Integrand size = 33, antiderivative size = 429 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (B c d-(A c-a C) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{3/2} d \sqrt {c d^2+a e^2}}+\frac {C \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c \sqrt {e}}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (B c d-(A c-a C) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{3/2} d \sqrt {c d^2+a e^2}} \] Output:
1/4*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(B*c*d-(A*c-C*a)*(e-(a*e^2+c*d^2 )^(1/2)/a^(1/2)))*arctan(2^(1/2)*a^(1/4)*c^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^ (1/2))^(1/2)*x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))-c* d*x^2))*2^(1/2)/a^(1/4)/c^(3/2)/d/(a*e^2+c*d^2)^(1/2)+C*arctanh(e^(1/2)*x/ (e*x^2+d)^(1/2))/c/e^(1/2)+1/4*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(B*c *d-(A*c-C*a)*(e+(a*e^2+c*d^2)^(1/2)/a^(1/2)))*arctanh(2^(1/2)*a^(1/4)*c^(1 /2)*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^( 1/2)*e-(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a^(1/4)/c^(3/2)/d/(a*e^2+c*d ^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.31 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=\frac {-2 C \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )+e \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {B c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-2 B c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 A c e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a C e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+B c \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{2 c \sqrt {e}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(Sqrt[d + e*x^2]*(a + c*x^4)),x]
Output:
(-2*C*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]] + e*RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 + 16*a*e^2*#1^2 - 4*c*d*#1^3 + c*#1^4 & , (B*c*d^2*Log[d + 2 *e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2*B*c*d*Log[d + 2*e*x^2 - 2*S qrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 4*A*c*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x* Sqrt[d + e*x^2] - #1]*#1 - 4*a*C*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + B*c*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]* #1^2)/(c*d^3 - 3*c*d^2*#1 - 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ])/(2*c*Sq rt[e])
Time = 0.63 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.54, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2257, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+c x^4\right ) \sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 2257 |
| \(\displaystyle \int \left (\frac {-a C+A c+B c x^2}{c \left (a+c x^4\right ) \sqrt {d+e x^2}}+\frac {C}{c \sqrt {d+e x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\sqrt {-a} B \sqrt {c}+a C-A c\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} c \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\left (\sqrt {-a} B \sqrt {c}-a C+A c\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} c \sqrt {\sqrt {-a} e+\sqrt {c} d}}+\frac {C \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c \sqrt {e}}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(Sqrt[d + e*x^2]*(a + c*x^4)),x]
Output:
((Sqrt[-a]*B*Sqrt[c] - A*c + a*C)*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/ ((-a)^(1/4)*Sqrt[d + e*x^2])])/(2*(-a)^(3/4)*c*Sqrt[Sqrt[c]*d - Sqrt[-a]*e ]) + (C*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c*Sqrt[e]) - ((Sqrt[-a]*B*S qrt[c] + A*c - a*C)*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*x)/((-a)^(1/4)*S qrt[d + e*x^2])])/(2*(-a)^(3/4)*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a , c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(912\) vs. \(2(343)=686\).
Time = 0.69 (sec) , antiderivative size = 913, normalized size of antiderivative = 2.13
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\left (\left (\sqrt {e}\, \left (A c -C a \right ) \sqrt {a \,e^{2}+c \,d^{2}}-C \,a^{\frac {3}{2}} e^{\frac {3}{2}}+\sqrt {a}\, c \left (-B d \sqrt {e}+A \,e^{\frac {3}{2}}\right )\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-e^{\frac {3}{2}} a \left (A c -C a \right ) \sqrt {a \,e^{2}+c \,d^{2}}-c \left (-B d \,e^{\frac {3}{2}}+e^{\frac {5}{2}} A \right ) a^{\frac {3}{2}}+C \,a^{\frac {5}{2}} e^{\frac {5}{2}}\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )-\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +x^{2} \sqrt {a \,e^{2}+c \,d^{2}}}{x^{2}}\right )}{4}+\frac {\left (\left (\sqrt {e}\, \left (A c -C a \right ) \sqrt {a \,e^{2}+c \,d^{2}}-C \,a^{\frac {3}{2}} e^{\frac {3}{2}}+\sqrt {a}\, c \left (-B d \sqrt {e}+A \,e^{\frac {3}{2}}\right )\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-e^{\frac {3}{2}} a \left (A c -C a \right ) \sqrt {a \,e^{2}+c \,d^{2}}-c \left (-B d \,e^{\frac {3}{2}}+e^{\frac {5}{2}} A \right ) a^{\frac {3}{2}}+C \,a^{\frac {5}{2}} e^{\frac {5}{2}}\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+x^{2} \sqrt {a \,e^{2}+c \,d^{2}}+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x^{2}}\right )}{4}+c \,d^{2} \left (2 a^{\frac {3}{2}} \sqrt {a \,e^{2}+c \,d^{2}}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, C +\left (\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right ) \left (-\sqrt {e}\, a \left (A c -C a \right ) \sqrt {a \,e^{2}+c \,d^{2}}+c \left (-B d \sqrt {e}+A \,e^{\frac {3}{2}}\right ) a^{\frac {3}{2}}-C \,a^{\frac {5}{2}} e^{\frac {3}{2}}\right )\right )}{2 \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, a^{\frac {3}{2}} \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {e}\, c^{2} d^{2}}\) | \(913\) |
