Integrand size = 34, antiderivative size = 382 \[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=-\frac {(C d-B e) x \sqrt {a-c x^4}}{3 e^2}+\frac {C x^3 \sqrt {a-c x^4}}{5 e}+\frac {a^{3/4} \left (2 a C e^2-5 c \left (C d^2-e (B d-A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{3/4} e^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (3 \sqrt {a} e \left (2 a C e^2-5 c \left (C d^2-e (B d-A e)\right )\right )+5 \sqrt {c} \left (2 a e^2 (C d-B e)-3 c d \left (C d^2-e (B d-A e)\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15 c^{3/4} e^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (c d^2-a e^2\right ) \left (C d^2-B d e+A e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d e^4 \sqrt {a-c x^4}} \] Output:
-1/3*(-B*e+C*d)*x*(-c*x^4+a)^(1/2)/e^2+1/5*C*x^3*(-c*x^4+a)^(1/2)/e+1/5*a^ (3/4)*(2*C*a*e^2-5*c*(C*d^2-e*(-A*e+B*d)))*(1-c*x^4/a)^(1/2)*EllipticE(c^( 1/4)*x/a^(1/4),I)/c^(3/4)/e^3/(-c*x^4+a)^(1/2)-1/15*a^(1/4)*(3*a^(1/2)*e*( 2*C*a*e^2-5*c*(C*d^2-e*(-A*e+B*d)))+5*c^(1/2)*(2*a*e^2*(-B*e+C*d)-3*c*d*(C *d^2-e*(-A*e+B*d))))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(3 /4)/e^4/(-c*x^4+a)^(1/2)-a^(1/4)*(-a*e^2+c*d^2)*(A*e^2-B*d*e+C*d^2)*(1-c*x ^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/c^(1/4)/d /e^4/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.91 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=\frac {-3 i \sqrt {a} d e \left (2 a C e^2-5 c \left (C d^2+e (-B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i d \left (6 a^{3/2} C e^3+10 a \sqrt {c} e^2 (C d-B e)-15 \sqrt {a} c e \left (C d^2+e (-B d+A e)\right )-15 c^{3/2} \left (C d^3+d e (-B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {c} \left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d e^2 x \left (-5 C d+5 B e+3 C e x^2\right ) \left (a-c x^4\right )-15 i \left (-c d^2+a e^2\right ) \left (C d^2+e (-B d+A e)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{15 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \sqrt {c} d e^4 \sqrt {a-c x^4}} \] Input:
Integrate[(Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4))/(d + e*x^2),x]
Output:
((-3*I)*Sqrt[a]*d*e*(2*a*C*e^2 - 5*c*(C*d^2 + e*(-(B*d) + A*e)))*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*d*(6*a ^(3/2)*C*e^3 + 10*a*Sqrt[c]*e^2*(C*d - B*e) - 15*Sqrt[a]*c*e*(C*d^2 + e*(- (B*d) + A*e)) - 15*c^(3/2)*(C*d^3 + d*e*(-(B*d) + A*e)))*Sqrt[1 - (c*x^4)/ a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + Sqrt[c]*(Sqrt[-( Sqrt[c]/Sqrt[a])]*d*e^2*x*(-5*C*d + 5*B*e + 3*C*e*x^2)*(a - c*x^4) - (15*I )*(-(c*d^2) + a*e^2)*(C*d^2 + e*(-(B*d) + A*e))*Sqrt[1 - (c*x^4)/a]*Ellipt icPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1 ]))/(15*Sqrt[-(Sqrt[c]/Sqrt[a])]*Sqrt[c]*d*e^4*Sqrt[a - c*x^4])
Time = 0.88 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.57, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2259, 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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {-a e^2 (C d-B e)-c d e (B d-A e)+c C d^3}{e^4 \sqrt {a-c x^4}}-\frac {x^2 \left (-a C e^2-c e (B d-A e)+c C d^2\right )}{e^3 \sqrt {a-c x^4}}+\frac {a A e^4-a B d e^3+a C d^2 e^2-A c d^2 e^2+B c d^3 e-c C d^4}{e^4 \sqrt {a-c x^4} \left (d+e x^2\right )}+\frac {c x^4 (C d-B e)}{e^2 \sqrt {a-c x^4}}-\frac {c C x^6}{e \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (-a C e^2-c e (B d-A e)+c C d^2\right )}{c^{3/4} e^3 \sqrt {a-c x^4}}-\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (-a C e^2-c e (B d-A e)+c C d^2\right )}{c^{3/4} e^3 \sqrt {a-c x^4}}+\frac {a^{5/4} \sqrt {1-\frac {c x^4}{a}} (C d-B e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt [4]{c} e^2 \sqrt {a-c x^4}}+\frac {3 a^{7/4} C \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{3/4} e \sqrt {a-c x^4}}-\frac {3 a^{7/4} C \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{3/4} e \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (-a e^2 (C d-B e)-c d e (B d-A e)+c C d^3\right )}{\sqrt [4]{c} e^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (c d^2-a e^2\right ) \left (A e^2-B d e+C d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d e^4 \sqrt {a-c x^4}}-\frac {x \sqrt {a-c x^4} (C d-B e)}{3 e^2}+\frac {C x^3 \sqrt {a-c x^4}}{5 e}\) |
Input:
Int[(Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4))/(d + e*x^2),x]
Output:
-1/3*((C*d - B*e)*x*Sqrt[a - c*x^4])/e^2 + (C*x^3*Sqrt[a - c*x^4])/(5*e) - (3*a^(7/4)*C*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], - 1])/(5*c^(3/4)*e*Sqrt[a - c*x^4]) - (a^(3/4)*(c*C*d^2 - a*C*e^2 - c*e*(B*d - A*e))*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/( c^(3/4)*e^3*Sqrt[a - c*x^4]) + (3*a^(7/4)*C*Sqrt[1 - (c*x^4)/a]*EllipticF[ ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*e*Sqrt[a - c*x^4]) + (a^(5/4) *(C*d - B*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1 ])/(3*c^(1/4)*e^2*Sqrt[a - c*x^4]) + (a^(3/4)*(c*C*d^2 - a*C*e^2 - c*e*(B* d - A*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/ (c^(3/4)*e^3*Sqrt[a - c*x^4]) + (a^(1/4)*(c*C*d^3 - c*d*e*(B*d - A*e) - a* e^2*(C*d - B*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)] , -1])/(c^(1/4)*e^4*Sqrt[a - c*x^4]) - (a^(1/4)*(c*d^2 - a*e^2)*(C*d^2 - B *d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), A rcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*e^4*Sqrt[a - c*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 2] && IntegerQ[q]
