Integrand size = 32, antiderivative size = 293 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (c+d x^2\right )} \, dx=-\frac {A \sqrt {a-c x^4}}{3 c x^3}-\frac {(B c-A d) \sqrt {a-c x^4}}{c^2 x}-\frac {a^{3/4} (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{7/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (3 \sqrt {a} d (B c-A d)-c^{3/2} (3 B c-A d)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{7/4} d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} (B c-A d) \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{13/4} d \sqrt {a-c x^4}} \] Output:
-1/3*A*(-c*x^4+a)^(1/2)/c/x^3-(-A*d+B*c)*(-c*x^4+a)^(1/2)/c^2/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^(7/4)/(-c*x^4+ a)^(1/2)+1/3*a^(1/4)*(3*a^(1/2)*d*(-A*d+B*c)-c^(3/2)*(-A*d+3*B*c))*(1-c*x^ 4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(7/4)/d/(-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)*EllipticPi(c^(1/4)*x/a^(1/4), -a^(1/2)*d/c^(3/2),I)/c^(13/4)/d/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.22 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.09 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (c+d x^2\right )} \, dx=\frac {-a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d-3 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d x^2+3 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^2 x^2+A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d x^4+3 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d x^6-3 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^2 x^6+3 i \sqrt {a} c^{3/2} d (B c-A d) x^3 \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i c^{3/2} \left (A d \left (c^{3/2}-3 \sqrt {a} d\right )-3 B \left (c^{5/2}-\sqrt {a} c d\right )\right ) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i B c^4 x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i A c^3 d x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i a B c d^2 x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i a A d^3 x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{3 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d x^3 \sqrt {a-c x^4}} \] Input:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(x^4*(c + d*x^2)),x]
Output:
(-(a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d) - 3*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^ 2*d*x^2 + 3*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2*x^2 + A*Sqrt[-(Sqrt[c]/Sqrt [a])]*c^3*d*x^4 + 3*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*d*x^6 - 3*A*Sqrt[-(Sqrt [c]/Sqrt[a])]*c^2*d^2*x^6 + (3*I)*Sqrt[a]*c^(3/2)*d*(B*c - A*d)*x^3*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) - 3*Sqrt[a]*d) - 3*B*(c^(5/2) - Sqrt[a]*c*d))*x^3*Sqrt[ 1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I )*B*c^4*x^3*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSi nh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (3*I)*A*c^3*d*x^3*Sqrt[1 - (c*x^4)/a ]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x] , -1] + (3*I)*a*B*c*d^2*x^3*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c ^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*a*A*d^3*x^3*Sq rt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt [c]/Sqrt[a])]*x], -1])/(3*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*d*x^3*Sqrt[a - c*x^ 4])
Time = 0.63 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.29, 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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{x^4 \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (\frac {a (B c-A d)}{c^2 x^2 \sqrt {a-c x^4}}+\frac {\left (c^3-a d^2\right ) (B c-A d)}{c^2 d \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {a A}{c x^4 \sqrt {a-c x^4}}-\frac {B c}{d \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{7/4} \sqrt {a-c x^4}}-\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{7/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{13/4} d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} A \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt [4]{c} \sqrt {a-c x^4}}-\frac {\sqrt {a-c x^4} (B c-A d)}{c^2 x}-\frac {A \sqrt {a-c x^4}}{3 c x^3}-\frac {\sqrt [4]{a} B c^{3/4} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-c x^4}}\) |
Input:
Int[((A + B*x^2)*Sqrt[a - c*x^4])/(x^4*(c + d*x^2)),x]
Output:
-1/3*(A*Sqrt[a - c*x^4])/(c*x^3) - ((B*c - A*d)*Sqrt[a - c*x^4])/(c^2*x) - (a^(3/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^( 1/4)], -1])/(c^(7/4)*Sqrt[a - c*x^4]) + (a^(1/4)*A*Sqrt[1 - (c*x^4)/a]*Ell ipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(1/4)*Sqrt[a - c*x^4]) - (a^ (1/4)*B*c^(3/4)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(d*Sqrt[a - c*x^4]) + (a^(3/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*Ellip ticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(7/4)*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^(13/4)*d*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 = 4.39 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.38
| method | result | size |
| risch | \(-\frac {\sqrt {-c \,x^{4}+a}\, \left (-3 A d \,x^{2}+3 B c \,x^{2}+A c \right )}{3 c^{2} x^{3}}+\frac {\frac {c \left (-\frac {3 d \left (A d -B c \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {A c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 B \,c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d}+\frac {3 \left (A a \,d^{3}-A \,c^{3} d -B a c \,d^{2}+B \,c^{4}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 c^{2}}\) | \(404\) |
| default | \(\frac {A \left (-\frac {\sqrt {-c \,x^{4}+a}}{3 x^{3}}-\frac {2 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 )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c}-\frac {\left (A d -B c \right ) \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^{2}}+\frac {d \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^{2}}\) | \(623\) |
| elliptic | \(-\frac {A \sqrt {-c \,x^{4}+a}}{3 c \,x^{3}}+\frac {\left (A d -B c \right ) \sqrt {-c \,x^{4}+a}}{c^{2} x}-\frac {\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 c}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, d}+\frac {\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}{3 \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}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) A d}{c^{\frac {3}{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}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{\sqrt {c}\, \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}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) A d}{c^{\frac {3}{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}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{\sqrt {c}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {d^{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 ) A a}{c^{3} \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}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\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 ) B 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 ) B}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(846\) |
Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^4/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
-1/3*(-c*x^4+a)^(1/2)*(-3*A*d*x^2+3*B*c*x^2+A*c)/c^2/x^3+1/3/c^2*(c/d*(-3* 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)-3*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))+3*(A*a*d^3-A*c ^3*d-B*a*c*d^2+B*c^4)/d/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. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^4/(d*x^2+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (c+d x^2\right )} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{x^{4} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/x**4/(d*x**2+c),x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)/(x**4*(c + d*x**2)), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (c+d x^2\right )} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^4/(d*x^2+c),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/((d*x^2 + c)*x^4), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (c+d x^2\right )} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^4/(d*x^2+c),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/((d*x^2 + c)*x^4), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (c+d x^2\right )} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{x^4\,\left (d\,x^2+c\right )} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(x^4*(c + d*x^2)),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(x^4*(c + d*x^2)), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (c+d x^2\right )} \, dx=\frac {-\sqrt {-c \,x^{4}+a}\, a -3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{8}-c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a^{2} d \,x^{3}+3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{8}-c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a b c \,x^{3}-2 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a \,c^{2} x^{3}+\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a c d \,x^{3}-3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{2} x^{3}}{3 c \,x^{3}} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^4/(d*x^2+c),x)
Output:
( - sqrt(a - c*x**4)*a - 3*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*d*x**3 + 3*int(sqrt(a - c*x**4)/(a*c*x**2 + a*d *x**4 - c**2*x**6 - c*d*x**8),x)*a*b*c*x**3 - 2*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*c**2*x**3 + int((sqrt(a - c*x**4)* x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*c*d*x**3 - 3*int((sqrt( a - c*x**4)*x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*b*c**2*x**3)/ (3*c*x**3)