Integrand size = 19, antiderivative size = 342 \[ \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {x^3}{3 a}+\frac {b \arctan \left (\frac {\sqrt {-\sqrt {a} c+\sqrt {b+a c^2}}-\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a} c+\sqrt {b+a c^2}}}\right )}{2 \sqrt {2} a^{7/4} \sqrt {\sqrt {a} c+\sqrt {b+a c^2}} d^{3/2}}-\frac {b \arctan \left (\frac {\sqrt {-\sqrt {a} c+\sqrt {b+a c^2}}+\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a} c+\sqrt {b+a c^2}}}\right )}{2 \sqrt {2} a^{7/4} \sqrt {\sqrt {a} c+\sqrt {b+a c^2}} d^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} c+\sqrt {b+a c^2}} \sqrt {d} x}{\sqrt {b+a c^2}+\sqrt {a} d x^2}\right )}{2 \sqrt {2} a^{7/4} \sqrt {-\sqrt {a} c+\sqrt {b+a c^2}} d^{3/2}} \] Output:
1/3*x^3/a+1/4*b*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^(7/4)/(a^(1/2)*c+ (a*c^2+b)^(1/2))^(1/2)/d^(3/2)-1/4*b*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^(7/4)/(a^(1/2)*c+(a*c^2+b)^(1/2))^(1/2)/d^(3/2)+1/4*b*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^(7/4)/(-a^(1/2)*c+(a*c^2+b)^(1/2))^(1/2)/d^(3/2)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.54 \[ \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {2 a d^{3/2} x^3+3 i \sqrt {b} \sqrt {-i \sqrt {a} \sqrt {b}+a c} \arctan \left (\frac {\sqrt {-i \sqrt {a} \sqrt {b}+a c} \sqrt {d} x}{i \sqrt {b}-\sqrt {a} c}\right )+3 i \sqrt {b} \sqrt {i \sqrt {a} \sqrt {b}+a c} \arctan \left (\frac {\sqrt {i \sqrt {a} \sqrt {b}+a c} \sqrt {d} x}{i \sqrt {b}+\sqrt {a} c}\right )}{6 a^2 d^{3/2}} \] Input:
Integrate[x^2/(a + b/(c + d*x^2)^2),x]
Output:
(2*a*d^(3/2)*x^3 + (3*I)*Sqrt[b]*Sqrt[(-I)*Sqrt[a]*Sqrt[b] + a*c]*ArcTan[( Sqrt[(-I)*Sqrt[a]*Sqrt[b] + a*c]*Sqrt[d]*x)/(I*Sqrt[b] - Sqrt[a]*c)] + (3* I)*Sqrt[b]*Sqrt[I*Sqrt[a]*Sqrt[b] + a*c]*ArcTan[(Sqrt[I*Sqrt[a]*Sqrt[b] + a*c]*Sqrt[d]*x)/(I*Sqrt[b] + Sqrt[a]*c)])/(6*a^2*d^(3/2))
Time = 1.38 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.31, 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.
| \(\displaystyle \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2}{a}-\frac {b x^2}{a \left (a c^2+2 a c d x^2+a d^2 x^4+b\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \arctan \left (\frac {\sqrt {\sqrt {a c^2+b}-\sqrt {a} c}-\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}\right )}{2 \sqrt {2} a^{7/4} d^{3/2} \sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}-\frac {b \arctan \left (\frac {\sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}\right )}{2 \sqrt {2} a^{7/4} d^{3/2} \sqrt {\sqrt {a c^2+b}+\sqrt {a} c}}-\frac {b \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt {d} x \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {a c^2+b}+\sqrt {a} d x^2\right )}{4 \sqrt {2} a^{7/4} d^{3/2} \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}}+\frac {b \log \left (\sqrt {2} \sqrt [4]{a} \sqrt {d} x \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}+\sqrt {a c^2+b}+\sqrt {a} d x^2\right )}{4 \sqrt {2} a^{7/4} d^{3/2} \sqrt {\sqrt {a c^2+b}-\sqrt {a} c}}+\frac {x^3}{3 a}\) |
Input:
Int[x^2/(a + b/(c + d*x^2)^2),x]
Output:
