Integrand size = 19, antiderivative size = 448 \[ \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {2 b c x}{a^2 d^3}-\frac {b x^3}{3 a^2 d^2}+\frac {x^7}{7 a}-\frac {b \left (b-3 a c^2-2 \sqrt {a} c \sqrt {b+a c^2}\right ) \arctan \left (\frac {\sqrt {-\sqrt {a} c+\sqrt {b+a c^2}}-\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a} c+\sqrt {b+a c^2}}}\right )}{2 \sqrt {2} a^{11/4} \sqrt {\sqrt {a} c+\sqrt {b+a c^2}} d^{7/2}}+\frac {b \left (b-3 a c^2-2 \sqrt {a} c \sqrt {b+a c^2}\right ) \arctan \left (\frac {\sqrt {-\sqrt {a} c+\sqrt {b+a c^2}}+\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a} c+\sqrt {b+a c^2}}}\right )}{2 \sqrt {2} a^{11/4} \sqrt {\sqrt {a} c+\sqrt {b+a c^2}} d^{7/2}}-\frac {b \left (b-3 a c^2+2 \sqrt {a} c \sqrt {b+a c^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} c+\sqrt {b+a c^2}} \sqrt {d} x}{\sqrt {b+a c^2}+\sqrt {a} d x^2}\right )}{2 \sqrt {2} a^{11/4} \sqrt {-\sqrt {a} c+\sqrt {b+a c^2}} d^{7/2}} \] Output:
2*b*c*x/a^2/d^3-1/3*b*x^3/a^2/d^2+1/7*x^7/a-1/4*b*(b-3*a*c^2-2*a^(1/2)*c*( a*c^2+b)^(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^(11/4)/(a^(1/2)*c +(a*c^2+b)^(1/2))^(1/2)/d^(7/2)+1/4*b*(b-3*a*c^2-2*a^(1/2)*c*(a*c^2+b)^(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^(11/4)/(a^(1/2)*c+(a*c^2+b)^( 1/2))^(1/2)/d^(7/2)-1/4*b*(b-3*a*c^2+2*a^(1/2)*c*(a*c^2+b)^(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^(11/4)/(-a^(1/2)*c+(a*c^2+b)^(1/2))^(1/2)/d^ (7/2)
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.50 \[ \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {2 b c x}{a^2 d^3}-\frac {b x^3}{3 a^2 d^2}+\frac {x^7}{7 a}+\frac {\sqrt {b} \left (\sqrt {b}+i \sqrt {a} c\right )^3 \arctan \left (\frac {\sqrt {a} \sqrt {d} x}{\sqrt {-i \sqrt {a} \sqrt {b}+a c}}\right )}{2 a^{5/2} \sqrt {-i \sqrt {a} \sqrt {b}+a c} d^{7/2}}+\frac {\sqrt {b} \left (\sqrt {b}-i \sqrt {a} c\right )^3 \arctan \left (\frac {\sqrt {a} \sqrt {d} x}{\sqrt {i \sqrt {a} \sqrt {b}+a c}}\right )}{2 a^{5/2} \sqrt {i \sqrt {a} \sqrt {b}+a c} d^{7/2}} \] Input:
Integrate[x^6/(a + b/(c + d*x^2)^2),x]
Output:
(2*b*c*x)/(a^2*d^3) - (b*x^3)/(3*a^2*d^2) + x^7/(7*a) + (Sqrt[b]*(Sqrt[b] + I*Sqrt[a]*c)^3*ArcTan[(Sqrt[a]*Sqrt[d]*x)/Sqrt[(-I)*Sqrt[a]*Sqrt[b] + a* c]])/(2*a^(5/2)*Sqrt[(-I)*Sqrt[a]*Sqrt[b] + a*c]*d^(7/2)) + (Sqrt[b]*(Sqrt [b] - I*Sqrt[a]*c)^3*ArcTan[(Sqrt[a]*Sqrt[d]*x)/Sqrt[I*Sqrt[a]*Sqrt[b] + a *c]])/(2*a^(5/2)*Sqrt[I*Sqrt[a]*Sqrt[b] + a*c]*d^(7/2))
Time = 3.38 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.41, 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.
