a*d^2)*x^2*Sqrt[a + b*x^2] 
) + (4*d^5*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - 
a*d^2]])/(-(b*c^2) - a*d^2)^(3/2) + (2*(3*b*c^2 - 2*a*d^2)*ArcTanh[(Sqrt[b 
]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(5/2))/c^3
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 2009}

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.

\(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2} (c+d x)} \, dx\)

\(\Big \downarrow \) 617

\(\displaystyle \int \left (-\frac {d^3}{c^3 \left (a+b x^2\right )^{3/2} (c+d x)}+\frac {d^2}{c^3 x \left (a+b x^2\right )^{3/2}}-\frac {d}{c^2 x^2 \left (a+b x^2\right )^{3/2}}+\frac {1}{c x^3 \left (a+b x^2\right )^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2} c^3}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2} c}+\frac {2 b d x}{a^2 c^2 \sqrt {a+b x^2}}-\frac {3 b}{2 a^2 c \sqrt {a+b x^2}}+\frac {d^5 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 \left (a d^2+b c^2\right )^{3/2}}+\frac {d^2}{a c^3 \sqrt {a+b x^2}}+\frac {d}{a c^2 x \sqrt {a+b x^2}}-\frac {d^3 (a d+b c x)}{a c^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}-\frac {1}{2 a c x^2 \sqrt {a+b x^2}}\)

Input:

Int[1/(x^3*(c + d*x)*(a + b*x^2)^(3/2)),x]
 

Output:

(-3*b)/(2*a^2*c*Sqrt[a + b*x^2]) + d^2/(a*c^3*Sqrt[a + b*x^2]) - 1/(2*a*c* 
x^2*Sqrt[a + b*x^2]) + d/(a*c^2*x*Sqrt[a + b*x^2]) + (2*b*d*x)/(a^2*c^2*Sq 
rt[a + b*x^2]) - (d^3*(a*d + b*c*x))/(a*c^3*(b*c^2 + a*d^2)*Sqrt[a + b*x^2 
]) + (d^5*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c 
^3*(b*c^2 + a*d^2)^(3/2)) + (3*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(5 
/2)*c) - (d^2*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(a^(3/2)*c^3)
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(405\) vs. \(2(165)=330\).

Time = 0.42 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.17

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 d x +c \right )}{2 a^{2} c^{2} x^{2}}-\frac {\frac {\left (2 a \,d^{2}-3 b \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}+\frac {b^{2} c^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{\left (\sqrt {-a b}\, d -b c \right ) \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {b^{2} c^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{\left (\sqrt {-a b}\, d +b c \right ) \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {2 b \,a^{2} d^{4} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (\sqrt {-a b}\, d +b c \right ) \left (\sqrt {-a b}\, d -b c \right ) c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 a^{2} c^{2}}\)	\(406\)
default	\(\frac {-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}}{c}+\frac {d^{2} \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{c^{3}}-\frac {d \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{c^{2}}-\frac {d^{2} \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{c^{3}}\)	\(479\)

Input:

int(1/x^3/(d*x+c)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(b*x^2+a)^(1/2)*(-2*d*x+c)/a^2/c^2/x^2-1/2/a^2/c^2*((2*a*d^2-3*b*c^2) 
/c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+b^2*c^2/((-a*b)^(1/2)*d-b 
*c)/(-a*b)^(1/2)/(x+(-a*b)^(1/2)/b)*((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2) 
*(x+(-a*b)^(1/2)/b))^(1/2)+b^2*c^2/((-a*b)^(1/2)*d+b*c)/(-a*b)^(1/2)/(x-(- 
a*b)^(1/2)/b)*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^( 
1/2)+2*b*a^2*d^4/((-a*b)^(1/2)*d+b*c)/((-a*b)^(1/2)*d-b*c)/c/((a*d^2+b*c^2 
)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2) 
^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (166) = 332\).

