\(\int \frac {1}{x (c+d x)^3 (a+b x^2)^{5/2}} \, dx\) [1304]

Optimal result

Integrand size = 22, antiderivative size = 374 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {b^2 \left (c \left (b c^2-3 a d^2\right )-d \left (3 b c^2-a d^2\right ) x\right )}{3 a \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^{3/2}}+\frac {b^2 \left (3 c \left (b^2 c^4+4 a b c^2 d^2-9 a^2 d^4\right )-2 d \left (3 b^2 c^4+17 a b c^2 d^2-4 a^2 d^4\right ) x\right )}{3 a^2 \left (b c^2+a d^2\right )^4 \sqrt {a+b x^2}}+\frac {d^6 \sqrt {a+b x^2}}{2 c \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {d^6 \left (13 b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{2 c^2 \left (b c^2+a d^2\right )^4 (c+d x)}+\frac {d^5 \left (42 b^2 c^4+9 a b c^2 d^2+2 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 c^3 \left (b c^2+a d^2\right )^{9/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2} c^3} \] Output:

1/3*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)^(3/2)+1/3*b^2*(3*c*(-9*a^2*d^4+4*a*b*c^2*d^2+b^2*c^4)-2*d*(-4*a^2*d^4+ 
17*a*b*c^2*d^2+3*b^2*c^4)*x)/a^2/(a*d^2+b*c^2)^4/(b*x^2+a)^(1/2)+1/2*d^6*( 
b*x^2+a)^(1/2)/c/(a*d^2+b*c^2)^3/(d*x+c)^2+1/2*d^6*(2*a*d^2+13*b*c^2)*(b*x 
^2+a)^(1/2)/c^2/(a*d^2+b*c^2)^4/(d*x+c)+1/2*d^5*(2*a^2*d^4+9*a*b*c^2*d^2+4 
2*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)^(9/2)-arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)/c^3
 

Mathematica [A] (verified)

Time = 11.00 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (\frac {3 d^6 \left (b c^2+a d^2\right )}{c (c+d x)^2}+\frac {39 b c^2 d^6+6 a d^8}{c^2 (c+d x)}+\frac {2 b^2 \left (b c^2+a d^2\right ) \left (b c^2 (c-3 d x)+a d^2 (-3 c+d x)\right )}{a \left (a+b x^2\right )^2}+\frac {2 b^2 \left (2 a b c^2 d^2 (6 c-17 d x)+3 b^2 c^4 (c-2 d x)+a^2 d^4 (-27 c+8 d x)\right )}{a^2 \left (a+b x^2\right )}\right )}{\left (b c^2+a d^2\right )^4}+\frac {6 \log (x)}{a^{5/2} c^3}-\frac {3 d^5 \left (42 b^2 c^4+9 a b c^2 d^2+2 a^2 d^4\right ) \log (c+d x)}{c^3 \left (b c^2+a d^2\right )^{9/2}}-\frac {6 \log \left (a+\sqrt {a} \sqrt {a+b x^2}\right )}{a^{5/2} c^3}+\frac {3 d^5 \left (42 b^2 c^4+9 a b c^2 d^2+2 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{c^3 \left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

Integrate[1/(x*(c + d*x)^3*(a + b*x^2)^(5/2)),x]
 

Output:

((Sqrt[a + b*x^2]*((3*d^6*(b*c^2 + a*d^2))/(c*(c + d*x)^2) + (39*b*c^2*d^6 
 + 6*a*d^8)/(c^2*(c + d*x)) + (2*b^2*(b*c^2 + a*d^2)*(b*c^2*(c - 3*d*x) + 
a*d^2*(-3*c + d*x)))/(a*(a + b*x^2)^2) + (2*b^2*(2*a*b*c^2*d^2*(6*c - 17*d 
*x) + 3*b^2*c^4*(c - 2*d*x) + a^2*d^4*(-27*c + 8*d*x)))/(a^2*(a + b*x^2))) 
)/(b*c^2 + a*d^2)^4 + (6*Log[x])/(a^(5/2)*c^3) - (3*d^5*(42*b^2*c^4 + 9*a* 
b*c^2*d^2 + 2*a^2*d^4)*Log[c + d*x])/(c^3*(b*c^2 + a*d^2)^(9/2)) - (6*Log[ 
a + Sqrt[a]*Sqrt[a + b*x^2]])/(a^(5/2)*c^3) + (3*d^5*(42*b^2*c^4 + 9*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)^(9/2)))/6
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(832\) vs. \(2(374)=748\).

