Integrand size = 34, antiderivative size = 226 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {a^{3/4} C \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} e \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (B e-C \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} e^2 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (C d^2-B d e+A e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d e^2 \sqrt {a-c x^4}} \] Output:
a^(3/4)*C*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(3/4)/e/(-c*x ^4+a)^(1/2)+a^(1/4)*(B*e-C*(d+a^(1/2)*e/c^(1/2)))*(1-c*x^4/a)^(1/2)*Ellipt icF(c^(1/4)*x/a^(1/4),I)/c^(1/4)/e^2/(-c*x^4+a)^(1/2)+a^(1/4)*(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^2/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.57 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {i \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} C d e E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-d \left (\sqrt {a} C e+\sqrt {c} (C d-B e)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {c} \left (C d^2+e (-B d+A e)\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {a} \left (-\frac {\sqrt {c}}{\sqrt {a}}\right )^{3/2} d e^2 \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)*Sqrt[a - c*x^4]),x]
Output:
(I*Sqrt[1 - (c*x^4)/a]*(Sqrt[a]*C*d*e*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/S qrt[a])]*x], -1] - d*(Sqrt[a]*C*e + Sqrt[c]*(C*d - B*e))*EllipticF[I*ArcSi nh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + Sqrt[c]*(C*d^2 + e*(-(B*d) + A*e))*E llipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x ], -1]))/(Sqrt[a]*(-(Sqrt[c]/Sqrt[a]))^(3/2)*d*e^2*Sqrt[a - c*x^4])
Time = 0.60 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2235, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}
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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\sqrt {a-c x^4} \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2235 |
| \(\displaystyle \frac {\left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {\int \frac {-C e x^2+C d-B e}{\sqrt {a-c x^4}}dx}{e^2}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {-\left (\left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right ) \int \frac {1}{\sqrt {a-c x^4}}dx\right )-\frac {\sqrt {a} C e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {-\left (\left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right ) \int \frac {1}{\sqrt {a-c x^4}}dx\right )-\frac {C e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{e^2}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {-\frac {\sqrt {1-\frac {c x^4}{a}} \left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}-\frac {C e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{e^2}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {-\frac {C e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{e^2}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {-\frac {C e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{e^2}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {-\frac {\sqrt {a} C e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {-\frac {a^{3/4} C e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{e^2}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \left (A e^2-B d e+C d^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{e^2 \sqrt {a-c x^4}}-\frac {-\frac {a^{3/4} C e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{e^2}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (A e^2-B d e+C d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d e^2 \sqrt {a-c x^4}}-\frac {-\frac {a^{3/4} C e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (B e-C \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right )\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{e^2}\) |
Input:
Int[(A + B*x^2 + C*x^4)/((d + e*x^2)*Sqrt[a - c*x^4]),x]
Output:
-((-((a^(3/4)*C*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4) ], -1])/(c^(3/4)*Sqrt[a - c*x^4])) - (a^(1/4)*(B*e - C*(d + (Sqrt[a]*e)/Sq rt[c]))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c ^(1/4)*Sqrt[a - c*x^4]))/e^2) + (a^(1/4)*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/ 4)], -1])/(c^(1/4)*d*e^2*Sqrt[a - c*x^4])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Si mp[-(e^2)^(-1) Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x], x] + Simp[( C*d^2 - B*d*e + A*e^2)/e^2 Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] / ; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0]
Time = 0.94 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.51
| method | result | size |
| default | \(\frac {\frac {B e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {C e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {C d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{e^{2}}+\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{e^{2} d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(342\) |
| elliptic | \(\frac {\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 ) B}{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 {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) C d}{e^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {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}\, \sqrt {c}}+\frac {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}\, \sqrt {c}}+\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 {\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 ) B}{e \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}}\) | \(578\) |
Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/e^2*(B*e/(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)-C*e*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))-C*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)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I))+(A*e^2-B*d*e+C*d^2)/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))
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="fricas" )
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )}\, dx \] Input:
integrate((C*x**4+B*x**2+A)/(e*x**2+d)/(-c*x**4+a)**(1/2),x)
Output:
Integral((A + B*x**2 + C*x**4)/(sqrt(a - c*x**4)*(d + e*x**2)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(-c*x^4 + a)*(e*x^2 + d)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(-c*x^4 + a)*(e*x^2 + d)), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{\sqrt {a-c\,x^4}\,\left (e\,x^2+d\right )} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(1/2)*(d + e*x^2)),x)
Output:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(1/2)*(d + e*x^2)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b \] Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(1/2),x)
Output:
int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a + int((sq rt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*c + int((sq rt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*b