Integrand size = 32, antiderivative size = 259 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (c+d x^2\right )} \, dx=-\frac {A \sqrt {a-c x^4}}{c x}-\frac {a^{3/4} (B c+A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (c^{3/2} (B c-A d)+\sqrt {a} d (B c+A d)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} d^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} (B c-A d) \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{9/4} d^2 \sqrt {a-c x^4}} \] Output:
-A*(-c*x^4+a)^(1/2)/c/x-a^(3/4)*(A*d+B*c)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1 /4)*x/a^(1/4),I)/c^(3/4)/d/(-c*x^4+a)^(1/2)+a^(1/4)*(c^(3/2)*(-A*d+B*c)+a^ (1/2)*d*(A*d+B*c))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(3/4 )/d^2/(-c*x^4+a)^(1/2)-a^(1/4)*(-A*d+B*c)*(-a*d^2+c^3)*(1-c*x^4/a)^(1/2)*E llipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*d/c^(3/2),I)/c^(9/4)/d^2/(-c*x^4+a)^( 1/2)
Result contains complex when optimal does not.
Time = 11.06 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.89 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (c+d x^2\right )} \, dx=\frac {-a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^2+A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^2 x^4+i \sqrt {a} c^{3/2} d (B c+A d) x \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i c^{3/2} \left (A d \left (-c^{3/2}+\sqrt {a} d\right )+B \left (c^{5/2}+\sqrt {a} c d\right )\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i B c^4 x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-i A c^3 d x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-i a B c d^2 x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i a A d^3 x \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^2 x \sqrt {a-c x^4}} \] Input:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(x^2*(c + d*x^2)),x]
Output:
(-(a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2) + A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^ 2*x^4 + I*Sqrt[a]*c^(3/2)*d*(B*c + A*d)*x*Sqrt[1 - (c*x^4)/a]*EllipticE[I* ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*c^(3/2)*(A*d*(-c^(3/2) + Sqrt [a]*d) + B*(c^(5/2) + Sqrt[a]*c*d))*x*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcS inh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*B*c^4*x*Sqrt[1 - (c*x^4)/a]*Ellip ticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*A*c^3*d*x*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcS inh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*a*B*c*d^2*x*Sqrt[1 - (c*x^4)/a]*E llipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], - 1] + I*a*A*d^3*x*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I* ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^ 2*x*Sqrt[a - c*x^4])
Time = 0.70 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.54, 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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{x^2 \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (-\frac {\left (c^3-a d^2\right ) (B c-A d)}{c d^2 \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {c (B c-A d)}{d^2 \sqrt {a-c x^4}}+\frac {a A}{c x^2 \sqrt {a-c x^4}}-\frac {B c x^2}{d \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} A \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {a^{3/4} A \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {a^{3/4} B \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-c x^4}}-\frac {a^{3/4} B \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{3/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{9/4} d^2 \sqrt {a-c x^4}}-\frac {A \sqrt {a-c x^4}}{c x}\) |
Input:
Int[((A + B*x^2)*Sqrt[a - c*x^4])/(x^2*(c + d*x^2)),x]
Output:
-((A*Sqrt[a - c*x^4])/(c*x)) - (a^(3/4)*A*Sqrt[1 - (c*x^4)/a]*EllipticE[Ar cSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) - (a^(3/4)*B*c^( 1/4)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(d*Sq rt[a - c*x^4]) + (a^(3/4)*A*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)* x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(3/4)*B*c^(1/4)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(d*Sqrt[a - c*x^4]) + (a^(1/4)*c^(3/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1 /4)*x)/a^(1/4)], -1])/(d^2*Sqrt[a - c*x^4]) - (a^(1/4)*(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])/(c^(9/4)*d^2*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 = 2.50 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.49
| method | result | size |
| risch | \(-\frac {A \sqrt {-c \,x^{4}+a}}{c x}-\frac {\frac {c \left (-\frac {d \left (A d +B c \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {A c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {B \,c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d^{2}}+\frac {\left (A a \,d^{3}-A \,c^{3} d -B a c \,d^{2}+B \,c^{4}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{c}\) | \(385\) |
| default | \(\frac {A \left (-\frac {\sqrt {-c \,x^{4}+a}}{x}+\frac {2 \sqrt {c}\, \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 )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c}-\frac {\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 )}{c}\) | \(529\) |
| elliptic | \(-\frac {A \sqrt {-c \,x^{4}+a}}{c x}-\frac {c \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 ) A}{d \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 {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, B \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, d}-\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, B \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, d}+\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, A \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, A \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {d \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 a}{c^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {c \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}{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 ) B 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 ) B}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(833\) |
Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
-A*(-c*x^4+a)^(1/2)/c/x-1/c*(c/d^2*(-d*(A*d+B*c)*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)/c^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^( 1/2)/a^(1/2))^(1/2),I))+A*c*d/(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)-B*c^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))+(A*a*d^3-A*c^3*d-B*a*c*d^2+B*c^4)/d^2/c/(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. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(d*x^2+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (c+d x^2\right )} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{x^{2} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/x**2/(d*x**2+c),x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)/(x**2*(c + d*x**2)), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (c+d x^2\right )} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(d*x^2+c),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/((d*x^2 + c)*x^2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (c+d x^2\right )} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(d*x^2+c),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/((d*x^2 + c)*x^2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (c+d x^2\right )} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{x^2\,\left (d\,x^2+c\right )} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(x^2*(c + d*x^2)),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(x^2*(c + d*x^2)), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (c+d x^2\right )} \, dx=\frac {\sqrt {-c \,x^{4}+a}\, a +2 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{8}-c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a^{2} c x +\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} d x +\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b c x +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a c d x -\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{2} x}{c x} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(d*x^2+c),x)
Output:
(sqrt(a - c*x**4)*a + 2*int(sqrt(a - c*x**4)/(a*c*x**2 + a*d*x**4 - c**2*x **6 - c*d*x**8),x)*a**2*c*x + int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2* x**4 - c*d*x**6),x)*a**2*d*x + int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2 *x**4 - c*d*x**6),x)*a*b*c*x + 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*d*x - 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**2*x)/(c*x)