\(\int \frac {1}{x^3 (c+d x)^3 \sqrt {a+b x^2}} \, dx\) [1207]

Optimal result

Integrand size = 22, antiderivative size = 274 \[ \int \frac {1}{x^3 (c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {a+b x^2}}{2 a c^3 x^2}+\frac {3 d \sqrt {a+b x^2}}{a c^4 x}+\frac {d^4 \sqrt {a+b x^2}}{2 c^3 \left (b c^2+a d^2\right ) (c+d x)^2}+\frac {3 d^4 \left (3 b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{2 c^4 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {d^3 \left (20 b^2 c^4+29 a b c^2 d^2+12 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 c^5 \left (b c^2+a d^2\right )^{5/2}}+\frac {\left (b c^2-12 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2} c^5} \] Output:

-1/2*(b*x^2+a)^(1/2)/a/c^3/x^2+3*d*(b*x^2+a)^(1/2)/a/c^4/x+1/2*d^4*(b*x^2+ 
a)^(1/2)/c^3/(a*d^2+b*c^2)/(d*x+c)^2+3/2*d^4*(2*a*d^2+3*b*c^2)*(b*x^2+a)^( 
1/2)/c^4/(a*d^2+b*c^2)^2/(d*x+c)+1/2*d^3*(12*a^2*d^4+29*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^5/(a*d^2 
+b*c^2)^(5/2)+1/2*(-12*a*d^2+b*c^2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/ 
2)/c^5
 

Mathematica [A] (verified)

Time = 10.70 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 (c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {c \sqrt {a+b x^2} \left (-\frac {c}{a x^2}+\frac {6 d}{a x}+\frac {c d^4}{\left (b c^2+a d^2\right ) (c+d x)^2}+\frac {9 b c^2 d^4+6 a d^6}{\left (b c^2+a d^2\right )^2 (c+d x)}\right )+\frac {\left (-b c^2+12 a d^2\right ) \log (x)}{a^{3/2}}-\frac {d^3 \left (20 b^2 c^4+29 a b c^2 d^2+12 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{5/2}}+\frac {\left (b c^2-12 a d^2\right ) \log \left (a+\sqrt {a} \sqrt {a+b x^2}\right )}{a^{3/2}}+\frac {d^3 \left (20 b^2 c^4+29 a b c^2 d^2+12 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{5/2}}}{2 c^5} \] Input:

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

Output:

(c*Sqrt[a + b*x^2]*(-(c/(a*x^2)) + (6*d)/(a*x) + (c*d^4)/((b*c^2 + a*d^2)* 
(c + d*x)^2) + (9*b*c^2*d^4 + 6*a*d^6)/((b*c^2 + a*d^2)^2*(c + d*x))) + (( 
-(b*c^2) + 12*a*d^2)*Log[x])/a^(3/2) - (d^3*(20*b^2*c^4 + 29*a*b*c^2*d^2 + 
 12*a^2*d^4)*Log[c + d*x])/(b*c^2 + a*d^2)^(5/2) + ((b*c^2 - 12*a*d^2)*Log 
[a + Sqrt[a]*Sqrt[a + b*x^2]])/a^(3/2) + (d^3*(20*b^2*c^4 + 29*a*b*c^2*d^2 
 + 12*a^2*d^4)*Log[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(b* 
c^2 + a*d^2)^(5/2))/(2*c^5)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.57, 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.

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

\(\Big \downarrow \) 617

\(\displaystyle \int \left (-\frac {6 d^3}{c^5 \sqrt {a+b x^2} (c+d x)}+\frac {6 d^2}{c^5 x \sqrt {a+b x^2}}-\frac {3 d^3}{c^4 \sqrt {a+b x^2} (c+d x)^2}-\frac {3 d}{c^4 x^2 \sqrt {a+b x^2}}-\frac {d^3}{c^3 \sqrt {a+b x^2} (c+d x)^3}+\frac {1}{c^3 x^3 \sqrt {a+b x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2} c^3}-\frac {6 d^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a} c^5}+\frac {6 d^3 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^5 \sqrt {a d^2+b c^2}}+\frac {3 b d^3 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 \left (a d^2+b c^2\right )^{3/2}}+\frac {b d^3 \left (2 b c^2-a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 c^3 \left (a d^2+b c^2\right )^{5/2}}+\frac {3 d \sqrt {a+b x^2}}{a c^4 x}-\frac {\sqrt {a+b x^2}}{2 a c^3 x^2}+\frac {3 b d^4 \sqrt {a+b x^2}}{2 c^2 (c+d x) \left (a d^2+b c^2\right )^2}+\frac {3 d^4 \sqrt {a+b x^2}}{c^4 (c+d x) \left (a d^2+b c^2\right )}+\frac {d^4 \sqrt {a+b x^2}}{2 c^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/2*Sqrt[a + b*x^2]/(a*c^3*x^2) + (3*d*Sqrt[a + b*x^2])/(a*c^4*x) + (d^4* 
Sqrt[a + b*x^2])/(2*c^3*(b*c^2 + a*d^2)*(c + d*x)^2) + (3*b*d^4*Sqrt[a + b 
*x^2])/(2*c^2*(b*c^2 + a*d^2)^2*(c + d*x)) + (3*d^4*Sqrt[a + b*x^2])/(c^4* 
(b*c^2 + a*d^2)*(c + d*x)) + (b*d^3*(2*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^3*(b*c^2 + a*d^2)^(5/2)) + 
(3*b*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)) + (6*d^3*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^ 
2]*Sqrt[a + b*x^2])])/(c^5*Sqrt[b*c^2 + a*d^2]) + (b*ArcTanh[Sqrt[a + b*x^ 
2]/Sqrt[a]])/(2*a^(3/2)*c^3) - (6*d^2*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(S 
qrt[a]*c^5)
 

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. \(730\) vs. \(2(244)=488\).

Time = 0.50 (sec) , antiderivative size = 731, normalized size of antiderivative = 2.67

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-6 d x +c \right )}{2 a \,c^{4} x^{2}}-\frac {6 d^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c^{5} \sqrt {a}}+\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) b}{2 a^{\frac {3}{2}} c^{3}}+\frac {3 d^{3} \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}}}}{c^{4} \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}+\frac {5 d^{2} b \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 )}{2 c^{3} \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {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}}}}{2 c^{3} \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b \,d^{3} \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}}}}{2 c^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (x +\frac {c}{d}\right )}+\frac {3 b^{2} 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 )}{2 c \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {6 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 )}{c^{5} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(731\)
default	\(\frac {-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}}{c^{3}}-\frac {6 d^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c^{5} \sqrt {a}}+\frac {3 d \sqrt {b \,x^{2}+a}}{a \,c^{4} x}+\frac {6 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 )}{c^{5} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {3 d \left (-\frac {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}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \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^{4}}-\frac {-\frac {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}}}}{2 \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b c d \left (-\frac {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}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \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 )}{2 \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b \,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 )}{2 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{c^{3}}\)	\(897\)

Input:

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

Output:

-1/2*(b*x^2+a)^(1/2)*(-6*d*x+c)/a/c^4/x^2-6/c^5*d^2/a^(1/2)*ln((2*a+2*a^(1 
/2)*(b*x^2+a)^(1/2))/x)+1/2/a^(3/2)/c^3*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2)) 
/x)*b+3/c^4*d^3/(a*d^2+b*c^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)+5/2/c^3*d^2*b/(a*d^2+b*c^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))+1/2/c^3*d^2/(a*d^2 
+b*c^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)+3/ 
2/c^2*b*d^3/(a*d^2+b*c^2)^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)+3/2/c*b^2*d^2/(a*d^2+b*c^2)^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))+6*d^2/c^5/((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. 795 vs. \(2 (245) = 490\).

Time = 2.67 (sec) , antiderivative size = 3249, normalized size of antiderivative = 11.86 \[ \int \frac {1}{x^3 (c+d x)^3 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 (c+d x)^3 \sqrt {a+b x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2202, normalized size of antiderivative = 8.04 \[ \int \frac {1}{x^3 (c+d x)^3 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(24*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*c**2*d**7*x**2 + 48*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*c*d**8*x**3 + 24*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**9*x**4 + 58*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*c**4*d**5*x**2 + 116*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*c**3*d**6*x**3 + 58*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*c**2*d**7*x**4 + 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**2*b 
**2*c**6*d**3*x**2 + 80*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**2*b**2*c**5*d**4*x**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* 
*2*b**2*c**4*d**5*x**4 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*c**2*d 
**7*x**2 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*c*d**8*x**3 - 24*sqr 
t(a*d**2 + b*c**2)*log(c + d*x)*a**4*d**9*x**4 - 58*sqrt(a*d**2 + b*c**2)* 
log(c + d*x)*a**3*b*c**4*d**5*x**2 - 116*sqrt(a*d**2 + b*c**2)*log(c + d*x 
)*a**3*b*c**3*d**6*x**3 - 58*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*b*c** 
2*d**7*x**4 - 40*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**2*c**6*d**3*x* 
*2 - 80*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**2*c**5*d**4*x**3 - 4...