Integrand size = 22, antiderivative size = 250 \[ \int \frac {1}{x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {b^2 \left (b c^2-a d^2-2 b c d x\right )}{a^2 \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {\sqrt {a+b x^2}}{2 a^2 c^2 x^2}+\frac {2 d \sqrt {a+b x^2}}{a^2 c^3 x}+\frac {d^6 \sqrt {a+b x^2}}{c^3 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {3 d^5 \left (2 b c^2+a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^4 \left (b c^2+a d^2\right )^{5/2}}+\frac {3 \left (b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2} c^4} \] Output:
-b^2*(-2*b*c*d*x-a*d^2+b*c^2)/a^2/(a*d^2+b*c^2)^2/(b*x^2+a)^(1/2)-1/2*(b*x ^2+a)^(1/2)/a^2/c^2/x^2+2*d*(b*x^2+a)^(1/2)/a^2/c^3/x+d^6*(b*x^2+a)^(1/2)/ c^3/(a*d^2+b*c^2)^2/(d*x+c)+3*d^5*(a*d^2+2*b*c^2)*arctanh((-b*c*x+a*d)/(a* d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/c^4/(a*d^2+b*c^2)^(5/2)+3/2*(-2*a*d^2+b* c^2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)/c^4
Time = 1.92 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {c \left (a^3 d^4 \left (-c^2+3 c d x+6 d^2 x^2\right )+b^3 c^4 x^2 \left (-3 c^2+5 c d x+8 d^2 x^2\right )+a^2 b d^2 \left (-2 c^4+6 c^3 d x+7 c^2 d^2 x^2+3 c d^3 x^3+6 d^4 x^4\right )+a b^2 c^2 \left (-c^4+3 c^3 d x+4 c^2 d^2 x^2+8 c d^3 x^3+8 d^4 x^4\right )\right )}{a^2 \left (b c^2+a d^2\right )^2 x^2 (c+d x) \sqrt {a+b x^2}}+\frac {12 d^5 \left (2 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {6 \left (-b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}}{2 c^4} \] Input:
Integrate[1/(x^3*(c + d*x)^2*(a + b*x^2)^(3/2)),x]
Output:
((c*(a^3*d^4*(-c^2 + 3*c*d*x + 6*d^2*x^2) + b^3*c^4*x^2*(-3*c^2 + 5*c*d*x + 8*d^2*x^2) + a^2*b*d^2*(-2*c^4 + 6*c^3*d*x + 7*c^2*d^2*x^2 + 3*c*d^3*x^3 + 6*d^4*x^4) + a*b^2*c^2*(-c^4 + 3*c^3*d*x + 4*c^2*d^2*x^2 + 8*c*d^3*x^3 + 8*d^4*x^4)))/(a^2*(b*c^2 + a*d^2)^2*x^2*(c + d*x)*Sqrt[a + b*x^2]) + (12 *d^5*(2*b*c^2 + a*d^2)*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)^(5/2) + (6*(-(b*c^2) + 2*a*d^2)*Ar cTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(5/2))/(2*c^4)
Time = 1.09 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.78, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2} (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 617 |
| \(\displaystyle \int \left (-\frac {3 d^3}{c^4 \left (a+b x^2\right )^{3/2} (c+d x)}+\frac {3 d^2}{c^4 x \left (a+b x^2\right )^{3/2}}-\frac {d^3}{c^3 \left (a+b x^2\right )^{3/2} (c+d x)^2}-\frac {2 d}{c^3 x^2 \left (a+b x^2\right )^{3/2}}+\frac {1}{c^2 x^3 \left (a+b x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 d^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2} c^4}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2} c^2}+\frac {4 b d x}{a^2 c^3 \sqrt {a+b x^2}}-\frac {3 b}{2 a^2 c^2 \sqrt {a+b x^2}}+\frac {3 b d^5 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^2 \left (a d^2+b c^2\right )^{5/2}}+\frac {3 d^5 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^4 \left (a d^2+b c^2\right )^{3/2}}+\frac {3 d^2}{a c^4 \sqrt {a+b x^2}}+\frac {2 d}{a c^3 x \sqrt {a+b x^2}}-\frac {1}{2 a c^2 x^2 \sqrt {a+b x^2}}-\frac {3 d^3 (a d+b c x)}{a c^4 \sqrt {a+b x^2} \left (a d^2+b c^2\right )}-\frac {d^4 \sqrt {a+b x^2} \left (b c^2-2 a d^2\right )}{a c^3 (c+d x) \left (a d^2+b c^2\right )^2}-\frac {d^3 (a d+b c x)}{a c^3 \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}\) |
Input:
Int[1/(x^3*(c + d*x)^2*(a + b*x^2)^(3/2)),x]
Output:
(-3*b)/(2*a^2*c^2*Sqrt[a + b*x^2]) + (3*d^2)/(a*c^4*Sqrt[a + b*x^2]) - 1/( 2*a*c^2*x^2*Sqrt[a + b*x^2]) + (2*d)/(a*c^3*x*Sqrt[a + b*x^2]) + (4*b*d*x) /(a^2*c^3*Sqrt[a + b*x^2]) - (3*d^3*(a*d + b*c*x))/(a*c^4*(b*c^2 + a*d^2)* Sqrt[a + b*x^2]) - (d^3*(a*d + b*c*x))/(a*c^3*(b*c^2 + a*d^2)*(c + d*x)*Sq rt[a + b*x^2]) - (d^4*(b*c^2 - 2*a*d^2)*Sqrt[a + b*x^2])/(a*c^3*(b*c^2 + a *d^2)^2*(c + d*x)) + (3*b*d^5*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*S qrt[a + b*x^2])])/(c^2*(b*c^2 + a*d^2)^(5/2)) + (3*d^5*ArcTanh[(a*d - b*c* x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c^4*(b*c^2 + a*d^2)^(3/2)) + ( 3*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(5/2)*c^2) - (3*d^2*ArcTanh[Sqr t[a + b*x^2]/Sqrt[a]])/(a^(3/2)*c^4)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(670\) vs. \(2(226)=452\).
