Integrand size = 32, antiderivative size = 405 \[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\frac {\left (7 B c^3-7 A c^2 d-2 a B d^2\right ) x \sqrt {a-c x^4}}{21 c d^3}-\frac {(B c-A d) x^3 \sqrt {a-c x^4}}{5 d^2}+\frac {B x^5 \sqrt {a-c x^4}}{7 d}+\frac {a^{3/4} (B c-A d) \left (5 c^3-2 a d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{3/4} d^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (21 \sqrt {a} \sqrt {c} d (B c-A d) \left (5 c^3-2 a d^2\right )-5 \left (7 A c^2 d \left (3 c^3-2 a d^2\right )-B \left (21 c^6-14 a c^3 d^2-2 a^2 d^4\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{105 c^{5/4} d^5 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{3/4} (B c-A d) \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^5 \sqrt {a-c x^4}} \] Output:
1/21*(-7*A*c^2*d-2*B*a*d^2+7*B*c^3)*x*(-c*x^4+a)^(1/2)/c/d^3-1/5*(-A*d+B*c )*x^3*(-c*x^4+a)^(1/2)/d^2+1/7*B*x^5*(-c*x^4+a)^(1/2)/d+1/5*a^(3/4)*(-A*d+ B*c)*(-2*a*d^2+5*c^3)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^( 3/4)/d^4/(-c*x^4+a)^(1/2)-1/105*a^(1/4)*(21*a^(1/2)*c^(1/2)*d*(-A*d+B*c)*( -2*a*d^2+5*c^3)-35*A*c^2*d*(-2*a*d^2+3*c^3)+5*B*(-2*a^2*d^4-14*a*c^3*d^2+2 1*c^6))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(5/4)/d^5/(-c*x ^4+a)^(1/2)+a^(1/4)*c^(3/4)*(-A*d+B*c)*(-a*d^2+c^3)*(1-c*x^4/a)^(1/2)*Elli pticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*d/c^(3/2),I)/d^5/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.91 (sec) , antiderivative size = 817, normalized size of antiderivative = 2.02 \[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\frac {35 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d^2 x-35 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^3 x-10 a^2 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^4 x-21 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^3 x^3+21 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^4 x^3-35 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4 d^2 x^5+35 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d^3 x^5+25 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^4 x^5+21 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d^3 x^7-21 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^4 x^7-15 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^4 x^9+21 i \sqrt {a} \sqrt {c} d (B c-A d) \left (-5 c^3+2 a d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \left (7 A \sqrt {c} d \left (15 c^{9/2}+15 \sqrt {a} c^3 d-10 a c^{3/2} d^2-6 a^{3/2} d^3\right )+B \left (-105 c^6-105 \sqrt {a} c^{9/2} d+70 a c^3 d^2+42 a^{3/2} c^{3/2} d^3+10 a^2 d^4\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-105 i B c^6 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+105 i A c^5 d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+105 i a B c^3 d^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-105 i a A c^2 d^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{105 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^5 \sqrt {a-c x^4}} \] Input:
Integrate[(x^4*(A + B*x^2)*Sqrt[a - c*x^4])/(c + d*x^2),x]
Output:
(35*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*d^2*x - 35*a*A*Sqrt[-(Sqrt[c]/Sqrt[a] )]*c^2*d^3*x - 10*a^2*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^4*x - 21*a*B*Sqrt[-(Sqr t[c]/Sqrt[a])]*c^2*d^3*x^3 + 21*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^4*x^3 - 3 5*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^4*d^2*x^5 + 35*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c ^3*d^3*x^5 + 25*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^4*x^5 + 21*B*Sqrt[-(Sqrt[ c]/Sqrt[a])]*c^3*d^3*x^7 - 21*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^4*x^7 - 15* B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^4*x^9 + (21*I)*Sqrt[a]*Sqrt[c]*d*(B*c - A *d)*(-5*c^3 + 2*a*d^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt [c]/Sqrt[a])]*x], -1] - I*(7*A*Sqrt[c]*d*(15*c^(9/2) + 15*Sqrt[a]*c^3*d - 10*a*c^(3/2)*d^2 - 6*a^(3/2)*d^3) + B*(-105*c^6 - 105*Sqrt[a]*c^(9/2)*d + 70*a*c^3*d^2 + 42*a^(3/2)*c^(3/2)*d^3 + 10*a^2*d^4))*Sqrt[1 - (c*x^4)/a]*E llipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (105*I)*B*c^6*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/ Sqrt[a])]*x], -1] + (105*I)*A*c^5*d*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt [a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (105*I)*a*B* c^3*d^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[S qrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (105*I)*a*A*c^2*d^3*Sqrt[1 - (c*x^4)/a]* EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(105*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^5*Sqrt[a - c*x^4])
Time = 0.93 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.74, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2249, 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.