x^3/(3*a) + (b*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^(7/4)*Sqrt [Sqrt[a]*c + Sqrt[b + a*c^2]]*d^(3/2)) - (b*ArcTan[(Sqrt[-(Sqrt[a]*c) + Sq rt[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^(7/4)*Sqrt[Sqrt[a]*c + Sqrt[b + a*c^2]]*d^(3/2)) - (b*L og[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^(7/4)*Sqrt[-(Sqrt[a]*c) + Sqrt[b + a*c^2]]*d^(3/2)) + (b*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^(7/4)*Sq rt[-(Sqrt[a]*c) + Sqrt[b + a*c^2]]*d^(3/2))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.20
| method | result | size |
| risch | \(\frac {x^{3}}{3 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,d^{2} \textit {\_Z}^{4}+2 a c d \,\textit {\_Z}^{2}+a \,c^{2}+b \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{d \,\textit {\_R}^{3}+c \textit {\_R}}\right )}{4 a^{2} d}\) | \(67\) |
| default | \(\frac {x^{3}}{3 a}-\frac {b \left (\frac {\sqrt {2 \sqrt {a^{2} c^{2} d^{2}+a b \,d^{2}}-2 a c d}\, \left (a c d +\sqrt {a^{2} c^{2} d^{2}+a b \,d^{2}}\right ) \left (\frac {\ln \left (\sqrt {a \,d^{2}}\, x^{2}-x \sqrt {2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}-2 a c d}+\sqrt {a \,c^{2}+b}\right )}{2 \sqrt {a \,d^{2}}}+\frac {\sqrt {2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}-2 a c d}\, \arctan \left (\frac {2 \sqrt {a \,d^{2}}\, x -\sqrt {2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}-2 a c d}}{\sqrt {4 \sqrt {a \,d^{2}}\, \sqrt {a \,c^{2}+b}-2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}+2 a c d}}\right )}{\sqrt {a \,d^{2}}\, \sqrt {4 \sqrt {a \,d^{2}}\, \sqrt {a \,c^{2}+b}-2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}+2 a c d}}\right )}{4 a b \,d^{2}}-\frac {\sqrt {2 \sqrt {a^{2} c^{2} d^{2}+a b \,d^{2}}-2 a c d}\, \left (a c d +\sqrt {a^{2} c^{2} d^{2}+a b \,d^{2}}\right ) \left (\frac {\ln \left (\sqrt {a \,d^{2}}\, x^{2}+x \sqrt {2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}-2 a c d}+\sqrt {a \,c^{2}+b}\right )}{2 \sqrt {a \,d^{2}}}-\frac {\sqrt {2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}-2 a c d}\, \arctan \left (\frac {2 \sqrt {a \,d^{2}}\, x +\sqrt {2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}-2 a c d}}{\sqrt {4 \sqrt {a \,d^{2}}\, \sqrt {a \,c^{2}+b}-2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}+2 a c d}}\right )}{\sqrt {a \,d^{2}}\, \sqrt {4 \sqrt {a \,d^{2}}\, \sqrt {a \,c^{2}+b}-2 \sqrt {a \,d^{2} \left (a \,c^{2}+b \right )}+2 a c d}}\right )}{4 a b \,d^{2}}\right )}{a}\) | \(571\) |
Input:
int(x^2/(a+b/(d*x^2+c)^2),x,method=_RETURNVERBOSE)
Output:
1/3/a*x^3-1/4/a^2*b/d*sum(_R^2/(_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))
Time = 0.11 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {4 \, x^{3} - 3 \, a \sqrt {\frac {a^{3} d^{3} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} + b c}{a^{3} d^{3}}} \log \left (a^{5} d^{4} \sqrt {\frac {a^{3} d^{3} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} + b c}{a^{3} d^{3}}} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} + b^{2} x\right ) + 3 \, a \sqrt {\frac {a^{3} d^{3} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} + b c}{a^{3} d^{3}}} \log \left (-a^{5} d^{4} \sqrt {\frac {a^{3} d^{3} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} + b