Integrand size = 32, antiderivative size = 460 \[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {A c^2-a B d+c (B c-A d) x^2}{2 a \left (c^3-a d^2\right ) x^3 \sqrt {a-c x^4}}-\frac {\left (5 A c^3-3 a B c d-2 a A d^2\right ) \sqrt {a-c x^4}}{6 a^2 c \left (c^3-a d^2\right ) x^3}-\frac {(B c-A d) \left (3 c^3-2 a d^2\right ) \sqrt {a-c x^4}}{2 a^2 c^2 \left (c^3-a d^2\right ) x}-\frac {(B c-A d) \left (3 c^3-2 a d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{5/4} c^{7/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {\left (5 A c^3+\sqrt {a} c^{3/2} (9 B c-4 A d)+6 a d (B c-A d)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{6 a^{7/4} c^{7/4} \left (c^{3/2}+\sqrt {a} d\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} d^3 (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{13/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}} \] Output:
1/2*(A*c^2-B*a*d+c*(-A*d+B*c)*x^2)/a/(-a*d^2+c^3)/x^3/(-c*x^4+a)^(1/2)-1/6 *(-2*A*a*d^2+5*A*c^3-3*B*a*c*d)*(-c*x^4+a)^(1/2)/a^2/c/(-a*d^2+c^3)/x^3-1/ 2*(-A*d+B*c)*(-2*a*d^2+3*c^3)*(-c*x^4+a)^(1/2)/a^2/c^2/(-a*d^2+c^3)/x-1/2* (-A*d+B*c)*(-2*a*d^2+3*c^3)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4), I)/a^(5/4)/c^(7/4)/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)+1/6*(5*A*c^3+a^(1/2)*c^(3 /2)*(-4*A*d+9*B*c)+6*a*d*(-A*d+B*c))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x /a^(1/4),I)/a^(7/4)/c^(7/4)/(c^(3/2)+a^(1/2)*d)/(-c*x^4+a)^(1/2)+a^(1/4)*d ^3*(-A*d+B*c)*(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)/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.68 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.58 \[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {2 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^5-2 a^2 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^2+6 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^5 x^2-6 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4 d x^2-6 a^2 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^2 x^2+6 a^2 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^3 x^2-5 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^6 x^4+3 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4 d x^4+2 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d^2 x^4-9 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^6 x^6+9 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^5 d x^6+6 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d^2 x^6-6 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^3 x^6+3 i \sqrt {a} c^{3/2} (B c-A d) \left (-3 c^3+2 a d^2\right ) x^3 \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i c^{3/2} \left (-c^{3/2}+\sqrt {a} d\right ) \left (5 A c^3+\sqrt {a} c^{3/2} (9 B c-4 A d)+6 a d (B c-A d)\right ) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+6 i a^2 B c d^3 x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-6 i a^2 A d^4 x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{6 a^2 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 \left (-c^3+a d^2\right ) x^3 \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2)/(x^4*(c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(2*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^5 - 2*a^2*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2 *d^2 + 6*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^5*x^2 - 6*a*A*Sqrt[-(Sqrt[c]/Sqrt[ a])]*c^4*d*x^2 - 6*a^2*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^2*x^2 + 6*a^2*A*Sq rt[-(Sqrt[c]/Sqrt[a])]*c*d^3*x^2 - 5*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^6*x^4 + 3*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^4*d*x^4 + 2*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]* c^3*d^2*x^4 - 9*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^6*x^6 + 9*A*Sqrt[-(Sqrt[c]/Sq rt[a])]*c^5*d*x^6 + 6*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*d^2*x^6 - 6*a*A*Sqr t[-(Sqrt[c]/Sqrt[a])]*c^2*d^3*x^6 + (3*I)*Sqrt[a]*c^(3/2)*(B*c - A*d)*(-3* c^3 + 2*a*d^2)*x^3*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/ Sqrt[a])]*x], -1] - I*c^(3/2)*(-c^(3/2) + Sqrt[a]*d)*(5*A*c^3 + Sqrt[a]*c^ (3/2)*(9*B*c - 4*A*d) + 6*a*d*(B*c - A*d))*x^3*Sqrt[1 - (c*x^4)/a]*Ellipti cF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (6*I)*a^2*B*c*d^3*x^3*Sqrt [1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c ]/Sqrt[a])]*x], -1] - (6*I)*a^2*A*d^4*x^3*Sqrt[1 - (c*x^4)/a]*EllipticPi[- ((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(6*a^2* Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*(-c^3 + a*d^2)*x^3*Sqrt[a - c*x^4])
Time = 0.89 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.19, 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.
