Integrand size = 28, antiderivative size = 1494 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \]
-1/4*e^(3/2)*(-A*e+B*d)*(a*e^2+3*c*d^2)*arctan(x*(a*e^2+c*d^2)^(1/2)/d^(1/ 2)/e^(1/2)/(c*x^4+a)^(1/2))/d^(3/2)/(a*e^2+c*d^2)^(5/2)-1/2*e^(3/2)*(-2*A* c*d*e-B*a*e^2+B*c*d^2)*arctan(x*(a*e^2+c*d^2)^(1/2)/d^(1/2)/e^(1/2)/(c*x^4 +a)^(1/2))/(a*e^2+c*d^2)^(5/2)/d^(1/2)+1/2*c*x*(A*c*d^2+2*a*B*d*e-a*A*e^2+ (-2*A*c*d*e-B*a*e^2+B*c*d^2)*x^2)/a/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)-1/2*e^ 3*(-A*e+B*d)*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)^2/(e*x^2+d)+1/2*e^2*(-A*e+B *d)*x*c^(1/2)*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)^2/(a^(1/2)+x^2*c^(1/2))-1/2* (-2*A*c*d*e-B*a*e^2+B*c*d^2)*x*c^(1/2)*(c*x^4+a)^(1/2)/a/(a*e^2+c*d^2)^2/( a^(1/2)+x^2*c^(1/2))-1/2*a^(1/4)*c^(1/4)*e^2*(-A*e+B*d)*(cos(2*arctan(c^(1 /4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2* arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/( a^(1/2)+x^2*c^(1/2))^2)^(1/2)/d/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)+1/2*c^(1/4 )*(-2*A*c*d*e-B*a*e^2+B*c*d^2)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/ cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4)) ),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^( 1/2)/a^(3/4)/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)-1/2*c^(1/4)*e*(-A*e+B*d)*(cos (2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*El lipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2) )*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/d/(a*e^2+c*d^2)/(-e*a^ (1/2)+d*c^(1/2))/(c*x^4+a)^(1/2)-1/2*c^(1/4)*e*(-2*A*c*d*e-B*a*e^2+B*c*...
Result contains complex when optimal does not.
Time = 11.14 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.29 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d \left (a e^3 (-B d+A e) x \left (a+c x^4\right )+c d x \left (d+e x^2\right ) \left (-a A e^2+B c d^2 x^2+A c d \left (d-2 e x^2\right )+a B e \left (2 d-e x^2\right )\right )\right )-\left (d+e x^2\right ) \sqrt {1+\frac {c x^4}{a}} \left (-\sqrt {a} \sqrt {c} d \left (-B c d^3+2 A c d^2 e+2 a B d e^2-a A e^3\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) \left (A c d^2+i \sqrt {a} \sqrt {c} d (B d-A e)+a e (2 B d-A e)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+a e \left (-5 B c d^3+7 A c d^2 e+a B d e^2+a A e^3\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (c d^3+a d e^2\right )^2 \left (d+e x^2\right ) \sqrt {a+c x^4}} \]
(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*(a*e^3*(-(B*d) + A*e)*x*(a + c*x^4) + c*d*x*( d + e*x^2)*(-(a*A*e^2) + B*c*d^2*x^2 + A*c*d*(d - 2*e*x^2) + a*B*e*(2*d - e*x^2))) - (d + e*x^2)*Sqrt[1 + (c*x^4)/a]*(-(Sqrt[a]*Sqrt[c]*d*(-(B*c*d^3 ) + 2*A*c*d^2*e + 2*a*B*d*e^2 - a*A*e^3)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[ c])/Sqrt[a]]*x], -1]) + I*(Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(A*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(2*B*d - A*e))*EllipticF[I*ArcSinh[S qrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + a*e*(-5*B*c*d^3 + 7*A*c*d^2*e + a*B*d*e ^2 + a*A*e^3)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*S qrt[c])/Sqrt[a]]*x], -1])))/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^3 + a*d*e^ 2)^2*(d + e*x^2)*Sqrt[a + c*x^4])
Time = 2.22 (sec) , antiderivative size = 1494, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int \frac {A+B x^2}{\left (a+c x^4\right )^{3/2} \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {e (A e-B d)}{\sqrt {a+c x^4} \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}+\frac {e \left (a B e^2+2 A c d e-B c d^2\right )}{\sqrt {a+c x^4} \left (d+e x^2\right ) \left (a e^2+c d^2\right )^2}+\frac {c \left (x^2 \left (-a B e^2-2 A c d e+B c d^2\right )-a A e^2+2 a B d e+A c d^2\right )}{\left (a+c x^4\right )^{3/2} \left (a e^2+c d^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(B d-A e) x \sqrt {c x^4+a} e^3}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e^2}{2 