| default | \(\frac {C \ln \left (\sqrt {e \,x^{2}+d}+x \sqrt {e}\right )}{c \sqrt {e}}-\frac {\left (-\sqrt {-a}\, B \sqrt {c}+A c -C a \right ) \left (-\frac {\ln \left (\frac {\frac {-2 \sqrt {-a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {e \left (x -\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2}+\frac {2 e \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {-2 \sqrt {-a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {e \left (x +\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2}-\frac {2 e \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}\right )}{2 c \sqrt {-a}}+\frac {\left (\sqrt {-a}\, B \sqrt {c}+A c -C a \right ) \left (-\frac {\ln \left (\frac {\frac {2 \sqrt {-a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {\sqrt {-a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {e \left (x -\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2}+\frac {2 e \sqrt {\sqrt {-a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {-a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {2 \sqrt {-a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {\sqrt {-a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {e \left (x +\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2}-\frac {2 e \sqrt {\sqrt {-a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {-a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}\right )}{2 c \sqrt {-a}}\) | \(939\) |
Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/2*(-1/4*((e^(1/2)*(A*c-C*a)*(a*e^2+c*d^2)^(1/2)-C*a^(3/2)*e^(3/2)+a^(1/2 )*c*(-B*d*e^(1/2)+A*e^(3/2)))*(a*(a*e^2+c*d^2))^(1/2)-e^(3/2)*a*(A*c-C*a)* (a*e^2+c*d^2)^(1/2)-c*(-B*d*e^(3/2)+e^(5/2)*A)*a^(3/2)+C*a^(5/2)*e^(5/2))* (2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*( a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)*ln((a^(1/2)*(e*x^2+d)-(e*x^2+d)^(1/2)* (2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x+x^2*(a*e^2+c*d^2)^(1/2))/x^2)+1/ 4*((e^(1/2)*(A*c-C*a)*(a*e^2+c*d^2)^(1/2)-C*a^(3/2)*e^(3/2)+a^(1/2)*c*(-B* d*e^(1/2)+A*e^(3/2)))*(a*(a*e^2+c*d^2))^(1/2)-e^(3/2)*a*(A*c-C*a)*(a*e^2+c *d^2)^(1/2)-c*(-B*d*e^(3/2)+e^(5/2)*A)*a^(3/2)+C*a^(5/2)*e^(5/2))*(2*(a*(a *e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2 +c*d^2))^(1/2)-2*a*e)^(1/2)*ln((a^(1/2)*(e*x^2+d)+x^2*(a*e^2+c*d^2)^(1/2)+ (e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x)/x^2)+c*d^2*(2*a ^(3/2)*(a*e^2+c*d^2)^(1/2)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))*(4*(a*e^2+c* d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)*C+(arctan((2*a^( 1/2)*(e*x^2+d)^(1/2)+(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x)/x/(4*(a*e^ 2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2))-arctan(((2* (a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x-2*a^(1/2)*(e*x^2+d)^(1/2))/x/(4*(a* e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)))*(-e^(1/2 )*a*(A*c-C*a)*(a*e^2+c*d^2)^(1/2)+c*(-B*d*e^(1/2)+A*e^(3/2))*a^(3/2)-C*a^( 5/2)*e^(3/2))))/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2...
Timed out. \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=\int \frac {A + B x^{2} + C x^{4}}{\left (a + c x^{4}\right ) \sqrt {d + e x^{2}}}\, dx \] Input:
integrate((C*x**4+B*x**2+A)/(e*x**2+d)**(1/2)/(c*x**4+a),x)
Output:
Integral((A + B*x**2 + C*x**4)/((a + c*x**4)*sqrt(d + e*x**2)), x)
\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (c x^{4} + a\right )} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)/((c*x^4 + a)*sqrt(e*x^2 + d)), x)
Exception generated. \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=\int \frac {C\,x^4+B\,x^2+A}{\left (c\,x^4+a\right )\,\sqrt {e\,x^2+d}} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a + c*x^4)*(d + e*x^2)^(1/2)),x)
Output:
int((A + B*x^2 + C*x^4)/((a + c*x^4)*(d + e*x^2)^(1/2)), x)
\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=\left (\int \frac {x^{4}}{\sqrt {e \,x^{2}+d}\, a +\sqrt {e \,x^{2}+d}\, c \,x^{4}}d x \right ) c +\left (\int \frac {x^{2}}{\sqrt {e \,x^{2}+d}\, a +\sqrt {e \,x^{2}+d}\, c \,x^{4}}d x \right ) b +\left (\int \frac {1}{\sqrt {e \,x^{2}+d}\, a +\sqrt {e \,x^{2}+d}\, c \,x^{4}}d x \right ) a \] Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+a),x)
Output:
int(x**4/(sqrt(d + e*x**2)*a + sqrt(d + e*x**2)*c*x**4),x)*c + int(x**2/(s qrt(d + e*x**2)*a + sqrt(d + e*x**2)*c*x**4),x)*b + int(1/(sqrt(d + e*x**2 )*a + sqrt(d + e*x**2)*c*x**4),x)*a