Integrand size = 28, antiderivative size = 453 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {e \left (21 B c d^2+21 A c d e-5 a B e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {e^2 (3 B d+A e) x^3 \sqrt {a+c x^4}}{5 c}+\frac {B e^3 x^5 \sqrt {a+c x^4}}{7 c}+\frac {\left (5 B c d^3+15 A c d^2 e-9 a B d e^2-3 a A e^3\right ) x \sqrt {a+c x^4}}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} \left (5 B c d^3+15 A c d^2 e-9 a B d e^2-3 a A e^3\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {\left (105 A c^2 d^3+25 a^2 B e^3-105 a c d e (B d+A e)-63 a^{3/2} \sqrt {c} e^2 (3 B d+A e)+105 \sqrt {a} c^{3/2} d^2 (B d+3 A e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{210 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}} \] Output:
1/21*e*(21*A*c*d*e-5*B*a*e^2+21*B*c*d^2)*x*(c*x^4+a)^(1/2)/c^2+1/5*e^2*(A* e+3*B*d)*x^3*(c*x^4+a)^(1/2)/c+1/7*B*e^3*x^5*(c*x^4+a)^(1/2)/c+1/5*(-3*A*a *e^3+15*A*c*d^2*e-9*B*a*d*e^2+5*B*c*d^3)*x*(c*x^4+a)^(1/2)/c^(3/2)/(a^(1/2 )+c^(1/2)*x^2)-1/5*a^(1/4)*(-3*A*a*e^3+15*A*c*d^2*e-9*B*a*d*e^2+5*B*c*d^3) *(a^(1/2)+c^(1/2)*x^2)*((c*x^4+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^(1/2))/c^(7/4)/(c*x^4+a)^(1/2)+1/2 10*(105*A*c^2*d^3+25*a^2*B*e^3-105*a*c*d*e*(A*e+B*d)-63*a^(3/2)*c^(1/2)*e^ 2*(A*e+3*B*d)+105*a^(1/2)*c^(3/2)*d^2*(3*A*e+B*d))*(a^(1/2)+c^(1/2)*x^2)*( (c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)* x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/c^(9/4)/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.31 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.48 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {e x \left (a+c x^4\right ) \left (-25 a B e^2+21 A c e \left (5 d+e x^2\right )+3 B c \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )+5 \left (21 A c d \left (c d^2-a e^2\right )+a B e \left (-21 c d^2+5 a e^2\right )\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+7 c \left (5 B c d^3+15 A c d^2 e-9 a B d e^2-3 a A e^3\right ) x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )}{105 c^2 \sqrt {a+c x^4}} \] Input:
Integrate[((A + B*x^2)*(d + e*x^2)^3)/Sqrt[a + c*x^4],x]
Output:
(e*x*(a + c*x^4)*(-25*a*B*e^2 + 21*A*c*e*(5*d + e*x^2) + 3*B*c*(35*d^2 + 2 1*d*e*x^2 + 5*e^2*x^4)) + 5*(21*A*c*d*(c*d^2 - a*e^2) + a*B*e*(-21*c*d^2 + 5*a*e^2))*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4 )/a)] + 7*c*(5*B*c*d^3 + 15*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3)*x^3*Sqrt[ 1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)])/(105*c^2*Sq rt[a + c*x^4])
