Integrand size = 31, antiderivative size = 481 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {x \left (a B (b d-2 a e)-A \left (b^2 d-2 a c d-a b e\right )-(A c (b d-2 a e)-a B (2 c d-b e)) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {(A c (b d-2 a e)-a B (2 c d-b e)) x \sqrt {a+b x^2+c x^4}}{a \sqrt {c} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(A c (b d-2 a e)-a B (2 c d-b e)) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{3/4} \sqrt {a+b x^2+c x^4}} \] Output:
-x*(a*B*(-2*a*e+b*d)-A*(-a*b*e-2*a*c*d+b^2*d)-(A*c*(-2*a*e+b*d)-a*B*(-b*e+ 2*c*d))*x^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)-(A*c*(-2*a*e+b*d)-a*B*(- b*e+2*c*d))*x*(c*x^4+b*x^2+a)^(1/2)/a/c^(1/2)/(-4*a*c+b^2)/(a^(1/2)+c^(1/2 )*x^2)+(A*c*(-2*a*e+b*d)-a*B*(-b*e+2*c*d))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b *x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^ (1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/c^(3/4)/(-4*a*c+b^2)/(c*x ^4+b*x^2+a)^(1/2)+1/2*(a^(1/2)*B-A*c^(1/2))*(c^(1/2)*d-a^(1/2)*e)*(a^(1/2) +c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacob iAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/( b-2*a^(1/2)*c^(1/2))/c^(3/4)/(c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.77 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.24 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (a B \left (-2 a e+2 c d x^2+b \left (d-e x^2\right )\right )+A \left (-b^2 d+b \left (a e-c d x^2\right )+2 a c \left (d+e x^2\right )\right )\right )+i \left (-b+\sqrt {b^2-4 a c}\right ) (A c (b d-2 a e)+a B (-2 c d+b e)) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (A c \left (-b^2 d+4 a c d+b \sqrt {b^2-4 a c} d-2 a \sqrt {b^2-4 a c} e\right )+a B \left (b \left (-b+\sqrt {b^2-4 a c}\right ) e+c \left (-2 \sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{4 a c \left (-b^2+4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(a*B*(-2*a*e + 2*c*d*x^2 + b*(d - e *x^2)) + A*(-(b^2*d) + b*(a*e - c*d*x^2) + 2*a*c*(d + e*x^2))) + I*(-b + S qrt[b^2 - 4*a*c])*(A*c*(b*d - 2*a*e) + a*B*(-2*c*d + b*e))*Sqrt[(b + Sqrt[ b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqr t[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4 *a*c])] - I*(A*c*(-(b^2*d) + 4*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*Sqrt[b^ 2 - 4*a*c]*e) + a*B*(b*(-b + Sqrt[b^2 - 4*a*c])*e + c*(-2*Sqrt[b^2 - 4*a*c ]*d + 4*a*e)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a* c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*E llipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b ^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(4*a*c*(-b^2 + 4*a*c)*Sqrt[c/(b + S qrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])
Time = 0.68 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2206, 25, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {\int -\frac {a (b B d-2 A c d+A b e-2 a B e)-(A c (b d-2 a e)-a B (2 c d-b e)) x^2}{\sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}-\frac {x \left (-A \left (-a b e-2 a c d+b^2 d\right )-\left (x^2 (A c (b d-2 a e)-a B (2 c d-b e))\right )+a B (b d-2 a e)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a (b B d-2 A c d+A b e-2 a B e)-(A c (b d-2 a e)-a B (2 c d-b e)) x^2}{\sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}-\frac {x \left (-A \left (-a b e-2 a c d+b^2 d\right )-\left (x^2 (A c (b d-2 a e)-a B (2 c d-b e))\right )+a B (b d-2 a e)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}+\frac {\sqrt {a} (-2 a A c e+a b B e-2 a B c d+A b c