Integrand size = 19, antiderivative size = 444 \[ \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=-\frac {c^2}{\left (b+a c^2\right ) x}-\frac {b \left (2 \sqrt {a} c+\sqrt {b+a c^2}\right ) \sqrt {d} \arctan \left (\frac {\sqrt {-\sqrt {a} c+\sqrt {b+a c^2}}-\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a} c+\sqrt {b+a c^2}}}\right )}{2 \sqrt {2} a^{3/4} \left (b+a c^2\right )^{3/2} \sqrt {\sqrt {a} c+\sqrt {b+a c^2}}}+\frac {b \left (2 \sqrt {a} c+\sqrt {b+a c^2}\right ) \sqrt {d} \arctan \left (\frac {\sqrt {-\sqrt {a} c+\sqrt {b+a c^2}}+\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a} c+\sqrt {b+a c^2}}}\right )}{2 \sqrt {2} a^{3/4} \left (b+a c^2\right )^{3/2} \sqrt {\sqrt {a} c+\sqrt {b+a c^2}}}+\frac {b \left (2 \sqrt {a} c-\sqrt {b+a c^2}\right ) \sqrt {d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} c+\sqrt {b+a c^2}} \sqrt {d} x}{\sqrt {b+a c^2}+\sqrt {a} d x^2}\right )}{2 \sqrt {2} a^{3/4} \left (b+a c^2\right )^{3/2} \sqrt {-\sqrt {a} c+\sqrt {b+a c^2}}} \] Output:
-c^2/(a*c^2+b)/x-1/4*b*(2*a^(1/2)*c+(a*c^2+b)^(1/2))*d^(1/2)*arctan(((-a^( 1/2)*c+(a*c^2+b)^(1/2))^(1/2)-2^(1/2)*a^(1/4)*d^(1/2)*x)/(a^(1/2)*c+(a*c^2 +b)^(1/2))^(1/2))*2^(1/2)/a^(3/4)/(a*c^2+b)^(3/2)/(a^(1/2)*c+(a*c^2+b)^(1/ 2))^(1/2)+1/4*b*(2*a^(1/2)*c+(a*c^2+b)^(1/2))*d^(1/2)*arctan(((-a^(1/2)*c+ (a*c^2+b)^(1/2))^(1/2)+2^(1/2)*a^(1/4)*d^(1/2)*x)/(a^(1/2)*c+(a*c^2+b)^(1/ 2))^(1/2))*2^(1/2)/a^(3/4)/(a*c^2+b)^(3/2)/(a^(1/2)*c+(a*c^2+b)^(1/2))^(1/ 2)+1/4*b*(2*a^(1/2)*c-(a*c^2+b)^(1/2))*d^(1/2)*arctanh(2^(1/2)*a^(1/4)*(-a ^(1/2)*c+(a*c^2+b)^(1/2))^(1/2)*d^(1/2)*x/((a*c^2+b)^(1/2)+a^(1/2)*d*x^2)) *2^(1/2)/a^(3/4)/(a*c^2+b)^(3/2)/(-a^(1/2)*c+(a*c^2+b)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\frac {-\frac {2 c^2}{x}+\frac {\sqrt {b} \left (\sqrt {b}-i \sqrt {a} c\right ) \sqrt {d} \arctan \left (\frac {\sqrt {a} \sqrt {d} x}{\sqrt {-i \sqrt {a} \sqrt {b}+a c}}\right )}{\sqrt {a} \sqrt {-i \sqrt {a} \sqrt {b}+a c}}+\frac {\sqrt {b} \left (\sqrt {b}+i \sqrt {a} c\right ) \sqrt {d} \arctan \left (\frac {\sqrt {a} \sqrt {d} x}{\sqrt {i \sqrt {a} \sqrt {b}+a c}}\right )}{\sqrt {a} \sqrt {i \sqrt {a} \sqrt {b}+a c}}}{2 \left (b+a c^2\right )} \] Input:
Integrate[1/(x^2*(a + b/(c + d*x^2)^2)),x]
Output:
((-2*c^2)/x + (Sqrt[b]*(Sqrt[b] - I*Sqrt[a]*c)*Sqrt[d]*ArcTan[(Sqrt[a]*Sqr t[d]*x)/Sqrt[(-I)*Sqrt[a]*Sqrt[b] + a*c]])/(Sqrt[a]*Sqrt[(-I)*Sqrt[a]*Sqrt [b] + a*c]) + (Sqrt[b]*(Sqrt[b] + I*Sqrt[a]*c)*Sqrt[d]*ArcTan[(Sqrt[a]*Sqr t[d]*x)/Sqrt[I*Sqrt[a]*Sqrt[b] + a*c]])/(Sqrt[a]*Sqrt[I*Sqrt[a]*Sqrt[b] + a*c]))/(2*(b + a*c^2))
