Integrand size = 22, antiderivative size = 241 \[ \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx=\frac {\left (A b^{2/3}+a^{2/3} C\right ) \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} b^{5/6}}+\frac {B \arctan \left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\left (A b^{2/3}+a^{2/3} C\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} b^{5/6}}+\frac {\left (A b^{2/3}+a^{2/3} C\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} b^{5/6}}+\frac {\left (A b^{2/3}-a^{2/3} C\right ) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt [3]{b} x^2}\right )}{2 \sqrt {3} a^{5/6} b^{5/6}} \] Output:
1/3*(A*b^(2/3)+a^(2/3)*C)*arctan(b^(1/6)*x/a^(1/6))/a^(5/6)/b^(5/6)+1/3*B* arctan(b^(1/2)*x^3/a^(1/2))/a^(1/2)/b^(1/2)+1/6*(A*b^(2/3)+a^(2/3)*C)*arct an(-3^(1/2)+2*b^(1/6)*x/a^(1/6))/a^(5/6)/b^(5/6)+1/6*(A*b^(2/3)+a^(2/3)*C) *arctan(3^(1/2)+2*b^(1/6)*x/a^(1/6))/a^(5/6)/b^(5/6)+1/6*(A*b^(2/3)-a^(2/3 )*C)*arctanh(3^(1/2)*a^(1/6)*b^(1/6)*x/(a^(1/3)+b^(1/3)*x^2))*3^(1/2)/a^(5 /6)/b^(5/6)
Time = 0.09 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.22 \[ \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx=\frac {4 \sqrt [6]{a} \left (A b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} B+a^{2/3} C\right ) \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [6]{a} \left (A b^{2/3}+2 \sqrt [3]{a} \sqrt [3]{b} B+a^{2/3} C\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{a} \left (A b^{2/3}+2 \sqrt [3]{a} \sqrt [3]{b} B+a^{2/3} C\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+\sqrt {3} \left (-\sqrt [6]{a} A b^{2/3}+a^{5/6} C\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-\sqrt {3} \left (-\sqrt [6]{a} A b^{2/3}+a^{5/6} C\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 a b^{5/6}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(a + b*x^6),x]
Output:
(4*a^(1/6)*(A*b^(2/3) - a^(1/3)*b^(1/3)*B + a^(2/3)*C)*ArcTan[(b^(1/6)*x)/ a^(1/6)] - 2*a^(1/6)*(A*b^(2/3) + 2*a^(1/3)*b^(1/3)*B + a^(2/3)*C)*ArcTan[ Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*a^(1/6)*(A*b^(2/3) + 2*a^(1/3)*b^(1/3 )*B + a^(2/3)*C)*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] + Sqrt[3]*(-(a^(1 /6)*A*b^(2/3)) + a^(5/6)*C)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1 /3)*x^2] - Sqrt[3]*(-(a^(1/6)*A*b^(2/3)) + a^(5/6)*C)*Log[a^(1/3) + Sqrt[3 ]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(12*a*b^(5/6))
Time = 1.71 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2415, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \int \left (\frac {A}{a+b x^6}+\frac {B x^2}{a+b x^6}+\frac {C x^4}{a+b x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {A \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac {A \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}-\frac {A \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {A \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {C \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {C \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {C \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {B \arctan \left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{3 \sqrt {a} \sqrt {b}}+\frac {C \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}}-\frac {C \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(a + b*x^6),x]
Output:
(A*ArcTan[(b^(1/6)*x)/a^(1/6)])/(3*a^(5/6)*b^(1/6)) + (C*ArcTan[(b^(1/6)*x )/a^(1/6)])/(3*a^(1/6)*b^(5/6)) + (B*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]])/(3*Sqr t[a]*Sqrt[b]) - (A*ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(6*a^( 5/6)*b^(1/6)) - (C*ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(6*a^( 1/6)*b^(5/6)) + (A*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(6*a^( 5/6)*b^(1/6)) + (C*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(6*a^( 1/6)*b^(5/6)) - (A*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]) /(4*Sqrt[3]*a^(5/6)*b^(1/6)) + (C*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*a^(1/6)*b^(5/6)) + (A*Log[a^(1/3) + Sqrt[3]*a^( 1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*a^(5/6)*b^(1/6)) - (C*Log[a^(1/3 ) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*a^(1/6)*b^(5/6))
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.16
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\left (C \,\textit {\_R}^{4}+B \,\textit {\_R}^{2}+A \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 b}\) | \(39\) |
| default | \(-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) C \left (\frac {a}{b}\right )^{\frac {5}{6}} \sqrt {3}}{12 a}+\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) A \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {3}}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) C}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) A}{6 a}+\frac {\sqrt {\frac {a}{b}}\, \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) B}{3 a}+\frac {\left (\frac {a}{b}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right ) C}{3 a}-\frac {\sqrt {\frac {a}{b}}\, \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right ) B}{3 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right ) A}{3 a}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) C \left (\frac {a}{b}\right )^{\frac {5}{6}} \sqrt {3}}{12 a}-\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) A \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {3}}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) C}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) A}{6 a}+\frac {\sqrt {\frac {a}{b}}\, \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) B}{3 a}\) | \(407\) |
Input:
int((C*x^4+B*x^2+A)/(b*x^6+a),x,method=_RETURNVERBOSE)
Output:
1/6/b*sum((C*_R^4+B*_R^2+A)/_R^5*ln(x-_R),_R=RootOf(_Z^6*b+a))
Result contains complex when optimal does not.
Time = 25.70 (sec) , antiderivative size = 36914, normalized size of antiderivative = 153.17 \[ \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx=\text {Too large to display} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^6+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx=\text {Timed out} \] Input:
integrate((C*x**4+B*x**2+A)/(b*x**6+a),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx=-\frac {\sqrt {3} {\left (C a^{\frac {2}{3}} - A b^{\frac {2}{3}}\right )} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {5}{6}} b^{\frac {5}{6}}} + \frac {\sqrt {3} {\left (C a^{\frac {2}{3}} - A b^{\frac {2}{3}}\right )} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {5}{6}} b^{\frac {5}{6}}} + \frac {{\left (C a^{\frac {2}{3}} - B a^{\frac {1}{3}} b^{\frac {1}{3}} + A b^{\frac {2}{3}}\right )} \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{3 \, a^{\frac {2}{3}} b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {{\left (C a b^{\frac {1}{3}} - 2 \, A a^{\frac {1}{3}} b + {\left (2 \, B a^{\frac {2}{3}} + 3 \, A a^{\frac {1}{3}} b^{\frac {1}{3}}\right )} b^{\frac {2}{3}}\right )} \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, a b \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {{\left (C a b^{\frac {1}{3}} - 2 \, A a^{\frac {1}{3}} b + {\left (2 \, B a^{\frac {2}{3}} + 3 \, A a^{\frac {1}{3}} b^{\frac {1}{3}}\right )} b^{\frac {2}{3}}\right )} \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, a b \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^6+a),x, algorithm="maxima")
