\(\int \frac {1}{x (c+d x)^3 (a+b x^2)^{3/2}} \, dx\) [1254]

Optimal result

Integrand size = 22, antiderivative size = 276 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {b^2 \left (c \left (b c^2-3 a d^2\right )-d \left (3 b c^2-a d^2\right ) x\right )}{a \left (b c^2+a d^2\right )^3 \sqrt {a+b x^2}}+\frac {d^4 \sqrt {a+b x^2}}{2 c \left (b c^2+a d^2\right )^2 (c+d x)^2}+\frac {d^4 \left (9 b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{2 c^2 \left (b c^2+a d^2\right )^3 (c+d x)}+\frac {d^3 \left (20 b^2 c^4+7 a b c^2 d^2+2 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 c^3 \left (b c^2+a d^2\right )^{7/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2} c^3} \] Output:

b^2*(c*(-3*a*d^2+b*c^2)-d*(-a*d^2+3*b*c^2)*x)/a/(a*d^2+b*c^2)^3/(b*x^2+a)^ 
(1/2)+1/2*d^4*(b*x^2+a)^(1/2)/c/(a*d^2+b*c^2)^2/(d*x+c)^2+1/2*d^4*(2*a*d^2 
+9*b*c^2)*(b*x^2+a)^(1/2)/c^2/(a*d^2+b*c^2)^3/(d*x+c)+1/2*d^3*(2*a^2*d^4+7 
*a*b*c^2*d^2+20*b^2*c^4)*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)^(7/2)-arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)/ 
c^3
 

Mathematica [A] (verified)

Time = 10.53 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {\sqrt {a+b x^2} \left (\frac {d^4 \left (b c^2+a d^2\right )}{c (c+d x)^2}+\frac {d^4 \left (9 b c^2+2 a d^2\right )}{c^2 (c+d x)}+\frac {2 b^2 \left (b c^2 (c-3 d x)+a d^2 (-3 c+d x)\right )}{a \left (a+b x^2\right )}\right )}{\left (b c^2+a d^2\right )^3}+\frac {2 \log (x)}{a^{3/2} c^3}-\frac {d^3 \left (20 b^2 c^4+7 a b c^2 d^2+2 a^2 d^4\right ) \log (c+d x)}{c^3 \left (b c^2+a d^2\right )^{7/2}}-\frac {2 \log \left (a+\sqrt {a} \sqrt {a+b x^2}\right )}{a^{3/2} c^3}+\frac {d^3 \left (20 b^2 c^4+7 a b c^2 d^2+2 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{c^3 \left (b c^2+a d^2\right )^{7/2}}\right ) \] Input:

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

Output:

((Sqrt[a + b*x^2]*((d^4*(b*c^2 + a*d^2))/(c*(c + d*x)^2) + (d^4*(9*b*c^2 + 
 2*a*d^2))/(c^2*(c + d*x)) + (2*b^2*(b*c^2*(c - 3*d*x) + a*d^2*(-3*c + d*x 
)))/(a*(a + b*x^2))))/(b*c^2 + a*d^2)^3 + (2*Log[x])/(a^(3/2)*c^3) - (d^3* 
(20*b^2*c^4 + 7*a*b*c^2*d^2 + 2*a^2*d^4)*Log[c + d*x])/(c^3*(b*c^2 + a*d^2 
)^(7/2)) - (2*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(a^(3/2)*c^3) + (d^3*(20*b 
^2*c^4 + 7*a*b*c^2*d^2 + 2*a^2*d^4)*Log[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)^(7/2)))/2
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.98, 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.

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

\(\Big \downarrow \) 617

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2} c^3}+\frac {3 b d^3 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c \left (a d^2+b c^2\right )^{5/2}}+\frac {3 b d^3 \left (4 b c^2-a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 c \left (a d^2+b c^2\right )^{7/2}}+\frac {d^3 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 \left (a d^2+b c^2\right )^{3/2}}+\frac {1}{a c^3 \sqrt {a+b x^2}}-\frac {d^2 \sqrt {a+b x^2} \left (b c^2-2 a d^2\right )}{a c^2 (c+d x) \left (a d^2+b c^2\right )^2}-\frac {b d^2 \sqrt {a+b x^2} \left (2 b c^2-13 a d^2\right )}{2 a (c+d x) \left (a d^2+b c^2\right )^3}-\frac {d^2 \sqrt {a+b x^2} \left (2 b c^2-3 a d^2\right )}{2 a c (c+d x)^2 \left (a d^2+b c^2\right )^2}-\frac {d (a d+b c x)}{a c^2 \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}-\frac {d (a d+b c x)}{a c \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}-\frac {d (a d+b c x)}{a c^3 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\)