Time = 2.14 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.32, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {7293, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {b d \left (2 c+d x^2\right )}{\left (a c^2+b\right ) \left (a c^2+2 a c d x^2+a d^2 x^4+b\right )}+\frac {c^2}{x^2 \left (a c^2+b\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \sqrt {d} \left (\sqrt {a c^2+b}+2 \sqrt {a} c\right ) \arctan \left (\frac {\sqrt {\sqrt {a c^2+b}-\sqrt {a} c}-\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}\right )}{2 \sqrt {2} a^{3/4} \left (a c^2+b\right )^{3/2} \sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}+\frac {b \sqrt {d} \left (\sqrt {a c^2+b}+2 \sqrt {a} c\right ) \arctan \left (\frac {\sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}\right )}{2 \sqrt {2} a^{3/4} \left (a c^2+b\right )^{3/2} \sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}-\frac {b \sqrt {d} \left (2 \sqrt {a} c-\sqrt {a c^2+b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt {d} x \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {a c^2+b}+\sqrt {a} d x^2\right )}{4 \sqrt {2} a^{3/4} \left (a c^2+b\right )^{3/2} \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}}+\frac {b \sqrt {d} \left (2 \sqrt {a} c-\sqrt {a c^2+b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt {d} x \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {a c^2+b}+\sqrt {a} d x^2\right )}{4 \sqrt {2} a^{3/4} \left (a c^2+b\right )^{3/2} \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}}-\frac {c^2}{x \left (a c^2+b\right )}\) |
Input:
Int[1/(x^2*(a + b/(c + d*x^2)^2)),x]
Output:
-(c^2/((b + a*c^2)*x)) - (b*(2*Sqrt[a]*c + Sqrt[b + a*c^2])*Sqrt[d]*ArcTan [(Sqrt[-(Sqrt[a]*c) + Sqrt[b + a*c^2]] - Sqrt[2]*a^(1/4)*Sqrt[d]*x)/Sqrt[S qrt[a]*c + Sqrt[b + a*c^2]]])/(2*Sqrt[2]*a^(3/4)*(b + a*c^2)^(3/2)*Sqrt[Sq rt[a]*c + Sqrt[b + a*c^2]]) + (b*(2*Sqrt[a]*c + Sqrt[b + a*c^2])*Sqrt[d]*A rcTan[(Sqrt[-(Sqrt[a]*c) + Sqrt[b + a*c^2]] + Sqrt[2]*a^(1/4)*Sqrt[d]*x)/S qrt[Sqrt[a]*c + Sqrt[b + a*c^2]]])/(2*Sqrt[2]*a^(3/4)*(b + a*c^2)^(3/2)*Sq rt[Sqrt[a]*c + Sqrt[b + a*c^2]]) - (b*(2*Sqrt[a]*c - Sqrt[b + a*c^2])*Sqrt [d]*Log[Sqrt[b + a*c^2] - Sqrt[2]*a^(1/4)*Sqrt[-(Sqrt[a]*c) + Sqrt[b + a*c ^2]]*Sqrt[d]*x + Sqrt[a]*d*x^2])/(4*Sqrt[2]*a^(3/4)*(b + a*c^2)^(3/2)*Sqrt [-(Sqrt[a]*c) + Sqrt[b + a*c^2]]) + (b*(2*Sqrt[a]*c - Sqrt[b + a*c^2])*Sqr t[d]*Log[Sqrt[b + a*c^2] + Sqrt[2]*a^(1/4)*Sqrt[-(Sqrt[a]*c) + Sqrt[b + a* c^2]]*Sqrt[d]*x + Sqrt[a]*d*x^2])/(4*Sqrt[2]*a^(3/4)*(b + a*c^2)^(3/2)*Sqr t[-(Sqrt[a]*c) + Sqrt[b + a*c^2]])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.55
| method | result | size |
| risch | \(-\frac {c^{2}}{\left (a \,c^{2}+b \right ) x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{6} c^{6}+3 a^{5} b \,c^{4}+3 a^{4} b^{2} c^{2}+a^{3} b^{3}\right ) \textit {\_Z}^{4}+\left (-2 a^{3} b \,c^{3} d +6 a^{2} b^{2} c d \right ) \textit {\_Z}^{2}+b^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{7} c^{8}+2 a^{6} b \,c^{6}+12 a^{5} b^{2} c^{4}+14 a^{4} b^{3} c^{2}+5 a^{3} b^{4}\right ) \textit {\_R}^{4}+\left (a^{4} b \,c^{5} d -6 a^{3} b^{2} c^{3} d +25 a^{2} b^{3} c d \right ) \textit {\_R}^{2}+4 b^{3} d^{2}\right ) x +\left (3 a^{5} b \,c^{6}+5 a^{4} b^{2} c^{4}+a^{3} b^{3} c^{2}-a^{2} b^{4}\right ) \textit {\_R}^{3}\right )\right )}{4}\) | \(243\) |
| default | \(\text {Expression too large to display}\) | \(2261\) |
Input:
int(1/x^2/(a+b/(d*x^2+c)^2),x,method=_RETURNVERBOSE)
Output:
-c^2/(a*c^2+b)/x+1/4*sum(_R*ln(((-a^7*c^8+2*a^6*b*c^6+12*a^5*b^2*c^4+14*a^ 4*b^3*c^2+5*a^3*b^4)*_R^4+(a^4*b*c^5*d-6*a^3*b^2*c^3*d+25*a^2*b^3*c*d)*_R^ 2+4*b^3*d^2)*x+(3*a^5*b*c^6+5*a^4*b^2*c^4+a^3*b^3*c^2-a^2*b^4)*_R^3),_R=Ro otOf((a^6*c^6+3*a^5*b*c^4+3*a^4*b^2*c^2+a^3*b^3)*_Z^4+(-2*a^3*b*c^3*d+6*a^ 2*b^2*c*d)*_Z^2+b^2*d^2))
Leaf count of result is larger than twice the leaf count of optimal. 2314 vs. \(2 (334) = 668\).
Time = 0.10 (sec) , antiderivative size = 2314, normalized size of antiderivative = 5.21 \[ \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(a+b/(d*x^2+c)^2),x, algorithm="fricas")
Output:
1/4*((a*c^2 + b)*x*sqrt(((a*b*c^3 - 3*b^2*c)*d + (a^4*c^6 + 3*a^3*b*c^4 + 3*a^2*b^2*c^2 + a*b^3)*sqrt(-(9*a^2*b^3*c^4 - 6*a*b^4*c^2 + b^5)*d^2/(a^9* c^12 + 6*a^8*b*c^10 + 15*a^7*b^2*c^8 + 20*a^6*b^3*c^6 + 15*a^5*b^4*c^4 + 6 *a^4*b^5*c^2 + a^3*b^6)))/(a^4*c^6 + 3*a^3*b*c^4 + 3*a^2*b^2*c^2 + a*b^3)) *log((3*a*b^2*c^2 - b^3)*d^2*x + (2*(3*a^2*b^2*c^3 - a*b^3*c)*d + (a^6*c^8 + 2*a^5*b*c^6 - 2*a^3*b^3*c^2 - a^2*b^4)*sqrt(-(9*a^2*b^3*c^4 - 6*a*b^4*c ^2 + b^5)*d^2/(a^9*c^12 + 6*a^8*b*c^10 + 15*a^7*b^2*c^8 + 20*a^6*b^3*c^6 + 15*a^5*b^4*c^4 + 6*a^4*b^5*c^2 + a^3*b^6)))*sqrt(((a*b*c^3 - 3*b^2*c)*d + (a^4*c^6 + 3*a^3*b*c^4 + 3*a^2*b^2*c^2 + a*b^3)*sqrt(-(9*a^2*b^3*c^4 - 6* a*b^4*c^2 + b^5)*d^2/(a^9*c^12 + 6*a^8*b*c^10 + 15*a^7*b^2*c^8 + 20*a^6*b^ 3*c^6 + 15*a^5*b^4*c^4 + 6*a^4*b^5*c^2 + a^3*b^6)))/(a^4*c^6 + 3*a^3*b*c^4 + 3*a^2*b^2*c^2 + a*b^3))) - (a*c^2 + b)*x*sqrt(((a*b*c^3 - 3*b^2*c)*d + (a^4*c^6 + 3*a^3*b*c^4 + 3*a^2*b^2*c^2 + a*b^3)*sqrt(-(9*a^2*b^3*c^4 - 6*a *b^4*c^2 + b^5)*d^2/(a^9*c^12 + 6*a^8*b*c^10 + 15*a^7*b^2*c^8 + 20*a^6*b^3 *c^6 + 15*a^5*b^4*c^4 + 6*a^4*b^5*c^2 + a^3*b^6)))/(a^4*c^6 + 3*a^3*b*c^4 + 3*a^2*b^2*c^2 + a*b^3))*log((3*a*b^2*c^2 - b^3)*d^2*x - (2*(3*a^2*b^2*c^ 3 - a*b^3*c)*d + (a^6*c^8 + 2*a^5*b*c^6 - 2*a^3*b^3*c^2 - a^2*b^4)*sqrt(-( 9*a^2*b^3*c^4 - 6*a*b^4*c^2 + b^5)*d^2/(a^9*c^12 + 6*a^8*b*c^10 + 15*a^7*b ^2*c^8 + 20*a^6*b^3*c^6 + 15*a^5*b^4*c^4 + 6*a^4*b^5*c^2 + a^3*b^6)))*sqrt (((a*b*c^3 - 3*b^2*c)*d + (a^4*c^6 + 3*a^3*b*c^4 + 3*a^2*b^2*c^2 + a*b^...