Output:
-1/12*sqrt(3)*(C*a^(2/3) - A*b^(2/3))*log(b^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^ (1/6)*x + a^(1/3))/(a^(5/6)*b^(5/6)) + 1/12*sqrt(3)*(C*a^(2/3) - A*b^(2/3) )*log(b^(1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/(a^(5/6)*b^(5/6)) + 1/3*(C*a^(2/3) - B*a^(1/3)*b^(1/3) + A*b^(2/3))*arctan(b^(1/3)*x/sqrt(a ^(1/3)*b^(1/3)))/(a^(2/3)*b^(2/3)*sqrt(a^(1/3)*b^(1/3))) + 1/6*(C*a*b^(1/3 ) - 2*A*a^(1/3)*b + (2*B*a^(2/3) + 3*A*a^(1/3)*b^(1/3))*b^(2/3))*arctan((2 *b^(1/3)*x + sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/(a*b*sqrt(a^( 1/3)*b^(1/3))) + 1/6*(C*a*b^(1/3) - 2*A*a^(1/3)*b + (2*B*a^(2/3) + 3*A*a^( 1/3)*b^(1/3))*b^(2/3))*arctan((2*b^(1/3)*x - sqrt(3)*a^(1/6)*b^(1/6))/sqrt (a^(1/3)*b^(1/3)))/(a*b*sqrt(a^(1/3)*b^(1/3)))
Time = 0.25 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx=\frac {{\left (A b^{4} + 2 \, \left (a b^{5}\right )^{\frac {1}{3}} B b^{2} + \left (a b^{5}\right )^{\frac {2}{3}} C\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, \left (a b^{5}\right )^{\frac {5}{6}}} + \frac {{\left (A b^{4} + 2 \, \left (a b^{5}\right )^{\frac {1}{3}} B b^{2} + \left (a b^{5}\right )^{\frac {2}{3}} C\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, \left (a b^{5}\right )^{\frac {5}{6}}} + \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {5}{6}} - B \sqrt {\frac {a}{b}} + A \left (\frac {a}{b}\right )^{\frac {1}{6}}\right )} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, a} + \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {1}{6}} A b^{4} - \left (a b^{5}\right )^{\frac {5}{6}} C\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a b^{5}} - \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {1}{6}} A b^{4} - \left (a b^{5}\right )^{\frac {5}{6}} C\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a b^{5}} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^6+a),x, algorithm="giac")
Output:
1/6*(A*b^4 + 2*(a*b^5)^(1/3)*B*b^2 + (a*b^5)^(2/3)*C)*arctan((2*x + sqrt(3 )*(a/b)^(1/6))/(a/b)^(1/6))/(a*b^5)^(5/6) + 1/6*(A*b^4 + 2*(a*b^5)^(1/3)*B *b^2 + (a*b^5)^(2/3)*C)*arctan((2*x - sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a *b^5)^(5/6) + 1/3*(C*(a/b)^(5/6) - B*sqrt(a/b) + A*(a/b)^(1/6))*arctan(x/( a/b)^(1/6))/a + 1/12*sqrt(3)*((a*b^5)^(1/6)*A*b^4 - (a*b^5)^(5/6)*C)*log(x ^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a*b^5) - 1/12*sqrt(3)*((a*b^5)^ (1/6)*A*b^4 - (a*b^5)^(5/6)*C)*log(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/ 3))/(a*b^5)
Time = 6.72 (sec) , antiderivative size = 2594, normalized size of antiderivative = 10.76 \[ \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx=\text {Too large to display} \] Input:
int((A + B*x^2 + C*x^4)/(a + b*x^6),x)
Output:
symsum(log(A^3*B^2*b^4 - B^5*a*b^3 - A^4*C*b^4 + B^2*C^3*a^2*b^2 - 72*B^3* root(46656*a^5*b^5*z^6 + 7776*A*C*a^4*b^4*z^4 + 3888*B^2*a^4*b^4*z^4 + 324 *A^2*C^2*a^3*b^3*z^2 + 216*B*C^3*a^4*b^2*z^2 + 216*A^3*B*a^2*b^4*z^2 + 108 *B^4*a^3*b^3*z^2 + 9*A^2*B^2*C^2*a^2*b^2 + 6*A^4*B*C*a*b^3 + 6*A*B*C^4*a^3 *b - 6*A*B^4*C*a^2*b^2 - 2*B^3*C^3*a^3*b - 2*A^3*B^3*a*b^3 + 2*A^3*C^3*a^2 *b^2 + B^6*a^2*b^2 + C^6*a^4 + A^6*b^4, z, k)^2*a^2*b^4 - 36*C^3*root(4665 6*a^5*b^5*z^6 + 7776*A*C*a^4*b^4*z^4 + 3888*B^2*a^4*b^4*z^4 + 324*A^2*C^2* a^3*b^3*z^2 + 216*B*C^3*a^4*b^2*z^2 + 