c}{a^{3} d^{3}}} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} + b^{2} x\right ) + 3 \, a \sqrt {-\frac {a^{3} d^{3} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} - b c}{a^{3} d^{3}}} \log \left (a^{5} d^{4} \sqrt {-\frac {a^{3} d^{3} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} - b c}{a^{3} d^{3}}} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} + b^{2} x\right ) - 3 \, a \sqrt {-\frac {a^{3} d^{3} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} - b c}{a^{3} d^{3}}} \log \left (-a^{5} d^{4} \sqrt {-\frac {a^{3} d^{3} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} - b c}{a^{3} d^{3}}} \sqrt {-\frac {b^{3}}{a^{7} d^{6}}} + b^{2} x\right )}{12 \, a} \] Input:
integrate(x^2/(a+b/(d*x^2+c)^2),x, algorithm="fricas")
Output:
1/12*(4*x^3 - 3*a*sqrt((a^3*d^3*sqrt(-b^3/(a^7*d^6)) + b*c)/(a^3*d^3))*log (a^5*d^4*sqrt((a^3*d^3*sqrt(-b^3/(a^7*d^6)) + b*c)/(a^3*d^3))*sqrt(-b^3/(a ^7*d^6)) + b^2*x) + 3*a*sqrt((a^3*d^3*sqrt(-b^3/(a^7*d^6)) + b*c)/(a^3*d^3 ))*log(-a^5*d^4*sqrt((a^3*d^3*sqrt(-b^3/(a^7*d^6)) + b*c)/(a^3*d^3))*sqrt( -b^3/(a^7*d^6)) + b^2*x) + 3*a*sqrt(-(a^3*d^3*sqrt(-b^3/(a^7*d^6)) - b*c)/ (a^3*d^3))*log(a^5*d^4*sqrt(-(a^3*d^3*sqrt(-b^3/(a^7*d^6)) - b*c)/(a^3*d^3 ))*sqrt(-b^3/(a^7*d^6)) + b^2*x) - 3*a*sqrt(-(a^3*d^3*sqrt(-b^3/(a^7*d^6)) - b*c)/(a^3*d^3))*log(-a^5*d^4*sqrt(-(a^3*d^3*sqrt(-b^3/(a^7*d^6)) - b*c) /(a^3*d^3))*sqrt(-b^3/(a^7*d^6)) + b^2*x))/a
Time = 0.41 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.22 \[ \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{7} d^{6} - 32 t^{2} a^{4} b c d^{3} + a b^{2} c^{2} + b^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{5} d^{4} + 4 t a^{2} b c d}{b^{2}} \right )} \right )\right )} + \frac {x^{3}}{3 a} \] Input:
integrate(x**2/(a+b/(d*x**2+c)**2),x)
Output:
RootSum(256*_t**4*a**7*d**6 - 32*_t**2*a**4*b*c*d**3 + a*b**2*c**2 + b**3, Lambda(_t, _t*log(x + (-64*_t**3*a**5*d**4 + 4*_t*a**2*b*c*d)/b**2))) + x **3/(3*a)
\[ \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\int { \frac {x^{2}}{a + \frac {b}{{\left (d x^{2} + c\right )}^{2}}} \,d x } \] Input:
integrate(x^2/(a+b/(d*x^2+c)^2),x, algorithm="maxima")
Output:
1/3*x^3/a - b*integrate(x^2/(a*d^2*x^4 + 2*a*c*d*x^2 + a*c^2 + b), x)/a
Exception generated. \[ \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(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 = 9.19 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=2\,\mathrm {atanh}\left (\frac {a^2\,\left (x\,\left (4\,b^3\,d^4-4\,a\,b^2\,c^2\,d^4\right )+\frac {4\,b\,c\,d^4\,x\,\left (\sqrt {-a^7\,b^3}+a^4\,b\,c\right )}{a^3}\right )\,\sqrt {\frac {\sqrt {-a^7\,b^3}+a^4\,b\,c}{16\,a^7\,d^3}}}{b^4\,d^2+a\,b^3\,c^2\,d^2}\right )\,\sqrt {\frac {\sqrt {-a^7\,b^3}+a^4\,b\,c}{16\,a^7\,d^3}}+2\,\mathrm {atanh}\left (\frac {a^2\,\left (x\,\left (4\,b^3\,d^4-4\,a\,b^2\,c^2\,d^4\right )-\frac {4\,b\,c\,d^4\,x\,\left (\sqrt {-a^7\,b^3}-a^4\,b\,c\right )}{a^3}\right )\,\sqrt {-\frac {\sqrt {-a^7\,b^3}-a^4\,b\,c}{16\,a^7\,d^3}}}{b^4\,d^2+a\,b^3\,c^2\,d^2}\right )\,\sqrt {-\frac {\sqrt {-a^7\,b^3}-a^4\,b\,c}{16\,a^7\,d^3}}+\frac {x^3}{3\,a} \] Input:
int(x^2/(a + b/(c + d*x^2)^2),x)
Output:
2*atanh((a^2*(x*(4*b^3*d^4 - 4*a*b^2*c^2*d^4) + (4*b*c*d^4*x*((-a^7*b^3)^( 1/2) + a^4*b*c))/a^3)*(((-a^7*b^3)^(1/2) + a^4*b*c)/(16*a^7*d^3))^(1/2))/( b^4*d^2 + a*b^3*c^2*d^2))*(((-a^7*b^3)^(1/2) + a^4*b*c)/(16*a^7*d^3))^(1/2 ) + 2*atanh((a^2*(x*(4*b^3*d^4 - 4*a*b^2*c^2*d^4) - (4*b*c*d^4*x*((-a^7*b^ 3)^(1/2) - a^4*b*c))/a^3)*(-((-a^7*b^3)^(1/2) - a^4*b*c)/(16*a^7*d^3))^(1/ 2))/(b^4*d^2 + a*b^3*c^2*d^2))*(-((-a^7*b^3)^(1/2) - a^4*b*c)/(16*a^7*d^3) )^(1/2) + x^3/(3*a)
Time = 0.26 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.85 \[ \int \frac {x^2}{a+\frac {b}{\left (c+d x^2\right )^2}} \, dx=\frac {6 \sqrt {d}\, \sqrt {a \,c^{2}+b}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}\, \mathit {atan} \left (\frac {\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}}\right )-6 \sqrt {d}\, \sqrt {a}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}\, \mathit {atan} \left (\frac {\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}}\right ) c -6 \sqrt {d}\, \sqrt {a \,c^{2}+b}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}\, \mathit {atan} \left (\frac {\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}}\right )+6 \sqrt {d}\, \sqrt {a}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}+a c}\, \sqrt {2}\, \mathit {atan} \left (\frac {\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}}\right ) c -3 \sqrt {d}\, \sqrt {a \,c^{2}+b}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, x +\sqrt {a \,c^{2}+b}+\sqrt {a}\, d \,x^{2}\right )+3 \sqrt {d}\, \sqrt {a \,c^{2}+b}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, x +\sqrt {a \,c^{2}+b}+\sqrt {a}\, d \,x^{2}\right )-3 \sqrt {d}\, \sqrt {a}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, x +\sqrt {a \,c^{2}+b}+\sqrt {a}\, d \,x^{2}\right ) c +3 \sqrt {d}\, \sqrt {a}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {d}\, \sqrt {\sqrt {a}\, \sqrt {a \,c^{2}+b}-a c}\, \sqrt {2}\, x +\sqrt {a \,c^{2}+b}+\sqrt {a}\, d \,x^{2}\right ) c +8 a \,d^{2} x^{3}}{24 a^{2} d^{2}} \] Input:
int(x^2/(a+b/(d*x^2+c)^2),x)
Output:
(6*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))) - 6*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)*sq rt(a*c**2 + b) + a*c)*sqrt(2)))*c - 6*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))) + 6*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*sq rt(a)*d*x)/(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) + a*c)*sqrt(2)))*c - 3*s qrt(d)*sqrt(a*c**2 + b)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2)*log( - sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2)*x + sqrt(a*c**2 + b ) + sqrt(a)*d*x**2) + 3*sqrt(d)*sqrt(a*c**2 + b)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2)*log(sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt( 2)*x + sqrt(a*c**2 + b) + sqrt(a)*d*x**2) - 3*sqrt(d)*sqrt(a)*sqrt(sqrt(a) *sqrt(a*c**2 + b) - a*c)*sqrt(2)*log( - sqrt(d)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2)*x + sqrt(a*c**2 + b) + sqrt(a)*d*x**2)*c + 3*sqrt(d)*sq rt(a)*sqrt(sqrt(a)*sqrt(a*c**2 + b) - a*c)*sqrt(2)*log(sqrt(d)*sqrt(sqrt(a )*sqrt(a*c**2 + b) - a*c)*sqrt(2)*x + sqrt(a*c**2 + b) + sqrt(a)*d*x**2...