Time = 3.76 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.46 \[ \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=- \frac {c^{2}}{x \left (a c^{2} + b\right )} + \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{6} c^{6} + 768 a^{5} b c^{4} + 768 a^{4} b^{2} c^{2} + 256 a^{3} b^{3}\right ) + t^{2} \left (- 32 a^{3} b c^{3} d + 96 a^{2} b^{2} c d\right ) + b^{2} d^{2}, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a^{6} c^{8} + 128 t^{3} a^{5} b c^{6} - 128 t^{3} a^{3} b^{3} c^{2} - 64 t^{3} a^{2} b^{4} - 4 t a^{3} b c^{5} d + 40 t a^{2} b^{2} c^{3} d - 20 t a b^{3} c d}{3 a b^{2} c^{2} d^{2} - b^{3} d^{2}} \right )} \right )\right )} \] Input:
integrate(1/x**2/(a+b/(d*x**2+c)**2),x)
Output:
-c**2/(x*(a*c**2 + b)) + RootSum(_t**4*(256*a**6*c**6 + 768*a**5*b*c**4 + 768*a**4*b**2*c**2 + 256*a**3*b**3) + _t**2*(-32*a**3*b*c**3*d + 96*a**2*b **2*c*d) + b**2*d**2, Lambda(_t, _t*log(x + (64*_t**3*a**6*c**8 + 128*_t** 3*a**5*b*c**6 - 128*_t**3*a**3*b**3*c**2 - 64*_t**3*a**2*b**4 - 4*_t*a**3* b*c**5*d + 40*_t*a**2*b**2*c**3*d - 20*_t*a*b**3*c*d)/(3*a*b**2*c**2*d**2 - b**3*d**2))))
\[ \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\int { \frac {1}{{\left (a + \frac {b}{{\left (d x^{2} + c\right )}^{2}}\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b/(d*x^2+c)^2),x, algorithm="maxima")
Output:
b*d*integrate((d*x^2 + 2*c)/(a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + b), x)/(a*c ^2 + b) - c^2/((a*c^2 + b)*x)
Exception generated. \[ \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/x^2/(a+b/(d*x^2+c)^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%{[1,0]:[1,0,%%%{1,[1,1]%%%}]%%},[0,1]%%%}+%%%{%%%{1,[0,1 ]%%%},[0,
Time = 10.03 (sec) , antiderivative size = 4231, normalized size of antiderivative = 9.53 \[ \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*(a + b/(c + d*x^2)^2)),x)
Output:
atan(((x*(4*a^2*b^6*d^8 + 8*a^3*b^5*c^2*d^8 - 8*a^5*b^3*c^6*d^8 - 4*a^6*b^ 2*c^8*d^8) + ((b*d*(-a^3*b^3)^(1/2) - 3*a*c^2*d*(-a^3*b^3)^(1/2) - 3*a^2*b ^2*c*d + a^3*b*c^3*d)/(16*(a^3*b^3 + a^6*c^6 + 3*a^5*b*c^4 + 3*a^4*b^2*c^2 )))^(1/2)*(x*((b*d*(-a^3*b^3)^(1/2) - 3*a*c^2*d*(-a^3*b^3)^(1/2) - 3*a^2*b ^2*c*d + a^3*b*c^3*d)/(16*(a^3*b^3 + a^6*c^6 + 3*a^5*b*c^4 + 3*a^4*b^2*c^2 )))^(1/2)*(64*a^4*b^6*c*d^7 + 64*a^9*b*c^11*d^7 + 320*a^5*b^5*c^3*d^7 + 64 0*a^6*b^4*c^5*d^7 + 640*a^7*b^3*c^7*d^7 + 320*a^8*b^2*c^9*d^7) + 32*a^3*b^ 6*c*d^7 + 128*a^4*b^5*c^3*d^7 + 192*a^5*b^4*c^5*d^7 + 128*a^6*b^3*c^7*d^7 + 32*a^7*b^2*c^9*d^7))*((b*d*(-a^3*b^3)^(1/2) - 3*a*c^2*d*(-a^3*b^3)^(1/2) - 3*a^2*b^2*c*d + a^3*b*c^3*d)/(16*(a^3*b^3 + a^6*c^6 + 3*a^5*b*c^4 + 