d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{a \left (b^2-4 a c\right )}-\frac {x \left (-A \left (-a b e-2 a c d+b^2 d\right )-\left (x^2 (A c (b d-2 a e)-a B (2 c d-b e))\right )+a B (b d-2 a e)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}+\frac {(-2 a A c e+a b B e-2 a B c d+A b c d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{a \left (b^2-4 a c\right )}-\frac {x \left (-A \left (-a b e-2 a c d+b^2 d\right )-\left (x^2 (A c (b d-2 a e)-a B (2 c d-b e))\right )+a B (b d-2 a e)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {(-2 a A c e+a b B e-2 a B c d+A b c d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}}{a \left (b^2-4 a c\right )}-\frac {x \left (-A \left (-a b e-2 a c d+b^2 d\right )-\left (x^2 (A c (b d-2 a e)-a B (2 c d-b e))\right )+a B (b d-2 a e)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) (-2 a A c e+a b B e-2 a B c d+A b c d)}{\sqrt {c}}}{a \left (b^2-4 a c\right )}-\frac {x \left (-A \left (-a b e-2 a c d+b^2 d\right )-\left (x^2 (A c (b d-2 a e)-a B (2 c d-b e))\right )+a B (b d-2 a e)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
Input:
Int[((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]
Output:
-((x*(a*B*(b*d - 2*a*e) - A*(b^2*d - 2*a*c*d - a*b*e) - (A*c*(b*d - 2*a*e) - a*B*(2*c*d - b*e))*x^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4])) + ( ((A*b*c*d - 2*a*B*c*d + a*b*B*e - 2*a*A*c*e)*(-((x*Sqrt[a + b*x^2 + c*x^4] )/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b* x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^( 1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/S qrt[c] + (a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*(Sqrt[a]*B - A*Sqrt[c])*(Sqrt[c] *d - Sqrt[a]*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a] *Sqrt[c]))/4])/(2*c^(3/4)*Sqrt[a + b*x^2 + c*x^4]))/(a*(b^2 - 4*a*c))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Time = 1.44 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.23
| method | result | size |
| elliptic | \(-\frac {2 c \left (-\frac {\left (2 a A c e -A b c d -B a b e +2 a B c d \right ) x^{3}}{2 c a \left (4 a c -b^{2}\right )}-\frac {\left (A a b e +2 a A c d -A \,b^{2} d -2 B \,a^{2} e +B a b d \right ) x}{2 c a \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (\frac {B e}{c}+\frac {A c d -B a e}{a c}-\frac {A a b e +2 a A c d -A \,b^{2} d -2 B \,a^{2} e +B a b d}{a \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (2 a A c e -A b c d -B a b e +2 a B c d \right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(593\) |
| default | \(\text {Expression too large to display}\) | \(1390\) |
Input:
int((B*x^2+A)*(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*c*(-1/2/c*(2*A*a*c*e-A*b*c*d-B*a*b*e+2*B*a*c*d)/a/(4*a*c-b^2)*x^3-1/2/c *(A*a*b*e+2*A*a*c*d-A*b^2*d-2*B*a^2*e+B*a*b*d)/a/(4*a*c-b^2)*x)/((x^4+b/c* x^2+a/c)*c)^(1/2)+1/4*(B*e/c+(A*c*d-B*a*e)/a/c-(A*a*b*e+2*A*a*c*d-A*b^2*d- 2*B*a^2*e+B*a*b*d)/a/(4*a*c-b^2))*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2 )*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a* x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2) ^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/2*(2*A*a *c*e-A*b*c*d-B*a*b*e+2*B*a*c*d)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2 ))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2) ^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(Ellipti cF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+ b^2)^(1/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a )^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (409) = 818\).