Time = 2.86 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(\frac {x \left (3 C \,x^{2} e +5 B e -5 C d \right ) \sqrt {-c \,x^{4}+a}}{15 e^{2}}-\frac {-\frac {15 \left (A a \,e^{4}-A c \,d^{2} e^{2}-B a d \,e^{3}+B c \,d^{3} e +C a \,d^{2} e^{2}-C c \,d^{4}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{e^{2} d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {5 \left (3 A c d \,e^{2}+2 B a \,e^{3}-3 B c \,d^{2} e -2 a C d \,e^{2}+3 C c \,d^{3}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \left (5 A c \,e^{2}-5 B c d e -2 C a \,e^{2}+5 C c \,d^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}}{15 e^{2}}\) | \(412\) |
| default | \(\frac {B e \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+C e \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5}-\frac {2 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )-C d \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{e^{2}}+\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (\frac {c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c}{e^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{e^{2}}\) | \(708\) |
| elliptic | \(\text {Expression too large to display}\) | \(1592\) |
Input:
int((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d),x,method=_RETURNVERBOSE)
Output:
1/15*x*(3*C*e*x^2+5*B*e-5*C*d)*(-c*x^4+a)^(1/2)/e^2-1/15/e^2*(-15*(A*a*e^4 -A*c*d^2*e^2-B*a*d*e^3+B*c*d^3*e+C*a*d^2*e^2-C*c*d^4)/e^2/d/(c^(1/2)/a^(1/ 2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c* x^4+a)^(1/2)*EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*e/c^(1/2)/d,(-c ^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/2))-5*(3*A*c*d*e^2+2*B*a*e^3-3* B*c*d^2*e-2*C*a*d*e^2+3*C*c*d^3)/e^2/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^ 2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF( x*(c^(1/2)/a^(1/2))^(1/2),I)-3/e*(5*A*c*e^2-5*B*c*d*e-2*C*a*e^2+5*C*c*d^2) *a^(1/2)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)* x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2)) ^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)))
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a}}{e x^{2} + d} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d),x, algorithm="fricas" )
Output:
integral((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)/(e*x^2 + d), x)
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=\int \frac {\sqrt {a - c x^{4}} \left (A + B x^{2} + C x^{4}\right )}{d + e x^{2}}\, dx \] Input:
integrate((-c*x**4+a)**(1/2)*(C*x**4+B*x**2+A)/(e*x**2+d),x)
Output:
Integral(sqrt(a - c*x**4)*(A + B*x**2 + C*x**4)/(d + e*x**2), x)
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a}}{e x^{2} + d} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)/(e*x^2 + d), x)
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a}}{e x^{2} + d} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)/(e*x^2 + d), x)
Timed out. \[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=\int \frac {\sqrt {a-c\,x^4}\,\left (C\,x^4+B\,x^2+A\right )}{e\,x^2+d} \,d x \] Input:
int(((a - c*x^4)^(1/2)*(A + B*x^2 + C*x^4))/(d + e*x^2),x)
Output:
int(((a - c*x^4)^(1/2)*(A + B*x^2 + C*x^4))/(d + e*x^2), x)
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=\frac {5 \sqrt {-c \,x^{4}+a}\, b e x -5 \sqrt {-c \,x^{4}+a}\, c d x +3 \sqrt {-c \,x^{4}+a}\, c e \,x^{3}+15 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} e^{2}-5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b d e +5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,d^{2}-9 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,e^{2}+15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c d e -15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c^{2} d^{2}+10 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b \,e^{2}-4 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c d e}{15 e^{2}} \] Input:
int((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d),x)
Output:
(5*sqrt(a - c*x**4)*b*e*x - 5*sqrt(a - c*x**4)*c*d*x + 3*sqrt(a - c*x**4)* c*e*x**3 + 15*int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6), x)*a**2*e**2 - 5*int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x** 6),x)*a*b*d*e + 5*int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x* *6),x)*a*c*d**2 - 9*int((sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*e**2 + 15*int((sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*b*c*d*e - 15*int((sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*c**2*d**2 + 10*int((sqrt(a - c*x**4)*x* *2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*e**2 - 4*int((sqrt(a - c *x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*d*e)/(15*e**2)