| \(\displaystyle \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {b \left (d x^2 \left (b-3 a c^2\right )-2 c \left (a c^2+b\right )\right )}{a^2 d^3 \left (a c^2+2 a c d x^2+a d^2 x^4+b\right )}+\frac {2 b c}{a^2 d^3}-\frac {b x^2}{a^2 d^2}+\frac {x^6}{a}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (2 a^{3/2} c^3-\left (b-3 a c^2\right ) \sqrt {a c^2+b}+2 \sqrt {a} b c\right ) \arctan \left (\frac {\sqrt {\sqrt {a c^2+b}-\sqrt {a} c}-\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}\right )}{2 \sqrt {2} a^{11/4} d^{7/2} \sqrt {a c^2+b} \sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}-\frac {b \left (2 a^{3/2} c^3-\left (b-3 a c^2\right ) \sqrt {a c^2+b}+2 \sqrt {a} b c\right ) \arctan \left (\frac {\sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}\right )}{2 \sqrt {2} a^{11/4} d^{7/2} \sqrt {a c^2+b} \sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}+\frac {b \left (2 \sqrt {a} c \sqrt {a c^2+b}-3 a c^2+b\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt {d} x \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {a c^2+b}+\sqrt {a} d x^2\right )}{4 \sqrt {2} a^{11/4} d^{7/2} \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}}-\frac {b \left (2 \sqrt {a} c \sqrt {a c^2+b}-3 a c^2+b\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt {d} x \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {a c^2+b}+\sqrt {a} d x^2\right )}{4 \sqrt {2} a^{11/4} d^{7/2} \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}}+\frac {2 b c x}{a^2 d^3}-\frac {b x^3}{3 a^2 d^2}+\frac {x^7}{7 a}\) |
Input:
Int[x^6/(a + b/(c + d*x^2)^2),x]
Output:
(2*b*c*x)/(a^2*d^3) - (b*x^3)/(3*a^2*d^2) + x^7/(7*a) + (b*(2*Sqrt[a]*b*c + 2*a^(3/2)*c^3 - (b - 3*a*c^2)*Sqrt[b + a*c^2])*ArcTan[(Sqrt[-(Sqrt[a]*c) + Sqrt[b + a*c^2]] - Sqrt[2]*a^(1/4)*Sqrt[d]*x)/Sqrt[Sqrt[a]*c + Sqrt[b + a*c^2]]])/(2*Sqrt[2]*a^(11/4)*Sqrt[b + a*c^2]*Sqrt[Sqrt[a]*c + Sqrt[b + a *c^2]]*d^(7/2)) - (b*(2*Sqrt[a]*b*c + 2*a^(3/2)*c^3 - (b - 3*a*c^2)*Sqrt[b + a*c^2])*ArcTan[(Sqrt[-(Sqrt[a]*c) + Sqrt[b + a*c^2]] + Sqrt[2]*a^(1/4)* Sqrt[d]*x)/Sqrt[Sqrt[a]*c + Sqrt[b + a*c^2]]])/(2*Sqrt[2]*a^(11/4)*Sqrt[b + a*c^2]*Sqrt[Sqrt[a]*c + Sqrt[b + a*c^2]]*d^(7/2)) + (b*(b - 3*a*c^2 + 2* Sqrt[a]*c*Sqrt[b + a*c^2])*Log[Sqrt[b + a*c^2] - Sqrt[2]*a^(1/4)*Sqrt[-(Sq rt[a]*c) + Sqrt[b + a*c^2]]*Sqrt[d]*x + Sqrt[a]*d*x^2])/(4*Sqrt[2]*a^(11/4 )*Sqrt[-(Sqrt[a]*c) + Sqrt[b + a*c^2]]*d^(7/2)) - (b*(b - 3*a*c^2 + 2*Sqrt [a]*c*Sqrt[b + a*c^2])*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^(11/4)*Sq rt[-(Sqrt[a]*c) + Sqrt[b + a*c^2]]*d^(7/2))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.25