| \(\displaystyle \int \frac {x^4 \sqrt {a-c x^4} \left (A+B x^2\right )}{c+d x^2} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (-\frac {c \left (c^3-a d^2\right ) (B c-A d)}{d^5 \sqrt {a-c x^4}}+\frac {x^2 \left (c^3-a d^2\right ) (B c-A d)}{d^4 \sqrt {a-c x^4}}-\frac {x^4 \left (-a B d^2-A c^2 d+B c^3\right )}{d^3 \sqrt {a-c x^4}}+\frac {a A c^2 d^3-a B c^3 d^2-A c^5 d+B c^6}{d^5 \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {c x^6 (B c-A d)}{d^2 \sqrt {a-c x^4}}-\frac {B c x^8}{d \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 a^{7/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{3/4} d^2 \sqrt {a-c x^4}}+\frac {3 a^{7/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{3/4} d^2 \sqrt {a-c x^4}}-\frac {a^{3/4} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} d^4 \sqrt {a-c x^4}}+\frac {a^{3/4} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} d^4 \sqrt {a-c x^4}}-\frac {a^{5/4} \sqrt {1-\frac {c x^4}{a}} \left (-a B d^2-A c^2 d+B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{5/4} d^3 \sqrt {a-c x^4}}-\frac {5 a^{9/4} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{21 c^{5/4} d \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} c^{3/4} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^5 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{3/4} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^5 \sqrt {a-c x^4}}-\frac {x \sqrt {a-c x^4} \left (A c^2 d-B \left (c^3-a d^2\right )\right )}{3 c d^3}-\frac {x^3 \sqrt {a-c x^4} (B c-A d)}{5 d^2}+\frac {5 a B x \sqrt {a-c x^4}}{21 c d}+\frac {B x^5 \sqrt {a-c x^4}}{7 d}\) |
Input:
Int[(x^4*(A + B*x^2)*Sqrt[a - c*x^4])/(c + d*x^2),x]
Output:
(5*a*B*x*Sqrt[a - c*x^4])/(21*c*d) - ((A*c^2*d - B*(c^3 - a*d^2))*x*Sqrt[a - c*x^4])/(3*c*d^3) - ((B*c - A*d)*x^3*Sqrt[a - c*x^4])/(5*d^2) + (B*x^5* Sqrt[a - c*x^4])/(7*d) + (3*a^(7/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*Ellipt icE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*d^2*Sqrt[a - c*x^4]) + (a ^(3/4)*(B*c - A*d)*(c^3 - a*d^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^( 1/4)*x)/a^(1/4)], -1])/(c^(3/4)*d^4*Sqrt[a - c*x^4]) - (5*a^(9/4)*B*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(21*c^(5/4)*d*Sq rt[a - c*x^4]) - (3*a^(7/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcS in[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*d^2*Sqrt[a - c*x^4]) - (a^(1/4)*c ^(3/4)*(B*c - A*d)*(c^3 - a*d^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^( 1/4)*x)/a^(1/4)], -1])/(d^5*Sqrt[a - c*x^4]) - (a^(3/4)*(B*c - A*d)*(c^3 - a*d^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c ^(3/4)*d^4*Sqrt[a - c*x^4]) - (a^(5/4)*(B*c^3 - A*c^2*d - a*B*d^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(5/4)*d^3*Sq rt[a - c*x^4]) + (a^(1/4)*c^(3/4)*(B*c - A*d)*(c^3 - a*d^2)*Sqrt[1 - (c*x^ 4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1]) /(d^5*Sqrt[a - c*x^4])
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, m}, x] & & PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Time = 5.35 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(-\frac {x \left (-15 B \,x^{4} c \,d^{2}-21 A c \,d^{2} x^{2}+21 B \,c^{2} d \,x^{2}+35 A \,c^{2} d +10 B a \,d^{2}-35 B \,c^{3}\right ) \sqrt {-c \,x^{4}+a}}{105 c \,d^{3}}+\frac {-\frac {5 \left (14 a A \,c^{2} d^{3}-21 A \,c^{5} d -2 a^{2} B \,d^{4}-14 a B \,c^{3} d^{2}+21 B \,c^{6}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {21 \sqrt {c}\, \left (2 A a \,d^{3}-5 A \,c^{3} d -2 B a c \,d^{2}+5 B \,c^{4}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {105 c^{2} \left (A a \,d^{3}-A \,c^{3} d -B a c \,d^{2}+B \,c^{4}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{105 c \,d^{3}}\) | \(434\) |
| default | \(\frac {c^{2} \left (A d -B c \right ) \left (\frac {c^{2} \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 )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {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 )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {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 )}{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}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {c^{2} \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}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d^{3}}-\frac {-d \left (A d -B c \right ) \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5}-\frac {2 a^{\frac {3}{2}} \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 )}{5 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )+A c d \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 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 )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )-B \,c^{2} \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 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 )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )-B \,d^{2} \left (\frac {x^{5} \sqrt {-c \,x^{4}+a}}{7}-\frac {2 a x \sqrt {-c \,x^{4}+a}}{21 c}+\frac {2 a^{2} \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 )}{21 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d^{3}}\) | \(820\) |