Time = 0.09 (sec) , antiderivative size = 1031, normalized size of antiderivative = 2.14 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((B*x^2+A)*(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/2*(sqrt(1/2)*(((2*B*a*b - A*b^2)*c^2*d - (B*a*b^2*c - 2*A*a*b*c^2)*e)*x^ 4 + (2*B*a^2*b - A*a*b^2)*c*d + ((2*B*a*b^2 - A*b^3)*c*d - (B*a*b^3 - 2*A* a*b^2*c)*e)*x^2 - (B*a^2*b^2 - 2*A*a^2*b*c)*e - (((2*B*a^2 - A*a*b)*c^2*d - (B*a^2*b*c - 2*A*a^2*c^2)*e)*x^4 + (2*B*a^3 - A*a^2*b)*c*d + ((2*B*a^2*b - A*a*b^2)*c*d - (B*a^2*b^2 - 2*A*a^2*b*c)*e)*x^2 - (B*a^3*b - 2*A*a^3*c) *e)*sqrt((b^2 - 4*a*c)/a^2))*sqrt(a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/ a)*elliptic_e(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((b^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) - sqrt(1/2)*(((B*a *b^2*c - (2*(A - B)*a*b + A*b^2)*c^2)*d + (2*A*a*b*c^2 - (2*B*a^2*b - (A - B)*a*b^2)*c)*e)*x^4 + ((B*a*b^3 - (2*(A - B)*a*b^2 + A*b^3)*c)*d - (2*B*a ^2*b^2 - (A - B)*a*b^3 - 2*A*a*b^2*c)*e)*x^2 + (B*a^2*b^2 - (2*(A - B)*a^2 *b + A*a*b^2)*c)*d - (2*B*a^3*b - (A - B)*a^2*b^2 - 2*A*a^2*b*c)*e + (((B* a^2*b*c - (2*(A + B)*a^2 - A*a*b)*c^2)*d - (2*A*a^2*c^2 + (2*B*a^3 - (A + B)*a^2*b)*c)*e)*x^4 + ((B*a^2*b^2 - (2*(A + B)*a^2*b - A*a*b^2)*c)*d - (2* B*a^3*b - (A + B)*a^2*b^2 + 2*A*a^2*b*c)*e)*x^2 + (B*a^3*b - (2*(A + B)*a^ 3 - A*a^2*b)*c)*d - (2*B*a^4 - (A + B)*a^3*b + 2*A*a^3*c)*e)*sqrt((b^2 - 4 *a*c)/a^2))*sqrt(a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)*elliptic_f(arc sin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((b ^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) - 2*sqrt(c*x^4 + b*x^2 + a)*(((2*B* a^2 - A*a*b)*c^2*d - (B*a^2*b*c - 2*A*a^2*c^2)*e)*x^3 - ((2*B*a^3 - A*a...
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)/(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral((A + B*x**2)*(d + e*x**2)/(a + b*x**2 + c*x**4)**(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)/(c*x^4 + b*x^2 + a)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)/(c*x^4 + b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\left (e\,x^2+d\right )}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x)
Output:
int(((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-\sqrt {c \,x^{4}+b \,x^{2}+a}\, b e x +\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{2} b e +\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{2} c d +\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a \,b^{2} e \,x^{2}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b c d \,x^{2}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b c e \,x^{4}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a \,c^{2} d \,x^{4}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{2} c e +\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b c d +\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b c e \,x^{2}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a \,c^{2} e \,x^{4}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b^{2} c d \,x^{2}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b \,c^{2} d \,x^{4}}{c \left (c \,x^{4}+b \,x^{2}+a \right )} \] Input:
int((B*x^2+A)*(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x)
Output:
( - sqrt(a + b*x**2 + c*x**4)*b*e*x + int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a**2*b* e + int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x **4 + 2*b*c*x**6 + c**2*x**8),x)*a**2*c*d + int(sqrt(a + b*x**2 + c*x**4)/ (a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a *b**2*e*x**2 + int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x* *4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a*b*c*d*x**2 + int(sqrt(a + b* x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a*b*c*e*x**4 + int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x **2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a*c**2*d*x**4 + int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b** 2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a**2*c*e + int((sqrt(a + b*x**2 + c*x* *4)*x**2)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2* x**8),x)*a*b*c*d + int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a*b*c*e*x**2 + int( (sqrt(a + b*x**2 + c*x**4)*x**2)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x* *4 + 2*b*c*x**6 + c**2*x**8),x)*a*c**2*e*x**4 + int((sqrt(a + b*x**2 + c*x **4)*x**2)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2 *x**8),x)*b**2*c*d*x**2 + int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a**2 + 2*a *b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*b*c**2*d*...