| \(\displaystyle \int \frac {A+B x^2}{x^4 \left (a-c x^4\right )^{3/2} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (\frac {B c-A d}{a c^2 x^2 \sqrt {a-c x^4}}+\frac {c \left (-a B d+c x^2 (B c-A d)+A c^2\right )}{a \left (c^3-a d^2\right ) \left (a-c x^4\right )^{3/2}}+\frac {d^3 (B c-A d)}{c^2 \left (c^3-a d^2\right ) \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {A}{a c x^4 \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{a^{5/4} c^{7/4} \sqrt {a-c x^4}}+\frac {c^{3/4} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B+A \sqrt {c}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{7/4} \left (\sqrt {a} d+c^{3/2}\right ) \sqrt {a-c x^4}}-\frac {\sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{a^{5/4} c^{7/4} \sqrt {a-c x^4}}-\frac {c^{5/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{5/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {A \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 a^{7/4} \sqrt [4]{c} \sqrt {a-c x^4}}-\frac {\sqrt {a-c x^4} (B c-A d)}{a^2 c^2 x}+\frac {c x \left (-a B d+c x^2 (B c-A d)+A c^2\right )}{2 a^2 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {A \sqrt {a-c x^4}}{3 a^2 c x^3}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{13/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}\) |
Input:
Int[(A + B*x^2)/(x^4*(c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(c*x*(A*c^2 - a*B*d + c*(B*c - A*d)*x^2))/(2*a^2*(c^3 - a*d^2)*Sqrt[a - c* x^4]) - (A*Sqrt[a - c*x^4])/(3*a^2*c*x^3) - ((B*c - A*d)*Sqrt[a - c*x^4])/ (a^2*c^2*x) - ((B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x )/a^(1/4)], -1])/(a^(5/4)*c^(7/4)*Sqrt[a - c*x^4]) - (c^(5/4)*(B*c - A*d)* Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(5/4) *(c^3 - a*d^2)*Sqrt[a - c*x^4]) + (A*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[ (c^(1/4)*x)/a^(1/4)], -1])/(3*a^(7/4)*c^(1/4)*Sqrt[a - c*x^4]) + ((Sqrt[a] *B + A*Sqrt[c])*c^(3/4)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a ^(1/4)], -1])/(2*a^(7/4)*(c^(3/2) + Sqrt[a]*d)*Sqrt[a - c*x^4]) + ((B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(a^(5 /4)*c^(7/4)*Sqrt[a - c*x^4]) + (a^(1/4)*d^3*(B*c - A*d)*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)*(c^3 - a*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 = 7.48 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.44
| 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} a^{2} x^{3}}+\frac {\frac {A \,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}}+\frac {3 B \,c^{\frac {3}{2}} \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}}-\frac {3 A \sqrt {c}\, d \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}}-\frac {3 a \,c^{3} \left (\frac {2 c \left (-\frac {\left (A d -B c \right ) x^{3}}{4 a}+\frac {\left (A \,c^{2}-B a d \right ) x}{4 a c}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (A \,c^{2}-B a d \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 )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (A d -B c \right ) \sqrt {c}\, \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 )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{a \,d^{2}-c^{3}}+\frac {3 a^{2} d^{3} \left (A d -B c \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 )}{\left (a \,d^{2}-c^{3}\right ) c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 a^{2} c^{2}}\) | \(663\) |