216*A^3*B*a^2*b^4*z^2 + 108*B^4*a^3* b^3*z^2 + 9*A^2*B^2*C^2*a^2*b^2 + 6*A^4*B*C*a*b^3 + 6*A*B*C^4*a^3*b - 6*A* B^4*C*a^2*b^2 - 2*B^3*C^3*a^3*b - 2*A^3*B^3*a*b^3 + 2*A^3*C^3*a^2*b^2 + B^ 6*a^2*b^2 + C^6*a^4 + A^6*b^4, z, k)^2*a^3*b^3 - 6*A^4*root(46656*a^5*b^5* z^6 + 7776*A*C*a^4*b^4*z^4 + 3888*B^2*a^4*b^4*z^4 + 324*A^2*C^2*a^3*b^3*z^ 2 + 216*B*C^3*a^4*b^2*z^2 + 216*A^3*B*a^2*b^4*z^2 + 108*B^4*a^3*b^3*z^2 + 9*A^2*B^2*C^2*a^2*b^2 + 6*A^4*B*C*a*b^3 + 6*A*B*C^4*a^3*b - 6*A*B^4*C*a^2* b^2 - 2*B^3*C^3*a^3*b - 2*A^3*B^3*a*b^3 + 2*A^3*C^3*a^2*b^2 + B^6*a^2*b^2 + C^6*a^4 + A^6*b^4, z, k)*b^5*x - A*C^4*a^2*b^2 - 36*A^3*root(46656*a^5*b ^5*z^6 + 7776*A*C*a^4*b^4*z^4 + 3888*B^2*a^4*b^4*z^4 + 324*A^2*C^2*a^3*b^3 *z^2 + 216*B*C^3*a^4*b^2*z^2 + 216*A^3*B*a^2*b^4*z^2 + 108*B^4*a^3*b^3*z^2 + 9*A^2*B^2*C^2*a^2*b^2 + 6*A^4*B*C*a*b^3 + 6*A*B*C^4*a^3*b - 6*A*B^4*C*a ^2*b^2 - 2*B^3*C^3*a^3*b - 2*A^3*B^3*a*b^3 + 2*A^3*C^3*a^2*b^2 + B^6*a^...
Time = 0.18 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x^2+C x^4}{a+b x^6} \, dx=\frac {-2 b^{\frac {2}{3}} a^{\frac {2}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-2 a^{\frac {1}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c -4 b^{\frac {4}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+2 b^{\frac {2}{3}} a^{\frac {2}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+2 a^{\frac {1}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c +4 b^{\frac {4}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+4 b^{\frac {2}{3}} a^{\frac {2}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right )+4 a^{\frac {1}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) c -4 b^{\frac {4}{3}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right )-b^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathrm {log}\left (-b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{2}\right )+b^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathrm {log}\left (b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{2}\right )+a^{\frac {1}{3}} \sqrt {3}\, \mathrm {log}\left (-b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{2}\right ) c -a^{\frac {1}{3}} \sqrt {3}\, \mathrm {log}\left (b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{2}\right ) c}{12 b^{\frac {5}{6}} \sqrt {a}} \] Input:
int((C*x^4+B*x^2+A)/(b*x^6+a),x)
Output:
(b**(1/6)*a**(1/6)*( - 2*b**(2/3)*a**(2/3)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*b**(1/3)*x)/(b**(1/6)*a**(1/6))) - 2*a**(1/3)*atan((b**(1/6)*a**(1/6) *sqrt(3) - 2*b**(1/3)*x)/(b**(1/6)*a**(1/6)))*c - 4*b**(1/3)*atan((b**(1/6 )*a**(1/6)*sqrt(3) - 2*b**(1/3)*x)/(b**(1/6)*a**(1/6)))*b + 2*b**(2/3)*a** (2/3)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*b**(1/3)*x)/(b**(1/6)*a**(1/6))) + 2*a**(1/3)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*b**(1/3)*x)/(b**(1/6)*a* *(1/6)))*c + 4*b**(1/3)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*b**(1/3)*x)/(b **(1/6)*a**(1/6)))*b + 4*b**(2/3)*a**(2/3)*atan((b**(1/3)*x)/(b**(1/6)*a** (1/6))) + 4*a**(1/3)*atan((b**(1/3)*x)/(b**(1/6)*a**(1/6)))*c - 4*b**(1/3) *atan((b**(1/3)*x)/(b**(1/6)*a**(1/6)))*b - b**(2/3)*a**(2/3)*sqrt(3)*log( - b**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + b**(1/3)*x**2) + b**(2/3)*a**( 2/3)*sqrt(3)*log(b**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + b**(1/3)*x**2) + a**(1/3)*sqrt(3)*log( - b**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + b**(1/3) *x**2)*c - a**(1/3)*sqrt(3)*log(b**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + b **(1/3)*x**2)*c))/(12*a**(2/3)*b)