| elliptic | \(\text {Expression too large to display}\) | \(1431\) |
Input:
int(x^4*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
-1/105*x*(-15*B*c*d^2*x^4-21*A*c*d^2*x^2+21*B*c^2*d*x^2+35*A*c^2*d+10*B*a* d^2-35*B*c^3)*(-c*x^4+a)^(1/2)/c/d^3+1/105/c/d^3*(-5*(14*A*a*c^2*d^3-21*A* c^5*d-2*B*a^2*d^4-14*B*a*c^3*d^2+21*B*c^6)/d^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)* EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-21*c^(1/2)/d*(2*A*a*d^3-5*A*c^3*d-2 *B*a*c*d^2+5*B*c^4)*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)*(EllipticF(x*(c^(1/ 2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))+105*c^2*(A*a* d^3-A*c^3*d-B*a*c*d^2+B*c^4)/d^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)*EllipticPi(x*( c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*d/c^(3/2),(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2 )/a^(1/2))^(1/2)))
Timed out. \[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\text {Timed out} \] Input:
integrate(x^4*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int \frac {x^{4} \left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{c + d x^{2}}\, dx \] Input:
integrate(x**4*(B*x**2+A)*(-c*x**4+a)**(1/2)/(d*x**2+c),x)
Output:
Integral(x**4*(A + B*x**2)*sqrt(a - c*x**4)/(c + d*x**2), x)
\[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} x^{4}}{d x^{2} + c} \,d x } \] Input:
integrate(x^4*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*x^4/(d*x^2 + c), x)
\[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} x^{4}}{d x^{2} + c} \,d x } \] Input:
integrate(x^4*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*x^4/(d*x^2 + c), x)
Timed out. \[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int \frac {x^4\,\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{d\,x^2+c} \,d x \] Input:
int((x^4*(A + B*x^2)*(a - c*x^4)^(1/2))/(c + d*x^2),x)
Output:
int((x^4*(A + B*x^2)*(a - c*x^4)^(1/2))/(c + d*x^2), x)
\[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\frac {-10 \sqrt {-c \,x^{4}+a}\, a b \,d^{2} x -35 \sqrt {-c \,x^{4}+a}\, a \,c^{2} d x +21 \sqrt {-c \,x^{4}+a}\, a c \,d^{2} x^{3}+35 \sqrt {-c \,x^{4}+a}\, b \,c^{3} x -21 \sqrt {-c \,x^{4}+a}\, b \,c^{2} d \,x^{3}+15 \sqrt {-c \,x^{4}+a}\, b c \,d^{2} x^{5}+10 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} b c \,d^{2}+35 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} c^{3} d -35 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b \,c^{4}+42 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} c \,d^{3}-42 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b \,c^{2} d^{2}-105 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a \,c^{4} d +105 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{5}+10 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} b \,d^{3}-28 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} c^{2} d^{2}+28 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b \,c^{3} d}{105 c \,d^{3}} \] Input:
int(x^4*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x)
Output:
( - 10*sqrt(a - c*x**4)*a*b*d**2*x - 35*sqrt(a - c*x**4)*a*c**2*d*x + 21*s qrt(a - c*x**4)*a*c*d**2*x**3 + 35*sqrt(a - c*x**4)*b*c**3*x - 21*sqrt(a - c*x**4)*b*c**2*d*x**3 + 15*sqrt(a - c*x**4)*b*c*d**2*x**5 + 10*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a**2*b*c*d**2 + 35*i nt(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a**2*c**3*d - 35*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*b* c**4 + 42*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x* *6),x)*a**2*c*d**3 - 42*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x**2 - c**2 *x**4 - c*d*x**6),x)*a*b*c**2*d**2 - 105*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*c**4*d + 105*int((sqrt(a - c*x**4) *x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*b*c**5 + 10*int((sqrt(a - c*x**4)*x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a**2*b*d**3 - 2 8*int((sqrt(a - c*x**4)*x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a **2*c**2*d**2 + 28*int((sqrt(a - c*x**4)*x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*b*c**3*d)/(105*c*d**3)