Time = 0.95 (sec) , antiderivative size = 878, normalized size of antiderivative = 1.94, 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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {d^2 x^2 (3 A e+B d)}{\sqrt {a+c x^4}}+\frac {e^2 x^6 (A e+3 B d)}{\sqrt {a+c x^4}}+\frac {3 d e x^4 (A e+B d)}{\sqrt {a+c x^4}}+\frac {A d^3}{\sqrt {a+c x^4}}+\frac {B e^3 x^8}{\sqrt {a+c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {B e^3 \sqrt {c x^4+a} x^5}{7 c}+\frac {e^2 (3 B d+A e) \sqrt {c x^4+a} x^3}{5 c}-\frac {5 a B e^3 \sqrt {c x^4+a} x}{21 c^2}+\frac {d e (B d+A e) \sqrt {c x^4+a} x}{c}-\frac {3 a e^2 (3 B d+A e) \sqrt {c x^4+a} x}{5 c^{3/2} \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {d^2 (B d+3 A e) \sqrt {c x^4+a} x}{\sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {3 a^{5/4} e^2 (3 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 )}{5 c^{7/4} \sqrt {c x^4+a}}-\frac {\sqrt [4]{a} d^2 (B d+3 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 )}{c^{3/4} \sqrt {c x^4+a}}+\frac {A d^3 \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 )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {c x^4+a}}+\frac {5 a^{7/4} B e^3 \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 )}{42 c^{9/4} \sqrt {c x^4+a}}-\frac {a^{3/4} d e (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 )}{2 c^{5/4} \sqrt {c x^4+a}}-\frac {3 a^{5/4} e^2 (3 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 )}{10 c^{7/4} \sqrt {c x^4+a}}+\frac {\sqrt [4]{a} d^2 (B d+3 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 )}{2 c^{3/4} \sqrt {c x^4+a}}\) |
Input:
Int[((A + B*x^2)*(d + e*x^2)^3)/Sqrt[a + c*x^4],x]
Output:
(-5*a*B*e^3*x*Sqrt[a + c*x^4])/(21*c^2) + (d*e*(B*d + A*e)*x*Sqrt[a + c*x^ 4])/c + (e^2*(3*B*d + A*e)*x^3*Sqrt[a + c*x^4])/(5*c) + (B*e^3*x^5*Sqrt[a + c*x^4])/(7*c) - (3*a*e^2*(3*B*d + A*e)*x*Sqrt[a + c*x^4])/(5*c^(3/2)*(Sq rt[a] + Sqrt[c]*x^2)) + (d^2*(B*d + 3*A*e)*x*Sqrt[a + c*x^4])/(Sqrt[c]*(Sq rt[a] + Sqrt[c]*x^2)) + (3*a^(5/4)*e^2*(3*B*d + A*e)*(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])/(5*c^(7/4)*Sqrt[a + c*x^4]) - (a^(1/4)*d^2*(B*d + 3*A* e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ell ipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(3/4)*Sqrt[a + c*x^4]) + (A *d^3*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*E llipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4]) + (5*a^(7/4)*B*e^3*(Sqrt[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])/(42*c^ (9/4)*Sqrt[a + c*x^4]) - (a^(3/4)*d*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*ArcTan[(c^(1/4)*x) /a^(1/4)], 1/2])/(2*c^(5/4)*Sqrt[a + c*x^4]) - (3*a^(5/4)*e^2*(3*B*d + A*e )*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elli pticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(10*c^(7/4)*Sqrt[a + c*x^4]) + (a^(1/4)*d^2*(B*d + 3*A*e)*(Sqrt[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*...