| method | result | size |
| risch | \(\frac {x^{7}}{7 a}-\frac {b \,x^{3}}{3 a^{2} d^{2}}+\frac {2 b c x}{a^{2} d^{3}}+\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,d^{2} \textit {\_Z}^{4}+2 a c d \,\textit {\_Z}^{2}+a \,c^{2}+b \right )}{\sum }\frac {\left (d \left (-3 a \,c^{2}+b \right ) \textit {\_R}^{2}-2 c^{3} a -2 b c \right ) \ln \left (x -\textit {\_R} \right )}{d \,\textit {\_R}^{3}+c \textit {\_R}}\right )}{4 a^{3} d^{4}}\) | \(111\) |
| default | \(\text {Expression too large to display}\) | \(1742\) |
Input:
int(x^6/(a+b/(d*x^2+c)^2),x,method=_RETURNVERBOSE)
Output:
1/7*x^7/a-1/3*b*x^3/a^2/d^2+2*b*c*x/a^2/d^3+1/4/a^3/d^4*b*sum((d*(-3*a*c^2 +b)*_R^2-2*c^3*a-2*b*c)/(_R^3*d+_R*c)*ln(x-_R),_R=RootOf(_Z^4*a*d^2+2*_Z^2 *a*c*d+a*c^2+b))
Leaf count of result is larger than twice the leaf count of optimal. 1444 vs. \(2 (346) = 692\).
Time = 0.14 (sec) , antiderivative size = 1444, normalized size of antiderivative = 3.22 \[ \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\text {Too large to display} \] Input:
integrate(x^6/(a+b/(d*x^2+c)^2),x, algorithm="fricas")
Output:
1/84*(12*a*d^3*x^7 - 21*a^2*d^3*sqrt((a^5*d^7*sqrt(-(25*a^4*b^3*c^8 - 100* a^3*b^4*c^6 + 110*a^2*b^5*c^4 - 20*a*b^6*c^2 + b^7)/(a^11*d^14)) + a^2*b*c ^5 - 10*a*b^2*c^3 + 5*b^3*c)/(a^5*d^7))*log((5*a^4*b^2*c^8 - 14*a^2*b^4*c^ 4 - 8*a*b^5*c^2 + b^6)*x + ((a^9*c^2 - a^8*b)*d^10*sqrt(-(25*a^4*b^3*c^8 - 100*a^3*b^4*c^6 + 110*a^2*b^5*c^4 - 20*a*b^6*c^2 + b^7)/(a^11*d^14)) + 2* (5*a^5*b^2*c^5 - 10*a^4*b^3*c^3 + a^3*b^4*c)*d^3)*sqrt((a^5*d^7*sqrt(-(25* a^4*b^3*c^8 - 100*a^3*b^4*c^6 + 110*a^2*b^5*c^4 - 20*a*b^6*c^2 + b^7)/(a^1 1*d^14)) + a^2*b*c^5 - 10*a*b^2*c^3 + 5*b^3*c)/(a^5*d^7))) + 21*a^2*d^3*sq rt((a^5*d^7*sqrt(-(25*a^4*b^3*c^8 - 100*a^3*b^4*c^6 + 110*a^2*b^5*c^4 - 20 *a*b^6*c^2 + b^7)/(a^11*d^14)) + a^2*b*c^5 - 10*a*b^2*c^3 + 5*b^3*c)/(a^5* d^7))*log((5*a^4*b^2*c^8 - 14*a^2*b^4*c^4 - 8*a*b^5*c^2 + b^6)*x - ((a^9*c ^2 - a^8*b)*d^10*sqrt(-(25*a^4*b^3*c^8 - 100*a^3*b^4*c^6 + 110*a^2*b^5*c^4 - 20*a*b^6*c^2 + b^7)/(a^11*d^14)) + 2*(5*a^5*b^2*c^5 - 10*a^4*b^3*c^3 + a^3*b^4*c)*d^3)*sqrt((a^5*d^7*sqrt(-(25*a^4*b^3*c^8 - 100*a^3*b^4*c^6 + 11 0*a^2*b^5*c^4 - 20*a*b^6*c^2 + b^7)/(a^11*d^14)) + a^2*b*c^5 - 10*a*b^2*c^ 3 + 5*b^3*c)/(a^5*d^7))) + 21*a^2*d^3*sqrt(-(a^5*d^7*sqrt(-(25*a^4*b^3*c^8 - 100*a^3*b^4*c^6 + 110*a^2*b^5*c^4 - 20*a*b^6*c^2 + b^7)/(a^11*d^14)) - a^2*b*c^5 + 10*a*b^2*c^3 - 5*b^3*c)/(a^5*d^7))*log((5*a^4*b^2*c^8 - 14*a^2 *b^4*c^4 - 8*a*b^5*c^2 + b^6)*x + ((a^9*c^2 - a^8*b)*d^10*sqrt(-(25*a^4*b^ 3*c^8 - 100*a^3*b^4*c^6 + 110*a^2*b^5*c^4 - 20*a*b^6*c^2 + b^7)/(a^11*d...