Time = 0.50 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.68
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-4 d x +c \right )}{2 a^{2} c^{3} x^{2}}-\frac {\frac {3 \left (2 a \,d^{2}-b \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}+\frac {b^{3} c^{3} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{\left (\sqrt {-a b}\, d +b c \right )^{2} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {b^{3} c^{3} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{\left (\sqrt {-a b}\, d -b c \right )^{2} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {2 b \,d^{3} a^{2} \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{\left (\sqrt {-a b}\, d +b c \right ) \left (\sqrt {-a b}\, d -b c \right )}-\frac {2 b^{2} d^{4} a^{2} \left (3 a \,d^{2}+5 b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (\sqrt {-a b}\, d +b c \right )^{2} \left (\sqrt {-a b}\, d -b c \right )^{2} c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 a^{2} c^{3}}\) | \(671\) |
| 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^{2}}+\frac {3 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^{4}}-\frac {2 d \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{c^{3}}-\frac {3 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^{4}}-\frac {d \left (-\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\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 {3 b c d \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 )}{a \,d^{2}+b \,c^{2}}-\frac {4 b \,d^{2} \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}}}}\right )}{c^{3}}\) | \(996\) |
Input:
int(1/x^3/(d*x+c)^2/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(b*x^2+a)^(1/2)*(-4*d*x+c)/a^2/c^3/x^2-1/2/a^2/c^3*(3*(2*a*d^2-b*c^2) /c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+b^3*c^3/((-a*b)^(1/2)*d+b *c)^2/(-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^3*c^3/((-a*b)^(1/2)*d-b*c)^2/(-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*d^3*a^2/((-a*b)^(1/2)*d+b*c)/((-a*b)^(1/2)*d-b*c)*(-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)-b *c*d/(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)))-2*b^2*d^4*a^2*(3*a*d^2+5*b*c^2)/((-a*b)^(1/2 )*d+b*c)^2/((-a*b)^(1/2)*d-b*c)^2/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)))
Leaf count of result is larger than twice the leaf count of optimal. 964 vs. \(2 (226) = 452\).
Time = 2.32 (sec) , antiderivative size = 3922, normalized size of antiderivative = 15.69 \[ \int \frac {1}{x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^3/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{2}}\, dx \] Input:
integrate(1/x**3/(d*x+c)**2/(b*x**2+a)**(3/2),x)
Output:
Integral(1/(x**3*(a + b*x**2)**(3/2)*(c + d*x)**2), x)
\[ \int \frac {1}{x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{2} x^{3}} \,d x } \] Input:
integrate(1/x^3/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x + c)^2*x^3), x)
Timed out. \[ \int \frac {1}{x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^3/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2} \,d x \] Input:
int(1/(x^3*(a + b*x^2)^(3/2)*(c + d*x)^2),x)
Output:
int(1/(x^3*(a + b*x^2)^(3/2)*(c + d*x)^2), x)
Time = 0.24 (sec) , antiderivative size = 2581, normalized size of antiderivative = 10.32 \[ \int \frac {1}{x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/x^3/(d*x+c)^2/(b*x^2+a)^(3/2),x)
Output:
(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**5*c*d**7*x**2 + 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**5*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*b*c**3*d**5*x**2 + 24*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sq rt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b*c**2*d**6*x**3 + 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** 4*b*c*d**7*x**4 + 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**4*b*d**8*x**5 + 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**3*b**2*c** 3*d**5*x**4 + 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**3*b**2*c**2*d**6*x**5 - 12*sqrt(a*d**2 + b*c* *2)*log(c + d*x)*a**5*c*d**7*x**2 - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)* a**5*d**8*x**3 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*b*c**3*d**5*x* *2 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*b*c**2*d**6*x**3 - 12*sqrt (a*d**2 + b*c**2)*log(c + d*x)*a**4*b*c*d**7*x**4 - 12*sqrt(a*d**2 + b*c** 2)*log(c + d*x)*a**4*b*d**8*x**5 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a **3*b**2*c**3*d**5*x**4 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*b**2* c**2*d**6*x**5 - 2*sqrt(a + b*x**2)*a**5*c**3*d**6 + 6*sqrt(a + b*x**2)*a* *5*c**2*d**7*x + 12*sqrt(a + b*x**2)*a**5*c*d**8*x**2 - 6*sqrt(a + b*x*...