Time = 0.47 (sec) , antiderivative size = 1949, normalized size of antiderivative = 10.42 \[ \int \frac {1}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x^3/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

[1/4*(2*(a^3*b*d^5*x^4 + a^4*d^5*x^2)*sqrt(b*c^2 + a*d^2)*log((2*a*b*c*d*x 
 - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt(b*c^2 + a*d^2) 
*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - ((3*b^4*c^6 + 
 4*a*b^3*c^4*d^2 - a^2*b^2*c^2*d^4 - 2*a^3*b*d^6)*x^4 + (3*a*b^3*c^6 + 4*a 
^2*b^2*c^4*d^2 - a^3*b*c^2*d^4 - 2*a^4*d^6)*x^2)*sqrt(a)*log(-(b*x^2 - 2*s 
qrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a^2*b^2*c^6 + 2*a^3*b*c^4*d^2 + a^ 
4*c^2*d^4 - 2*(2*a*b^3*c^5*d + 3*a^2*b^2*c^3*d^3 + a^3*b*c*d^5)*x^3 + (3*a 
*b^3*c^6 + 4*a^2*b^2*c^4*d^2 + a^3*b*c^2*d^4)*x^2 - 2*(a^2*b^2*c^5*d + 2*a 
^3*b*c^3*d^3 + a^4*c*d^5)*x)*sqrt(b*x^2 + a))/((a^3*b^3*c^7 + 2*a^4*b^2*c^ 
5*d^2 + a^5*b*c^3*d^4)*x^4 + (a^4*b^2*c^7 + 2*a^5*b*c^5*d^2 + a^6*c^3*d^4) 
*x^2), 1/4*(4*(a^3*b*d^5*x^4 + a^4*d^5*x^2)*sqrt(-b*c^2 - a*d^2)*arctan(sq 
rt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2 
*c^2 + a*b*d^2)*x^2)) - ((3*b^4*c^6 + 4*a*b^3*c^4*d^2 - a^2*b^2*c^2*d^4 - 
2*a^3*b*d^6)*x^4 + (3*a*b^3*c^6 + 4*a^2*b^2*c^4*d^2 - a^3*b*c^2*d^4 - 2*a^ 
4*d^6)*x^2)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 
2*(a^2*b^2*c^6 + 2*a^3*b*c^4*d^2 + a^4*c^2*d^4 - 2*(2*a*b^3*c^5*d + 3*a^2* 
b^2*c^3*d^3 + a^3*b*c*d^5)*x^3 + (3*a*b^3*c^6 + 4*a^2*b^2*c^4*d^2 + a^3*b* 
c^2*d^4)*x^2 - 2*(a^2*b^2*c^5*d + 2*a^3*b*c^3*d^3 + a^4*c*d^5)*x)*sqrt(b*x 
^2 + a))/((a^3*b^3*c^7 + 2*a^4*b^2*c^5*d^2 + a^5*b*c^3*d^4)*x^4 + (a^4*b^2 
*c^7 + 2*a^5*b*c^5*d^2 + a^6*c^3*d^4)*x^2), -1/2*(((3*b^4*c^6 + 4*a*b^3...
 

Sympy [F]

\[ \int \frac {1}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:

integrate(1/x**3/(d*x+c)/(b*x**2+a)**(3/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/(x**3*(a + b*x**2)**(3/2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )} x^{3}} \,d x } \] Input:

integrate(1/x^3/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(3/2)*(d*x + c)*x^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (166) = 332\).

Time = 0.15 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.95 \[ \int \frac {1}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 \, d^{5} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{5} + a c^{3} d^{2}\right )} \sqrt {-b c^{2} - a d^{2}}} + \frac {\frac {{\left (a^{2} b^{3} c^{2} d + a^{3} b^{2} d^{3}\right )} x}{a^{4} b^{2} c^{4} + 2 \, a^{5} b c^{2} d^{2} + a^{6} d^{4}} - \frac {a^{2} b^{3} c^{3} + a^{3} b^{2} c d^{2}}{a^{4} b^{2} c^{4} + 2 \, a^{5} b c^{2} d^{2} + a^{6} d^{4}}}{\sqrt {b x^{2} + a}} - \frac {{\left (3 \, b c^{2} - 2 \, a d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} c^{3}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b c + 2 \, a^{2} \sqrt {b} d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2} c^{2}} \] Input:

integrate(1/x^3/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

-2*d^5*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - 
 a*d^2))/((b*c^5 + a*c^3*d^2)*sqrt(-b*c^2 - a*d^2)) + ((a^2*b^3*c^2*d + a^ 
3*b^2*d^3)*x/(a^4*b^2*c^4 + 2*a^5*b*c^2*d^2 + a^6*d^4) - (a^2*b^3*c^3 + a^ 
3*b^2*c*d^2)/(a^4*b^2*c^4 + 2*a^5*b*c^2*d^2 + a^6*d^4))/sqrt(b*x^2 + a) - 
(3*b*c^2 - 2*a*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt( 
-a)*a^2*c^3) + ((sqrt(b)*x - sqrt(b*x^2 + a))^3*b*c - 2*(sqrt(b)*x - sqrt( 
b*x^2 + a))^2*a*sqrt(b)*d + (sqrt(b)*x - sqrt(b*x^2 + a))*a*b*c + 2*a^2*sq 
rt(b)*d)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^2*a^2*c^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:

int(1/(x^3*(a + b*x^2)^(3/2)*(c + d*x)),x)
 

Output:

int(1/(x^3*(a + b*x^2)^(3/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 959, normalized size of antiderivative = 5.13 \[ \int \frac {1}{x^3 (c+d x) \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(1/x^3/(d*x+c)/(b*x^2+a)^(3/2),x)
 

Output:

(4*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a 
*d + b*c*x)*a**4*d**5*x**2 + 4*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x** 
2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b*d**5*x**4 - 4*sqrt(a*d**2 + 
 b*c**2)*log(c + d*x)*a**4*d**5*x**2 - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x 
)*a**3*b*d**5*x**4 - 2*sqrt(a + b*x**2)*a**4*c**2*d**4 + 4*sqrt(a + b*x**2 
)*a**4*c*d**5*x - 4*sqrt(a + b*x**2)*a**3*b*c**4*d**2 + 8*sqrt(a + b*x**2) 
*a**3*b*c**3*d**3*x - 2*sqrt(a + b*x**2)*a**3*b*c**2*d**4*x**2 + 4*sqrt(a 
+ b*x**2)*a**3*b*c*d**5*x**3 - 2*sqrt(a + b*x**2)*a**2*b**2*c**6 + 4*sqrt( 
a + b*x**2)*a**2*b**2*c**5*d*x - 8*sqrt(a + b*x**2)*a**2*b**2*c**4*d**2*x* 
*2 + 12*sqrt(a + b*x**2)*a**2*b**2*c**3*d**3*x**3 - 6*sqrt(a + b*x**2)*a*b 
**3*c**6*x**2 + 8*sqrt(a + b*x**2)*a*b**3*c**5*d*x**3 + 2*sqrt(a)*log(sqrt 
(a + b*x**2) - sqrt(a))*a**4*d**6*x**2 + sqrt(a)*log(sqrt(a + b*x**2) - sq 
rt(a))*a**3*b*c**2*d**4*x**2 + 2*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a 
**3*b*d**6*x**4 - 4*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*b**2*c**4 
*d**2*x**2 + sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*b**2*c**2*d**4*x 
**4 - 3*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a*b**3*c**6*x**2 - 4*sqrt( 
a)*log(sqrt(a + b*x**2) - sqrt(a))*a*b**3*c**4*d**2*x**4 - 3*sqrt(a)*log(s 
qrt(a + b*x**2) - sqrt(a))*b**4*c**6*x**4 - 2*sqrt(a)*log(sqrt(a + b*x**2) 
 + sqrt(a))*a**4*d**6*x**2 - sqrt(a)*log(sqrt(a + b*x**2) + sqrt(a))*a**3* 
b*c**2*d**4*x**2 - 2*sqrt(a)*log(sqrt(a + b*x**2) + sqrt(a))*a**3*b*d**...