Time = 2.85 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.64 \[ \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{11} d^{14} + t^{2} \left (- 32 a^{8} b c^{5} d^{7} + 320 a^{7} b^{2} c^{3} d^{7} - 160 a^{6} b^{3} c d^{7}\right ) + a^{5} b^{2} c^{10} + 5 a^{4} b^{3} c^{8} + 10 a^{3} b^{4} c^{6} + 10 a^{2} b^{5} c^{4} + 5 a b^{6} c^{2} + b^{7}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{9} c^{2} d^{10} + 64 t^{3} a^{8} b d^{10} + 4 t a^{6} b c^{7} d^{3} - 84 t a^{5} b^{2} c^{5} d^{3} + 140 t a^{4} b^{3} c^{3} d^{3} - 28 t a^{3} b^{4} c d^{3}}{5 a^{4} b^{2} c^{8} - 14 a^{2} b^{4} c^{4} - 8 a b^{5} c^{2} + b^{6}} \right )} \right )\right )} + \frac {x^{7}}{7 a} + \frac {2 b c x}{a^{2} d^{3}} - \frac {b x^{3}}{3 a^{2} d^{2}} \] Input:
integrate(x**6/(a+b/(d*x**2+c)**2),x)
Output:
RootSum(256*_t**4*a**11*d**14 + _t**2*(-32*a**8*b*c**5*d**7 + 320*a**7*b** 2*c**3*d**7 - 160*a**6*b**3*c*d**7) + a**5*b**2*c**10 + 5*a**4*b**3*c**8 + 10*a**3*b**4*c**6 + 10*a**2*b**5*c**4 + 5*a*b**6*c**2 + b**7, Lambda(_t, _t*log(x + (-64*_t**3*a**9*c**2*d**10 + 64*_t**3*a**8*b*d**10 + 4*_t*a**6* b*c**7*d**3 - 84*_t*a**5*b**2*c**5*d**3 + 140*_t*a**4*b**3*c**3*d**3 - 28* _t*a**3*b**4*c*d**3)/(5*a**4*b**2*c**8 - 14*a**2*b**4*c**4 - 8*a*b**5*c**2 + b**6)))) + x**7/(7*a) + 2*b*c*x/(a**2*d**3) - b*x**3/(3*a**2*d**2)
\[ \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\int { \frac {x^{6}}{a + \frac {b}{{\left (d x^{2} + c\right )}^{2}}} \,d x } \] Input:
integrate(x^6/(a+b/(d*x^2+c)^2),x, algorithm="maxima")
Output:
-b*integrate((2*a*c^3 + (3*a*c^2 - b)*d*x^2 + 2*b*c)/(a*d^2*x^4 + 2*a*c*d* x^2 + a*c^2 + b), x)/(a^2*d^3) + 1/21*(3*a*d^3*x^7 - 7*b*d*x^3 + 42*b*c*x) /(a^2*d^3)
Exception generated. \[ \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^6/(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 = 0.98 (sec) , antiderivative size = 3247, normalized size of antiderivative = 7.25 \[ \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\text {Too large to display} \] Input:
int(x^6/(a + b/(c + d*x^2)^2),x)
Output:
atan(((((32*a^6*b^3*c*d^7 + 32*a^7*b^2*c^3*d^7)/(a^5*d^4) - 64*a^4*b*c*d^7 *x*((b^2*(-a^11*b^3)^(1/2) + 5*a^2*c^4*(-a^11*b^3)^(1/2) + 5*a^6*b^3*c + a ^8*b*c^5 - 10*a^7*b^2*c^3 - 10*a*b*c^2*(-a^11*b^3)^(1/2))/(16*a^11*d^7))^( 1/2))*((b^2*(-a^11*b^3)^(1/2) + 5*a^2*c^4*(-a^11*b^3)^(1/2) + 5*a^6*b^3*c + a^8*b*c^5 - 10*a^7*b^2*c^3 - 10*a*b*c^2*(-a^11*b^3)^(1/2))/(16*a^11*d^7) )^(1/2) - (4*x*(b^5 - 15*a*b^4*c^2 + 15*a^2*b^3*c^4 - a^3*b^2*c^6))/a^2)*( (b^2*(-a^11*b^3)^(1/2) + 5*a^2*c^4*(-a^11*b^3)^(1/2) + 5*a^6*b^3*c + a^8*b *c^5 - 10*a^7*b^2*c^3 - 10*a*b*c^2*(-a^11*b^3)^(1/2))/(16*a^11*d^7))^(1/2) *1i - (((32*a^6*b^3*c*d^7 + 32*a^7*b^2*c^3*d^7)/(a^5*d^4) + 64*a^4*b*c*d^7 *x*((b^2*(-a^11*b^3)^(1/2) + 5*a^2*c^4*(-a^11*b^3)^(1/2) + 5*a^6*b^3*c + a ^8*b*c^5 - 10*a^7*b^2*c^3 - 10*a*b*c^2*(-a^11*b^3)^(1/2))/(16*a^11*d^7))^( 1/2))*((b^2*(-a^11*b^3)^(1/2) + 5*a^2*c^4*(-a^11*b^3)^(1/2) + 5*a^6*b^3*c + a^8*b*c^5 - 10*a^7*b^2*c^3 - 10*a*b*c^2*(-a^11*b^3)^(1/2))/(16*a^11*d^7) )^(1/2) + (4*x*(b^5 - 15*a*b^4*c^2 + 15*a^2*b^3*c^4 - a^3*b^2*c^6))/a^2)*( (b^2*(-a^11*b^3)^(1/2) + 5*a^2*c^4*(-a^11*b^3)^(1/2) + 5*a^6*b^3*c + a^8*b *c^5 - 10*a^7*b^2*c^3 - 10*a*b*c^2*(-a^11*b^3)^(1/2))/(16*a^11*d^7))^(1/2) *1i)/((((32*a^6*b^3*c*d^7 + 32*a^7*b^2*c^3*d^7)/(a^5*d^4) - 64*a^4*b*c*d^7 *x*((b^2*(-a^11*b^3)^(1/2) + 5*a^2*c^4*(-a^11*b^3)^(1/2) + 5*a^6*b^3*c + a ^8*b*c^5 - 10*a^7*b^2*c^3 - 10*a*b*c^2*(-a^11*b^3)^(1/2))/(16*a^11*d^7))^( 1/2))*((b^2*(-a^11*b^3)^(1/2) + 5*a^2*c^4*(-a^11*b^3)^(1/2) + 5*a^6*b^3...
Time = 0.21 (sec) , antiderivative size = 1301, normalized size of antiderivative = 2.90 \[ \int \frac {x^6}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx =\text {Too large to display} \] Input:
int(x^6/(a+b/(d*x^2+c)^2),x)
Output:
(42*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 - 42*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 - 42*sqrt(d)*sqrt(a)*s qrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)*atan((sqrt(d)*sqrt(sqrt(a)*sqr t(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 + 126*sqrt(d)*sqrt(a)*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*c - 42*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 + 42*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*sq rt(a)*d*x)/(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*b + 42* sqrt(d)*sqrt(a)*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)*sq rt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*a*c**3 - 126*sqrt(d)*sqrt(...