| default | \(\frac {A \left (-\frac {\sqrt {-c \,x^{4}+a}}{3 a^{2} x^{3}}+\frac {c x}{2 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {5 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 )}{6 a^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c}-\frac {\left (A d -B c \right ) \left (\frac {c \,x^{3}}{2 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {-c \,x^{4}+a}}{a^{2} x}+\frac {3 \sqrt {c}\, \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 )}{2 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c^{2}}+\frac {d \left (A d -B c \right ) \left (\frac {2 c \left (\frac {d \,x^{3}}{4 a \left (a \,d^{2}-c^{3}\right )}-\frac {c x}{4 a \left (a \,d^{2}-c^{3}\right )}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\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 )}{2 a \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {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 )}{2 \sqrt {a}\, \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, d \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 )}{2 \sqrt {a}\, \left (a \,d^{2}-c^{3}\right ) \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 )}{\left (a \,d^{2}-c^{3}\right ) c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c^{2}}\) | \(694\) |
| elliptic | \(\text {Expression too large to display}\) | \(1209\) |
Input:
int((B*x^2+A)/x^4/(d*x^2+c)/(-c*x^4+a)^(3/2),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/a^2/x^3+1/3/a^2/c^2*( A*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* B*c^(3/2)*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))-3*A*c^(1/2)*d*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)-Elliptic E(x*(c^(1/2)/a^(1/2))^(1/2),I))-3*a*c^3/(a*d^2-c^3)*(2*c*(-1/4*(A*d-B*c)/a *x^3+1/4*(A*c^2-B*a*d)/a/c*x)/(-(x^4-a/c)*c)^(1/2)+1/2*(A*c^2-B*a*d)/a/(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)-1/2*(A*d-B* c)*c^(1/2)/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)))+3*a^2*d^3*(A*d-B*c)/( a*d^2-c^3)/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)))
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((B*x^2+A)/x^4/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-c*x^4 + a)*(B*x^2 + A)/(c^2*d*x^14 + c^3*x^12 - 2*a*c*d*x^1 0 - 2*a*c^2*x^8 + a^2*d*x^6 + a^2*c*x^4), x)
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {A + B x^{2}}{x^{4} \left (a - c x^{4}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)/x**4/(d*x**2+c)/(-c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)/(x**4*(a - c*x**4)**(3/2)*(c + d*x**2)), x)
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((B*x^2+A)/x^4/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)/((-c*x^4 + a)^(3/2)*(d*x^2 + c)*x^4), x)
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((B*x^2+A)/x^4/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)/((-c*x^4 + a)^(3/2)*(d*x^2 + c)*x^4), x)
Timed out. \[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{x^4\,{\left (a-c\,x^4\right )}^{3/2}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((A + B*x^2)/(x^4*(a - c*x^4)^(3/2)*(c + d*x^2)),x)
Output:
int((A + B*x^2)/(x^4*(a - c*x^4)^(3/2)*(c + d*x^2)), x)
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((B*x^2+A)/x^4/(d*x^2+c)/(-c*x^4+a)^(3/2),x)
Output:
( - sqrt(a - c*x**4)*a - 3*sqrt(a - c*x**4)*b*x**2 - 3*int(sqrt(a - c*x**4 )/(a**2*c*x**2 + a**2*d*x**4 - 2*a*c**2*x**6 - 2*a*c*d*x**8 + c**3*x**10 + c**2*d*x**12),x)*a**3*d*x**3 + 3*int(sqrt(a - c*x**4)/(a**2*c*x**2 + a**2 *d*x**4 - 2*a*c**2*x**6 - 2*a*c*d*x**8 + c**3*x**10 + c**2*d*x**12),x)*a** 2*c*d*x**7 - 3*int(sqrt(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a**2*b*d*x**3 + 5*int(sqrt(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x** 8 + c**2*d*x**10),x)*a**2*c**2*x**3 + 3*int(sqrt(a - c*x**4)/(a**2*c + a** 2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a*b *c*d*x**7 - 5*int(sqrt(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a*c**3*x**7 + 9*int((sqrt(a - c*x**4)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3 *x**8 + c**2*d*x**10),x)*a*b*c*d*x**3 - 9*int((sqrt(a - c*x**4)*x**4)/(a** 2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**1 0),x)*b*c**2*d*x**7 + 5*int((sqrt(a - c*x**4)*x**2)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a**2*c*d*x** 3 + 9*int((sqrt(a - c*x**4)*x**2)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a*b*c**2*x**3 - 5*int((sqrt(a - c*x**4)*x**2)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c** 3*x**8 + c**2*d*x**10),x)*a*c**2*d*x**7 - 9*int((sqrt(a - c*x**4)*x**2)...