3*a ^4*b^2*c^2)))^(1/2)*1i + (x*(4*a^2*b^6*d^8 + 8*a^3*b^5*c^2*d^8 - 8*a^5*b^3 *c^6*d^8 - 4*a^6*b^2*c^8*d^8) - ((b*d*(-a^3*b^3)^(1/2) - 3*a*c^2*d*(-a^3*b ^3)^(1/2) - 3*a^2*b^2*c*d + a^3*b*c^3*d)/(16*(a^3*b^3 + a^6*c^6 + 3*a^5*b* c^4 + 3*a^4*b^2*c^2)))^(1/2)*(32*a^3*b^6*c*d^7 - x*((b*d*(-a^3*b^3)^(1/2) - 3*a*c^2*d*(-a^3*b^3)^(1/2) - 3*a^2*b^2*c*d + a^3*b*c^3*d)/(16*(a^3*b^3 + a^6*c^6 + 3*a^5*b*c^4 + 3*a^4*b^2*c^2)))^(1/2)*(64*a^4*b^6*c*d^7 + 64*a^9 *b*c^11*d^7 + 320*a^5*b^5*c^3*d^7 + 640*a^6*b^4*c^5*d^7 + 640*a^7*b^3*c^7* d^7 + 320*a^8*b^2*c^9*d^7) + 128*a^4*b^5*c^3*d^7 + 192*a^5*b^4*c^5*d^7 + 1 28*a^6*b^3*c^7*d^7 + 32*a^7*b^2*c^9*d^7))*((b*d*(-a^3*b^3)^(1/2) - 3*a*c^2 *d*(-a^3*b^3)^(1/2) - 3*a^2*b^2*c*d + a^3*b*c^3*d)/(16*(a^3*b^3 + a^6*c...
Time = 1.71 (sec) , antiderivative size = 1320, normalized size of antiderivative = 2.97 \[ \int \frac {1}{x^2 \left (a+\frac {b}{\left (c+d x^2\right )^2}\right )} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(a+b/(d*x^2+c)^2),x)
Output:
(2*sqrt(d)*sqrt(a*c**2 + b)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)*a tan((sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) - 2*sqrt(a)*d*x) /(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*a*c**2*x - 2*sqrt (d)*sqrt(a*c**2 + b)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)*atan((sq rt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) - 2*sqrt(a)*d*x)/(sqrt( d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*b*x - 2*sqrt(d)*sqrt(a)* sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)*atan((sqrt(d)*sqrt(sqrt(a)*sq rt(a*c**2 + b) - a*c)*sqrt(2) - 2*sqrt(a)*d*x)/(sqrt(d)*sqrt(sqrt(a)*sqrt( a*c**2 + b) + a*c)*sqrt(2)))*a*c**3*x - 2*sqrt(d)*sqrt(a)*sqrt(sqrt(a)*sqr t(a*c**2 + b) + a*c)*sqrt(2)*atan((sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) - 2*sqrt(a)*d*x)/(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a* c)*sqrt(2)))*b*c*x - 2*sqrt(d)*sqrt(a*c**2 + b)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)*atan((sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt (2) + 2*sqrt(a)*d*x)/(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2) ))*a*c**2*x + 2*sqrt(d)*sqrt(a*c**2 + b)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a *c)*sqrt(2)*atan((sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) + 2 *sqrt(a)*d*x)/(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*b*x + 2*sqrt(d)*sqrt(a)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)*atan((sqr t(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2) + 2*sqrt(a)*d*x)/(sqrt(d )*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*a*c**3*x + 2*sqrt(d)*s...