Input:

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

Output:

1/(a*c^3*Sqrt[a + b*x^2]) - (d*(a*d + b*c*x))/(a*c^3*(b*c^2 + a*d^2)*Sqrt[ 
a + b*x^2]) - (d*(a*d + b*c*x))/(a*c*(b*c^2 + a*d^2)*(c + d*x)^2*Sqrt[a + 
b*x^2]) - (d*(a*d + b*c*x))/(a*c^2*(b*c^2 + a*d^2)*(c + d*x)*Sqrt[a + b*x^ 
2]) - (d^2*(2*b*c^2 - 3*a*d^2)*Sqrt[a + b*x^2])/(2*a*c*(b*c^2 + a*d^2)^2*( 
c + d*x)^2) - (b*d^2*(2*b*c^2 - 13*a*d^2)*Sqrt[a + b*x^2])/(2*a*(b*c^2 + a 
*d^2)^3*(c + d*x)) - (d^2*(b*c^2 - 2*a*d^2)*Sqrt[a + b*x^2])/(a*c^2*(b*c^2 
 + a*d^2)^2*(c + d*x)) + (3*b*d^3*(4*b*c^2 - a*d^2)*ArcTanh[(a*d - b*c*x)/ 
(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(2*c*(b*c^2 + a*d^2)^(7/2)) + (3*b 
*d^3*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c*(b*c 
^2 + a*d^2)^(5/2)) + (d^3*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)) - 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. \(1816\) vs. \(2(252)=504\).

Time = 0.48 (sec) , antiderivative size = 1817, normalized size of antiderivative = 6.58

method	result	size
default	\(\text {Expression too large to display}\)	\(1817\)

Input:

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

Output:

1/c^3*(1/a/(b*x^2+a)^(1/2)-1/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x) 
)-1/d/c^2*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^ 
2+b*c^2)/d^2)^(1/2)+3*b*c*d/(a*d^2+b*c^2)*(1/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^ 
2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c 
/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x 
+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2)*d^2/((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)))-4*b/(a*d^2+b 
*c^2)*d^2*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*( 
x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-1/c^3*(1/(a*d^2+b*c^2)* 
d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b*c*d/(a*d^2+b 
*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/ 
d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2)*d^2/((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) 
))-1/d^2/c*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+ 
(a*d^2+b*c^2)/d^2)^(1/2)+5/2*b*c*d/(a*d^2+b*c^2)*(-1/(a*d^2+b*c^2)*d^2/(x+ 
c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+3*b*c*d/(a*d^2+ 
b*c^2)*(1/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2 
)^(1/2)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (254) = 508\).

Time = 5.60 (sec) , antiderivative size = 5036, normalized size of antiderivative = 18.25 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 789 vs. \(2 (254) = 508\).

Time = 0.21 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.86 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {{\left (3 \, a b^{6} c^{8} d + 8 \, a^{2} b^{5} c^{6} d^{3} + 6 \, a^{3} b^{4} c^{4} d^{5} - a^{5} b^{2} d^{9}\right )} x}{a^{2} b^{6} c^{12} + 6 \, a^{3} b^{5} c^{10} d^{2} + 15 \, a^{4} b^{4} c^{8} d^{4} + 20 \, a^{5} b^{3} c^{6} d^{6} + 15 \, a^{6} b^{2} c^{4} d^{8} + 6 \, a^{7} b c^{2} d^{10} + a^{8} d^{12}} - \frac {a b^{6} c^{9} - 6 \, a^{3} b^{4} c^{5} d^{4} - 8 \, a^{4} b^{3} c^{3} d^{6} - 3 \, a^{5} b^{2} c d^{8}}{a^{2} b^{6} c^{12} + 6 \, a^{3} b^{5} c^{10} d^{2} + 15 \, a^{4} b^{4} c^{8} d^{4} + 20 \, a^{5} b^{3} c^{6} d^{6} + 15 \, a^{6} b^{2} c^{4} d^{8} + 6 \, a^{7} b c^{2} d^{10} + a^{8} d^{12}}}{\sqrt {b x^{2} + a}} - \frac {{\left (20 \, b^{2} c^{4} d^{3} + 7 \, a b c^{2} d^{5} + 2 \, a^{2} d^{7}\right )} \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^{3} c^{9} + 3 \, a b^{2} c^{7} d^{2} + 3 \, a^{2} b c^{5} d^{4} + a^{3} c^{3} d^{6}\right )} \sqrt {-b c^{2} - a d^{2}}} + \frac {8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{3} d^{4} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c d^{6} + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{4} d^{3} - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{2} d^{5} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} d^{7} - 28 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{3} d^{4} - 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c d^{6} + 9 \, a^{2} b^{\frac {3}{2}} c^{2} d^{5} + 2 \, a^{3} \sqrt {b} d^{7}}{{\left (b^{3} c^{8} + 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{4} d^{4} + a^{3} c^{2} d^{6}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} + \frac {2 \, \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a c^{3}} \] Input:

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

Output:

-((3*a*b^6*c^8*d + 8*a^2*b^5*c^6*d^3 + 6*a^3*b^4*c^4*d^5 - a^5*b^2*d^9)*x/ 
(a^2*b^6*c^12 + 6*a^3*b^5*c^10*d^2 + 15*a^4*b^4*c^8*d^4 + 20*a^5*b^3*c^6*d 
^6 + 15*a^6*b^2*c^4*d^8 + 6*a^7*b*c^2*d^10 + a^8*d^12) - (a*b^6*c^9 - 6*a^ 
3*b^4*c^5*d^4 - 8*a^4*b^3*c^3*d^6 - 3*a^5*b^2*c*d^8)/(a^2*b^6*c^12 + 6*a^3 
*b^5*c^10*d^2 + 15*a^4*b^4*c^8*d^4 + 20*a^5*b^3*c^6*d^6 + 15*a^6*b^2*c^4*d 
^8 + 6*a^7*b*c^2*d^10 + a^8*d^12))/sqrt(b*x^2 + a) - (20*b^2*c^4*d^3 + 7*a 
*b*c^2*d^5 + 2*a^2*d^7)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b) 
*c)/sqrt(-b*c^2 - a*d^2))/((b^3*c^9 + 3*a*b^2*c^7*d^2 + 3*a^2*b*c^5*d^4 + 
a^3*c^3*d^6)*sqrt(-b*c^2 - a*d^2)) + (8*(sqrt(b)*x - sqrt(b*x^2 + a))^3*b^ 
2*c^3*d^4 + (sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b*c*d^6 + 18*(sqrt(b)*x - sq 
rt(b*x^2 + a))^2*b^(5/2)*c^4*d^3 - 5*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^( 
3/2)*c^2*d^5 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*d^7 - 28*(sqr 
t(b)*x - sqrt(b*x^2 + a))*a*b^2*c^3*d^4 - 7*(sqrt(b)*x - sqrt(b*x^2 + a))* 
a^2*b*c*d^6 + 9*a^2*b^(3/2)*c^2*d^5 + 2*a^3*sqrt(b)*d^7)/((b^3*c^8 + 3*a*b 
^2*c^6*d^2 + 3*a^2*b*c^4*d^4 + a^3*c^2*d^6)*((sqrt(b)*x - sqrt(b*x^2 + a)) 
^2*d + 2*(sqrt(b)*x - sqrt(b*x^2 + a))*sqrt(b)*c - a*d)^2) + 2*arctan(-(sq 
rt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a*c^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 3699, normalized size of antiderivative = 13.40 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*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**5*c**2*d**7 + 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**5*c*d**8*x + 2*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**5*d* 
*9*x**2 + 7*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*b*c**4*d**5 + 14*sqrt(a*d**2 + b*c**2)*log( - sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b*c**3*d**6*x + 9 
*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*b*c**2*d**7*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**4*b*c*d**8*x**3 + 2*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*b*d**9*x**4 + 20*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqr 
t(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b**2*c**6*d**3 + 40*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 
**2*c**5*d**4*x + 27*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**2*c**4*d**5*x**2 + 14*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**2*c**3*d**6*x**3 + 7*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**2*c**2*d**7*x**4 + 20*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)...