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 = 3.73 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.72
| method | result | size |
| elliptic | \(\frac {B \,e^{3} x^{5} \sqrt {c \,x^{4}+a}}{7 c}+\frac {\left (A \,e^{3}+3 B d \,e^{2}\right ) x^{3} \sqrt {c \,x^{4}+a}}{5 c}+\frac {\left (3 A d \,e^{2}+3 B e \,d^{2}-\frac {5 e^{3} B a}{7 c}\right ) x \sqrt {c \,x^{4}+a}}{3 c}+\frac {\left (A \,d^{3}-\frac {\left (3 A d \,e^{2}+3 B e \,d^{2}-\frac {5 e^{3} B a}{7 c}\right ) a}{3 c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (3 A \,d^{2} e +B \,d^{3}-\frac {3 \left (A \,e^{3}+3 B d \,e^{2}\right ) a}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(327\) |
| risch | \(\frac {e x \left (15 e^{2} B \,x^{4} c +21 A c \,e^{2} x^{2}+63 B c d e \,x^{2}+105 A c d e -25 B a \,e^{2}+105 B c \,d^{2}\right ) \sqrt {c \,x^{4}+a}}{105 c^{2}}-\frac {\frac {21 i \sqrt {c}\, \left (3 A a \,e^{3}-15 A c \,d^{2} e +9 B a d \,e^{2}-5 B c \,d^{3}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {105 A \,c^{2} d^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {25 a^{2} B \,e^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {105 A a c d \,e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {105 B a c \,d^{2} e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{105 c^{2}}\) | \(509\) |
| default | \(\frac {A \,d^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+e^{2} \left (A e +3 B d \right ) \left (\frac {x^{3} \sqrt {c \,x^{4}+a}}{5 c}-\frac {3 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+3 d e \left (A e +B d \right ) \left (\frac {x \sqrt {c \,x^{4}+a}}{3 c}-\frac {a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\frac {i d^{2} \left (3 A e +B d \right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+e^{3} B \left (\frac {x^{5} \sqrt {c \,x^{4}+a}}{7 c}-\frac {5 a x \sqrt {c \,x^{4}+a}}{21 c^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 c^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )\) | \(524\) |
Input:
int((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/7*B*e^3*x^5*(c*x^4+a)^(1/2)/c+1/5*(A*e^3+3*B*d*e^2)/c*x^3*(c*x^4+a)^(1/2 )+1/3*(3*A*d*e^2+3*B*e*d^2-5/7*e^3*B/c*a)/c*x*(c*x^4+a)^(1/2)+(A*d^3-1/3*( 3*A*d*e^2+3*B*e*d^2-5/7*e^3*B/c*a)/c*a)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^( 1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*El lipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+I*(3*A*d^2*e+B*d^3-3/5*(A*e^3+3*B*d *e^2)/c*a)*a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/ 2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(I *c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.62 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {21 \, {\left (5 \, B a c d^{3} + 15 \, A a c d^{2} e - 9 \, B a^{2} d e^{2} - 3 \, A a^{2} e^{3}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (105 \, {\left (3 \, A + B\right )} a c d^{2} e - {\left (63 \, A + 25 \, B\right )} a^{2} e^{3} + 105 \, {\left (B a c - A c^{2}\right )} d^{3} - 21 \, {\left (9 \, B a^{2} - 5 \, A a c\right )} d e^{2}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (15 \, B a c e^{3} x^{6} + 105 \, B a c d^{3} + 315 \, A a c d^{2} e - 189 \, B a^{2} d e^{2} - 63 \, A a^{2} e^{3} + 21 \, {\left (3 \, B a c d e^{2} + A a c e^{3}\right )} x^{4} + 5 \, {\left (21 \, B a c d^{2} e + 21 \, A a c d e^{2} - 5 \, B a^{2} e^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + a}}{105 \, a c^{2} x} \] Input:
integrate((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
1/105*(21*(5*B*a*c*d^3 + 15*A*a*c*d^2*e - 9*B*a^2*d*e^2 - 3*A*a^2*e^3)*sqr t(c)*x*(-a/c)^(3/4)*elliptic_e(arcsin((-a/c)^(1/4)/x), -1) - (105*(3*A + B )*a*c*d^2*e - (63*A + 25*B)*a^2*e^3 + 105*(B*a*c - A*c^2)*d^3 - 21*(9*B*a^ 2 - 5*A*a*c)*d*e^2)*sqrt(c)*x*(-a/c)^(3/4)*elliptic_f(arcsin((-a/c)^(1/4)/ x), -1) + (15*B*a*c*e^3*x^6 + 105*B*a*c*d^3 + 315*A*a*c*d^2*e - 189*B*a^2* d*e^2 - 63*A*a^2*e^3 + 21*(3*B*a*c*d*e^2 + A*a*c*e^3)*x^4 + 5*(21*B*a*c*d^ 2*e + 21*A*a*c*d*e^2 - 5*B*a^2*e^3)*x^2)*sqrt(c*x^4 + a))/(a*c^2*x)
Result contains complex when optimal does not.
Time = 4.49 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.80 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {A d^{3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {3 A d^{2} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 A d e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {A e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {B d^{3} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 B d^{2} e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {3 B d e^{2} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {B e^{3} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate((B*x**2+A)*(e*x**2+d)**3/(c*x**4+a)**(1/2),x)
Output:
A*d**3*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4 *sqrt(a)*gamma(5/4)) + 3*A*d**2*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,) , c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(7/4)) + 3*A*d*e**2*x**5*gamma (5/4)*hyper((1/2, 5/4), (9/4,), c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma (9/4)) + A*e**3*x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), c*x**4*exp_pola r(I*pi)/a)/(4*sqrt(a)*gamma(11/4)) + B*d**3*x**3*gamma(3/4)*hyper((1/2, 3/ 4), (7/4,), c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(7/4)) + 3*B*d**2*e* x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x**4*exp_polar(I*pi)/a)/(4*sqr t(a)*gamma(9/4)) + 3*B*d*e**2*x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), c *x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(11/4)) + B*e**3*x**9*gamma(9/4)* hyper((1/2, 9/4), (13/4,), c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(13/4 ))
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{\sqrt {c x^{4} + a}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^3/sqrt(c*x^4 + a), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{\sqrt {c x^{4} + a}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^3/sqrt(c*x^4 + a), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^3}{\sqrt {c\,x^4+a}} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2)^3)/(a + c*x^4)^(1/2),x)
Output:
int(((A + B*x^2)*(d + e*x^2)^3)/(a + c*x^4)^(1/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {-25 \sqrt {c \,x^{4}+a}\, a b \,e^{3} x +105 \sqrt {c \,x^{4}+a}\, a c d \,e^{2} x +21 \sqrt {c \,x^{4}+a}\, a c \,e^{3} x^{3}+105 \sqrt {c \,x^{4}+a}\, b c \,d^{2} e x +63 \sqrt {c \,x^{4}+a}\, b c d \,e^{2} x^{3}+15 \sqrt {c \,x^{4}+a}\, b c \,e^{3} x^{5}+25 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a^{2} b \,e^{3}-105 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a^{2} c d \,e^{2}-105 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a b c \,d^{2} e +105 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a \,c^{2} d^{3}-63 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) a^{2} c \,e^{3}-189 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) a b c d \,e^{2}+315 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) a \,c^{2} d^{2} e +105 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) b \,c^{2} d^{3}}{105 c^{2}} \] Input:
int((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(1/2),x)
Output:
( - 25*sqrt(a + c*x**4)*a*b*e**3*x + 105*sqrt(a + c*x**4)*a*c*d*e**2*x + 2 1*sqrt(a + c*x**4)*a*c*e**3*x**3 + 105*sqrt(a + c*x**4)*b*c*d**2*e*x + 63* sqrt(a + c*x**4)*b*c*d*e**2*x**3 + 15*sqrt(a + c*x**4)*b*c*e**3*x**5 + 25* int(sqrt(a + c*x**4)/(a + c*x**4),x)*a**2*b*e**3 - 105*int(sqrt(a + c*x**4 )/(a + c*x**4),x)*a**2*c*d*e**2 - 105*int(sqrt(a + c*x**4)/(a + c*x**4),x) *a*b*c*d**2*e + 105*int(sqrt(a + c*x**4)/(a + c*x**4),x)*a*c**2*d**3 - 63* int((sqrt(a + c*x**4)*x**2)/(a + c*x**4),x)*a**2*c*e**3 - 189*int((sqrt(a + c*x**4)*x**2)/(a + c*x**4),x)*a*b*c*d*e**2 + 315*int((sqrt(a + c*x**4)*x **2)/(a + c*x**4),x)*a*c**2*d**2*e + 105*int((sqrt(a + c*x**4)*x**2)/(a + c*x**4),x)*b*c**2*d**3)/(105*c**2)