Time = 2.41 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.22, 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.

\(\displaystyle \int \frac {1}{x \left (a+b x^2\right )^{5/2} (c+d x)^3} \, dx\)

\(\Big \downarrow \) 617

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right ) d^5}{c^3 \left (b c^2+a d^2\right )^{5/2}}+\frac {5 b \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right ) d^5}{c \left (b c^2+a d^2\right )^{7/2}}+\frac {5 b \left (6 b c^2-a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right ) d^5}{2 c \left (b c^2+a d^2\right )^{9/2}}-\frac {b \left (4 b^2 c^4+28 a b d^2 c^2-81 a^2 d^4\right ) \sqrt {b x^2+a} d^2}{6 a^2 \left (b c^2+a d^2\right )^4 (c+d x)}-\frac {\left (2 b^2 c^4+9 a b d^2 c^2-8 a^2 d^4\right ) \sqrt {b x^2+a} d^2}{3 a^2 c^2 \left (b c^2+a d^2\right )^3 (c+d x)}-\frac {\left (4 b^2 c^4+24 a b d^2 c^2-15 a^2 d^4\right ) \sqrt {b x^2+a} d^2}{6 a^2 c \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {\left (3 a^2 d^3+b c \left (2 b c^2+5 a d^2\right ) x\right ) d}{3 a^2 c^3 \left (b c^2+a d^2\right )^2 \sqrt {b x^2+a}}+\frac {\left (a d \left (b c^2-4 a d^2\right )-b c \left (2 b c^2+7 a d^2\right ) x\right ) d}{3 a^2 c^2 \left (b c^2+a d^2\right )^2 (c+d x) \sqrt {b x^2+a}}+\frac {\left (a d \left (2 b c^2-5 a d^2\right )-b c \left (2 b c^2+9 a d^2\right ) x\right ) d}{3 a^2 c \left (b c^2+a d^2\right )^2 (c+d x)^2 \sqrt {b x^2+a}}-\frac {(a d+b c x) d}{3 a c^3 \left (b c^2+a d^2\right ) \left (b x^2+a\right )^{3/2}}-\frac {(a d+b c x) d}{3 a c^2 \left (b c^2+a d^2\right ) (c+d x) \left (b x^2+a\right )^{3/2}}-\frac {(a d+b c x) d}{3 a c \left (b c^2+a d^2\right ) (c+d x)^2 \left (b x^2+a\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right )}{a^{5/2} c^3}+\frac {1}{a^2 c^3 \sqrt {b x^2+a}}+\frac {1}{3 a c^3 \left (b x^2+a\right )^{3/2}}\)

Input:

Int[1/(x*(c + d*x)^3*(a + b*x^2)^(5/2)),x]
 

Output:

1/(3*a*c^3*(a + b*x^2)^(3/2)) - (d*(a*d + b*c*x))/(3*a*c^3*(b*c^2 + a*d^2) 
*(a + b*x^2)^(3/2)) - (d*(a*d + b*c*x))/(3*a*c*(b*c^2 + a*d^2)*(c + d*x)^2 
*(a + b*x^2)^(3/2)) - (d*(a*d + b*c*x))/(3*a*c^2*(b*c^2 + a*d^2)*(c + d*x) 
*(a + b*x^2)^(3/2)) + 1/(a^2*c^3*Sqrt[a + b*x^2]) - (d*(3*a^2*d^3 + b*c*(2 
*b*c^2 + 5*a*d^2)*x))/(3*a^2*c^3*(b*c^2 + a*d^2)^2*Sqrt[a + b*x^2]) + (d*( 
a*d*(b*c^2 - 4*a*d^2) - b*c*(2*b*c^2 + 7*a*d^2)*x))/(3*a^2*c^2*(b*c^2 + a* 
d^2)^2*(c + d*x)*Sqrt[a + b*x^2]) + (d*(a*d*(2*b*c^2 - 5*a*d^2) - b*c*(2*b 
*c^2 + 9*a*d^2)*x))/(3*a^2*c*(b*c^2 + a*d^2)^2*(c + d*x)^2*Sqrt[a + b*x^2] 
) - (d^2*(4*b^2*c^4 + 24*a*b*c^2*d^2 - 15*a^2*d^4)*Sqrt[a + b*x^2])/(6*a^2 
*c*(b*c^2 + a*d^2)^3*(c + d*x)^2) - (b*d^2*(4*b^2*c^4 + 28*a*b*c^2*d^2 - 8 
1*a^2*d^4)*Sqrt[a + b*x^2])/(6*a^2*(b*c^2 + a*d^2)^4*(c + d*x)) - (d^2*(2* 
b^2*c^4 + 9*a*b*c^2*d^2 - 8*a^2*d^4)*Sqrt[a + b*x^2])/(3*a^2*c^2*(b*c^2 + 
a*d^2)^3*(c + d*x)) + (5*b*d^5*(6*b*c^2 - a*d^2)*ArcTanh[(a*d - b*c*x)/(Sq 
rt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(2*c*(b*c^2 + a*d^2)^(9/2)) + (5*b*d^ 
5*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)^(7/2)) + (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)^(5/2)) - ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]/ 
(a^(5/2)*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. \(3175\) vs. \(2(344)=688\).

Time = 0.55 (sec) , antiderivative size = 3176, normalized size of antiderivative = 8.49

method	result	size
default	\(\text {Expression too large to display}\)	\(3176\)

Input:

int(1/x/(d*x+c)^3/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/c^3*(1/3/a/(b*x^2+a)^(3/2)+1/a*(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)^(3/2)+5*b*c*d/(a*d^2+b*c^2)*(1/3 
/(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)^(3/2)+b 
*c*d/(a*d^2+b*c^2)*(2/3*(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)^(3/2)+16/3*b/(4* 
b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(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*(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*(2/3*(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)^(3/2)+ 
16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(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*(1/3/(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)^(3/2)+b*c*d/(a*d^2+b 
*c^2)*(2/3*(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)^(3/2)+16/3*b/(4*b*(a*d^2+b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2073 vs. \(2 (345) = 690\).

Time = 16.89 (sec) , antiderivative size = 8361, normalized size of antiderivative = 22.36 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{x \left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x\right )^{3}}\, dx \] Input:

integrate(1/x/(d*x+c)**3/(b*x**2+a)**(5/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/(x*(a + b*x**2)**(5/2)*(c + d*x)**3), x)
 

Maxima [F]

\[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{3} x} \,d x } \] Input:

integrate(1/x/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(5/2)*(d*x + c)^3*x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2299 vs. \(2 (345) = 690\).

Time = 0.38 (sec) , antiderivative size = 2299, normalized size of antiderivative = 6.15 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

-(42*b^2*c^4*d^5 + 9*a*b*c^2*d^7 + 2*a^2*d^9)*arctan(-((sqrt(b)*x - sqrt(b 
*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^4*c^11 + 4*a*b^3*c^9*d 
^2 + 6*a^2*b^2*c^7*d^4 + 4*a^3*b*c^5*d^6 + a^4*c^3*d^8)*sqrt(-b*c^2 - a*d^ 
2)) - 1/3*(((2*(3*a^3*b^18*c^28*d + 53*a^4*b^17*c^26*d^3 + 398*a^5*b^16*c^ 
24*d^5 + 1734*a^6*b^15*c^22*d^7 + 4961*a^7*b^14*c^20*d^9 + 9911*a^8*b^13*c 
^18*d^11 + 14256*a^9*b^12*c^16*d^13 + 14916*a^10*b^11*c^14*d^15 + 11253*a^ 
11*b^10*c^12*d^17 + 5907*a^12*b^9*c^10*d^19 + 1958*a^13*b^8*c^8*d^21 + 278 
*a^14*b^7*c^6*d^23 - 57*a^15*b^6*c^4*d^25 - 31*a^16*b^5*c^2*d^27 - 4*a^17* 
b^4*d^29)*x/(a^5*b^17*c^32 + 16*a^6*b^16*c^30*d^2 + 120*a^7*b^15*c^28*d^4 
+ 560*a^8*b^14*c^26*d^6 + 1820*a^9*b^13*c^24*d^8 + 4368*a^10*b^12*c^22*d^1 
0 + 8008*a^11*b^11*c^20*d^12 + 11440*a^12*b^10*c^18*d^14 + 12870*a^13*b^9* 
c^16*d^16 + 11440*a^14*b^8*c^14*d^18 + 8008*a^15*b^7*c^12*d^20 + 4368*a^16 
*b^6*c^10*d^22 + 1820*a^17*b^5*c^8*d^24 + 560*a^18*b^4*c^6*d^26 + 120*a^19 
*b^3*c^4*d^28 + 16*a^20*b^2*c^2*d^30 + a^21*b*d^32) - 3*(a^3*b^18*c^29 + 1 
6*a^4*b^17*c^27*d^2 + 105*a^5*b^16*c^25*d^4 + 376*a^6*b^15*c^23*d^6 + 781* 
a^7*b^14*c^21*d^8 + 792*a^8*b^13*c^19*d^10 - 363*a^9*b^12*c^17*d^12 - 2640 
*a^10*b^11*c^15*d^14 - 4653*a^11*b^10*c^13*d^16 - 4928*a^12*b^9*c^11*d^18 
- 3509*a^13*b^8*c^9*d^20 - 1704*a^14*b^7*c^7*d^22 - 545*a^15*b^6*c^5*d^24 
- 104*a^16*b^5*c^3*d^26 - 9*a^17*b^4*c*d^28)/(a^5*b^17*c^32 + 16*a^6*b^16* 
c^30*d^2 + 120*a^7*b^15*c^28*d^4 + 560*a^8*b^14*c^26*d^6 + 1820*a^9*b^1...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{x\,{\left (b\,x^2+a\right )}^{5/2}\,{\left (c+d\,x\right )}^3} \,d x \] Input:

int(1/(x*(a + b*x^2)^(5/2)*(c + d*x)^3),x)
 

Output:

int(1/(x*(a + b*x^2)^(5/2)*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 6147, normalized size of antiderivative = 16.44 \[ \int \frac {1}{x (c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/x/(d*x+c)^3/(b*x^2+a)^(5/2),x)
 

Output:

(6*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**7*c**2*d**9 + 12*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**7*c*d**10*x + 6*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**7* 
d**11*x**2 + 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**6*b*c**4*d**7 + 54*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**6*b*c**3*d**8*x 
 + 39*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**6*b*c**2*d**9*x**2 + 24*sqrt(a*d**2 + b*c**2)*log( - sqr 
t(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**6*b*c*d**10*x**3 + 1 
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**6*b*d**11*x**4 + 126*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*b**2*c**6*d**5 + 252*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*b**2*c**5*d**6*x + 180*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*b**2*c**4*d**7*x**2 + 108*s 
qrt(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*b**2*c**3*d**8*x**3 + 60*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*b**2*c**2*d**9*x**4 + 
12*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) ...