d \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt {c} (B d-A e) x \sqrt {c x^4+a} e^2}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {(B d-A e) \left (3 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac {\left (B c d^2-2 A c e d-a B e^2\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {c x^4+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2+a e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {a^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2-\frac {\sqrt {c} \left (A c d^2+2 a B e d-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt {c} \left (B c d^2-2 A c e d-a B e^2\right ) x \sqrt {c x^4+a}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {c x \left (A c d^2+2 a B e d-a A e^2+\left (B c d^2-2 A c e d-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}\) |
(c*x*(A*c*d^2 + 2*a*B*d*e - a*A*e^2 + (B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x^2) )/(2*a*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (Sqrt[c]*e^2*(B*d - A*e)*x*Sqr t[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (Sqrt[c]*( B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)^2*( Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*c*d^2 + a*e^2)*ArcTan[(S qrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(4*d^(3/2)*(c*d^ 2 + a*e^2)^(5/2)) - (e^(3/2)*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*ArcTan[(Sqrt[ c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*d^2 + a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*S qrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/ a^(1/4)], 1/2])/(2*d*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (c^(1/4)*(B*c*d^ 2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/ 4)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTa n[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sq rt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF [2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e...
3.1.13.3.1 Defintions of rubi rules used
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]
Result contains complex when optimal does not.
Time = 1.50 (sec) , antiderivative size = 1384, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1384\) |
| elliptic | \(\text {Expression too large to display}\) | \(1664\) |
B/e*(-2*c*(1/4/a*e/(a*e^2+c*d^2)*x^3-1/4*d/a/(a*e^2+c*d^2)*x)/((x^4+a/c)*c )^(1/2)+1/2*c*d/a/(a*e^2+c*d^2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^( 1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF( x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2*I/a^(1/2)*c^(1/2)*e/(a*e^2+c*d^2)/(I/a^ (1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)* x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-1/2*I/ a^(1/2)*c^(1/2)*e/(a*e^2+c*d^2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^( 1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticE( x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/(a*e^2+c*d^2)*e^2/d/(I/a^(1/2)*c^(1/2))^( 1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^ 4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(- I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)))+(A*e-B*d)/e*(-2*c*(1/ 2/a*d*c*e/(a*e^2+c*d^2)^2*x^3+1/4/a*(a*e^2-c*d^2)/(a*e^2+c*d^2)^2*x)/((x^4 +a/c)*c)^(1/2)+1/2*e^4/(a*e^2+c*d^2)^2/d*x*(c*x^4+a)^(1/2)/(e*x^2+d)-1/(I/ a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2 )*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)*e^2* c/(a*e^2+c*d^2)^2+1/2/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^ (1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/ 2)*c^(1/2))^(1/2),I)*c^2/a/(a*e^2+c*d^2)^2*d^2+I/a^(1/2)/(I/a^(1/2)*c^(1/2 ))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2...
Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {A + B x^{2}}{\left (a + c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{2}}\, dx \]
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (c\,x^4+a\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \]