Integrand size = 28, antiderivative size = 471 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {x \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2\right )}{2 a c^2 \sqrt {a+c x^4}}+\frac {B e^3 x \sqrt {a+c x^4}}{3 c^2}-\frac {\left (B c d^3+3 A c d^2 e-9 a B d e^2-3 a A e^3\right ) x \sqrt {a+c x^4}}{2 a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\left (B c d^3+3 A c d^2 e-9 a B d e^2-3 a A e^3\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt {a+c x^4}}+\frac {\left (3 A c^2 d^3-5 a^2 B e^3+9 a c d e (B d+A e)+9 a^{3/2} \sqrt {c} e^2 (3 B d+A e)-3 \sqrt {a} c^{3/2} d^2 (B d+3 A e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 a^{5/4} c^{9/4} \sqrt {a+c x^4}} \] Output:
1/2*x*(A*c*d*(-3*a*e^2+c*d^2)-a*B*e*(-a*e^2+3*c*d^2)+c*(-A*a*e^3+3*A*c*d^2 *e-3*B*a*d*e^2+B*c*d^3)*x^2)/a/c^2/(c*x^4+a)^(1/2)+1/3*B*e^3*x*(c*x^4+a)^( 1/2)/c^2-1/2*(-3*A*a*e^3+3*A*c*d^2*e-9*B*a*d*e^2+B*c*d^3)*x*(c*x^4+a)^(1/2 )/a/c^(3/2)/(a^(1/2)+c^(1/2)*x^2)+1/2*(-3*A*a*e^3+3*A*c*d^2*e-9*B*a*d*e^2+ 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)*E llipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/c^(7/4)/(c* x^4+a)^(1/2)+1/12*(3*A*c^2*d^3-5*a^2*B*e^3+9*a*c*d*e*(A*e+B*d)+9*a^(3/2)*c ^(1/2)*e^2*(A*e+3*B*d)-3*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^(5/4)/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 = 10.30 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.47 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {3 A c x \left (c d^3+a e^2 \left (-3 d+2 e x^2\right )\right )+a B e x \left (5 a e^2+c \left (-9 d^2+18 d e x^2+2 e^2 x^4\right )\right )+\left (a B e \left (9 c d^2-5 a e^2\right )+3 A c d \left (c d^2+3 a e^2\right )\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+2 c \left (B c d^3+3 A c d^2 e-9 a B d e^2-3 a A e^3\right ) x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {c x^4}{a}\right )}{6 a c^2 \sqrt {a+c x^4}} \] Input:
Integrate[((A + B*x^2)*(d + e*x^2)^3)/(a + c*x^4)^(3/2),x]
Output:
(3*A*c*x*(c*d^3 + a*e^2*(-3*d + 2*e*x^2)) + a*B*e*x*(5*a*e^2 + c*(-9*d^2 + 18*d*e*x^2 + 2*e^2*x^4)) + (a*B*e*(9*c*d^2 - 5*a*e^2) + 3*A*c*d*(c*d^2 + 3*a*e^2))*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4) /a)] + 2*c*(B*c*d^3 + 3*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[3/4, 3/2, 7/4, -((c*x^4)/a)])/(6*a*c^2*Sqrt[a + c*x^4])
Time = 1.04 (sec) , antiderivative size = 912, 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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {e \left (-a B e^2+3 A c d e+3 B c d^2\right )}{c^2 \sqrt {a+c x^4}}+\frac {c x^2 \left (-a A e^3-3 a B d e^2+3 A c d^2 e+B c d^3\right )+A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )}{c^2 \left (a+c x^4\right )^{3/2}}+\frac {e^2 x^2 (A e+3 B d)}{c \sqrt {a+c x^4}}+\frac {B e^3 x^4}{c \sqrt {a+c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{3/4} B \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^3}{6 c^{9/4} \sqrt {c x^4+a}}+\frac {B x \sqrt {c x^4+a} e^3}{3 c^2}-\frac {\sqrt [4]{a} (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 ) e^2}{c^{7/4} \sqrt {c x^4+a}}+\frac {\sqrt [4]{a} (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 ) e^2}{2 c^{7/4} \sqrt {c x^4+a}}+\frac {(3 B d+A e) x \sqrt {c x^4+a} e^2}{c^{3/2} \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {\left (3 B c d^2+3 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} c^{9/4} \sqrt {c x^4+a}}+\frac {\left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\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} c^{7/4} \sqrt {c x^4+a}}+\frac {\left (A c^2 d^3-\sqrt {a} c^{3/2} (B d+3 A e) d^2-3 a c e (B d+A e) d+a^2 B e^3+a^{3/2} \sqrt {c} e^2 (3 B d+A e)\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^{5/4} c^{9/4} \sqrt {c x^4+a}}-\frac {\left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right ) x \sqrt {c x^4+a}}{2 a c^{3/2} \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {x \left (c \left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right ) x^2+A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )\right )}{2 a c^2 \sqrt {c x^4+a}}\) |
Input:
Int[((A + B*x^2)*(d + e*x^2)^3)/(a + c*x^4)^(3/2),x]
Output:
(x*(A*c*d*(c*d^2 - 3*a*e^2) - a*B*e*(3*c*d^2 - a*e^2) + c*(B*c*d^3 + 3*A*c *d^2*e - 3*a*B*d*e^2 - a*A*e^3)*x^2))/(2*a*c^2*Sqrt[a + c*x^4]) + (B*e^3*x *Sqrt[a + c*x^4])/(3*c^2) + (e^2*(3*B*d + A*e)*x*Sqrt[a + c*x^4])/(c^(3/2) *(Sqrt[a] + Sqrt[c]*x^2)) - ((B*c*d^3 + 3*A*c*d^2*e - 3*a*B*d*e^2 - a*A*e^ 3)*x*Sqrt[a + c*x^4])/(2*a*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*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])/(c^(7/4)*Sqrt[a + c*x^4]) + ((B*c*d^3 + 3*A*c*d^2*e - 3*a*B*d*e^2 - a*A*e^3)*(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^(7/4)*Sqrt[a + c*x^4]) - (a^(3/4)*B *e^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])/(6*c^(9/4)*Sqrt[a + c*x^4]) + (a^(1/4)*e^2*(3*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqr t[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c ^(7/4)*Sqrt[a + c*x^4]) + (e*(3*B*c*d^2 + 3*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]*EllipticF[2*ArcTa n[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(9/4)*Sqrt[a + c*x^4]) + ((A*c^ 2*d^3 + a^2*B*e^3 - 3*a*c*d*e*(B*d + A*e) + a^(3/2)*Sqrt[c]*e^2*(3*B*d + A *e) - Sqrt[a]*c^(3/2)*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/...
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 = 6.13 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.90
| method | result | size |
| elliptic | \(-\frac {2 c \left (\frac {\left (A a \,e^{3}-3 A c \,d^{2} e +3 B a d \,e^{2}-B c \,d^{3}\right ) x^{3}}{4 c^{2} a}+\frac {\left (3 A a c d \,e^{2}-A \,c^{2} d^{3}-a^{2} B \,e^{3}+3 B a c \,d^{2} e \right ) x}{4 a \,c^{3}}\right )}{\sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {B \,e^{3} x \sqrt {c \,x^{4}+a}}{3 c^{2}}+\frac {\left (\frac {e \left (3 A c d e -B a \,e^{2}+3 B c \,d^{2}\right )}{c^{2}}-\frac {3 A a c d \,e^{2}-A \,c^{2} d^{3}-a^{2} B \,e^{3}+3 B a c \,d^{2} e}{2 c^{2} a}-\frac {B \,e^{3} a}{3 c^{2}}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (\frac {e^{2} \left (A e +3 B d \right )}{c}+\frac {A a \,e^{3}-3 A c \,d^{2} e +3 B a d \,e^{2}-B c \,d^{3}}{2 c a}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(426\) |
| default | \(A \,d^{3} \left (\frac {x}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\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 )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+e^{2} \left (A e +3 B d \right ) \left (-\frac {x^{3}}{2 c \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {3 i \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 )}{2 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}{2 c \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\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 )}{2 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+d^{2} \left (3 A e +B d \right ) \left (\frac {x^{3}}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}-\frac {i \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 )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+e^{3} B \left (\frac {a x}{2 c^{2} \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {x \sqrt {c \,x^{4}+a}}{3 c^{2}}-\frac {5 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 )}{6 c^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )\) | \(579\) |
| risch | \(\frac {B \,e^{3} x \sqrt {c \,x^{4}+a}}{3 c^{2}}+\frac {3 c^{2} e^{2} \left (A e +3 B d \right ) \left (-\frac {x^{3}}{2 c \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {3 i \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 )}{2 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+3 c^{2} d^{2} \left (3 A e +B d \right ) \left (\frac {x^{3}}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}-\frac {i \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 )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+c e \left (9 A c d e -4 B a \,e^{2}+9 B c \,d^{2}\right ) \left (-\frac {x}{2 c \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\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 )}{2 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+3 A \,c^{2} d^{3} \left (\frac {x}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\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 )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )-a^{2} B \,e^{3} \left (\frac {x}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\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 )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{3 c^{2}}\) | \(616\) |
Input:
int((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*c*(1/4/c^2*(A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2-B*c*d^3)/a*x^3+1/4/a/c^3*(3 *A*a*c*d*e^2-A*c^2*d^3-B*a^2*e^3+3*B*a*c*d^2*e)*x)/(c*(a/c+x^4))^(1/2)+1/3 *B*e^3*x*(c*x^4+a)^(1/2)/c^2+(e*(3*A*c*d*e-B*a*e^2+3*B*c*d^2)/c^2-1/2/c^2/ a*(3*A*a*c*d*e^2-A*c^2*d^3-B*a^2*e^3+3*B*a*c*d^2*e)-1/3*B*e^3/c^2*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)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+I*(1/c *e^2*(A*e+3*B*d)+1/2/c*(A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2-B*c*d^3)/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 = 448, normalized size of antiderivative = 0.95 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=-\frac {3 \, {\left ({\left (B a c^{2} d^{3} + 3 \, A a c^{2} d^{2} e - 9 \, B a^{2} c d e^{2} - 3 \, A a^{2} c e^{3}\right )} x^{5} + {\left (B a^{2} c d^{3} + 3 \, A a^{2} c d^{2} e - 9 \, B a^{3} d e^{2} - 3 \, A a^{3} e^{3}\right )} x\right )} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (9 \, {\left (A + B\right )} a c^{2} d^{2} e - {\left (9 \, A + 5 \, B\right )} a^{2} c e^{3} + 3 \, {\left (B a c^{2} + A c^{3}\right )} d^{3} - 9 \, {\left (3 \, B a^{2} c - A a c^{2}\right )} d e^{2}\right )} x^{5} + {\left (9 \, {\left (A + B\right )} a^{2} c d^{2} e - {\left (9 \, A + 5 \, B\right )} a^{3} e^{3} + 3 \, {\left (B a^{2} c + A a c^{2}\right )} d^{3} - 9 \, {\left (3 \, B a^{3} - A a^{2} c\right )} d e^{2}\right )} x\right )} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (2 \, B a^{2} c e^{3} x^{6} - 3 \, B a^{2} c d^{3} - 9 \, A a^{2} c d^{2} e + 27 \, B a^{3} d e^{2} + 9 \, A a^{3} e^{3} + 6 \, {\left (3 \, B a^{2} c d e^{2} + A a^{2} c e^{3}\right )} x^{4} + {\left (3 \, A a c^{2} d^{3} - 9 \, B a^{2} c d^{2} e - 9 \, A a^{2} c d e^{2} + 5 \, B a^{3} e^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + a}}{6 \, {\left (a^{2} c^{3} x^{5} + a^{3} c^{2} x\right )}} \] Input:
integrate((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/6*(3*((B*a*c^2*d^3 + 3*A*a*c^2*d^2*e - 9*B*a^2*c*d*e^2 - 3*A*a^2*c*e^3) *x^5 + (B*a^2*c*d^3 + 3*A*a^2*c*d^2*e - 9*B*a^3*d*e^2 - 3*A*a^3*e^3)*x)*sq rt(c)*(-a/c)^(3/4)*elliptic_e(arcsin((-a/c)^(1/4)/x), -1) - ((9*(A + B)*a* c^2*d^2*e - (9*A + 5*B)*a^2*c*e^3 + 3*(B*a*c^2 + A*c^3)*d^3 - 9*(3*B*a^2*c - A*a*c^2)*d*e^2)*x^5 + (9*(A + B)*a^2*c*d^2*e - (9*A + 5*B)*a^3*e^3 + 3* (B*a^2*c + A*a*c^2)*d^3 - 9*(3*B*a^3 - A*a^2*c)*d*e^2)*x)*sqrt(c)*(-a/c)^( 3/4)*elliptic_f(arcsin((-a/c)^(1/4)/x), -1) - (2*B*a^2*c*e^3*x^6 - 3*B*a^2 *c*d^3 - 9*A*a^2*c*d^2*e + 27*B*a^3*d*e^2 + 9*A*a^3*e^3 + 6*(3*B*a^2*c*d*e ^2 + A*a^2*c*e^3)*x^4 + (3*A*a*c^2*d^3 - 9*B*a^2*c*d^2*e - 9*A*a^2*c*d*e^2 + 5*B*a^3*e^3)*x^2)*sqrt(c*x^4 + a))/(a^2*c^3*x^5 + a^3*c^2*x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{3}}{\left (a + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**3/(c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)*(d + e*x**2)**3/(a + c*x**4)**(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^3/(c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^3/(c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^3}{{\left (c\,x^4+a\right )}^{3/2}} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2)^3)/(a + c*x^4)^(3/2),x)
Output:
int(((A + B*x^2)*(d + e*x^2)^3)/(a + c*x^4)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((B*x^2+A)*(e*x^2+d)^3/(c*x^4+a)^(3/2),x)
Output:
(5*sqrt(a + c*x**4)*a*b*e**3*x - 9*sqrt(a + c*x**4)*a*c*d*e**2*x + 3*sqrt( a + c*x**4)*a*c*e**3*x**3 - 9*sqrt(a + c*x**4)*b*c*d**2*e*x + 9*sqrt(a + c *x**4)*b*c*d*e**2*x**3 + sqrt(a + c*x**4)*b*c*e**3*x**5 - 5*int(sqrt(a + c *x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**3*b*e**3 + 9*int(sqrt(a + c*x **4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**3*c*d*e**2 + 9*int(sqrt(a + c*x **4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*b*c*d**2*e - 5*int(sqrt(a + c *x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*b*c*e**3*x**4 + 3*int(sqrt( a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*d**3 + 9*int(sqrt (a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*d*e**2*x**4 + 9* int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*b*c**2*d**2*e*x* *4 + 3*int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c**3*d**3 *x**4 - 9*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a **3*c*e**3 - 27*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8 ),x)*a**2*b*c*d*e**2 + 9*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*d**2*e - 9*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a *c*x**4 + c**2*x**8),x)*a**2*c**2*e**3*x**4 + 3*int((sqrt(a + c*x**4)*x**2 )/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*b*c**2*d**3 - 27*int((sqrt(a + c*x* *4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*b*c**2*d*e**2*x**4 + 9*int( (sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c**3*d**2*e*x **4 + 3*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*...