Integrand size = 32, antiderivative size = 353 \[ \int \frac {A+B x^2}{x^6 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {A \sqrt {a-c x^4}}{5 a c x^5}-\frac {(B c-A d) \sqrt {a-c x^4}}{3 a c^2 x^3}-\frac {\left (3 A c^3-5 a B c d+5 a A d^2\right ) \sqrt {a-c x^4}}{5 a^2 c^3 x}-\frac {\left (3 A c^3-5 a B c d+5 a A d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} c^{11/4} \sqrt {a-c x^4}}+\frac {\left (9 A c^3+5 \sqrt {a} c^{3/2} (B c-A d)-15 a d (B c-A d)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15 a^{5/4} c^{11/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} d^2 (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{17/4} \sqrt {a-c x^4}} \] Output:
-1/5*A*(-c*x^4+a)^(1/2)/a/c/x^5-1/3*(-A*d+B*c)*(-c*x^4+a)^(1/2)/a/c^2/x^3- 1/5*(5*A*a*d^2+3*A*c^3-5*B*a*c*d)*(-c*x^4+a)^(1/2)/a^2/c^3/x-1/5*(5*A*a*d^ 2+3*A*c^3-5*B*a*c*d)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(5 /4)/c^(11/4)/(-c*x^4+a)^(1/2)+1/15*(9*A*c^3+5*a^(1/2)*c^(3/2)*(-A*d+B*c)-1 5*a*d*(-A*d+B*c))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(5/4) /c^(11/4)/(-c*x^4+a)^(1/2)+a^(1/4)*d^2*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*Ellipt icPi(c^(1/4)*x/a^(1/4),-a^(1/2)*d/c^(3/2),I)/c^(17/4)/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.52 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x^2}{x^6 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-3 a^2 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3-5 a^2 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 x^2+5 a^2 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d x^2-6 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4 x^4+15 a^2 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d x^4-15 a^2 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^2 x^4+5 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4 x^6-5 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d x^6+9 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^5 x^8-15 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d x^8+15 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^2 x^8-3 i \sqrt {a} c^{3/2} \left (-3 A c^3+5 a B c d-5 a A d^2\right ) x^5 \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \sqrt {a} c^{3/2} \left (-9 A c^3+15 a d (B c-A d)+5 \sqrt {a} c^{3/2} (-B c+A d)\right ) x^5 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-15 i a^2 B c d^2 x^5 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+15 i a^2 A d^3 x^5 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{15 a^2 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4 x^5 \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2)/(x^6*(c + d*x^2)*Sqrt[a - c*x^4]),x]
Output:
(-3*a^2*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3 - 5*a^2*B*Sqrt[-(Sqrt[c]/Sqrt[a])]* c^3*x^2 + 5*a^2*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d*x^2 - 6*a*A*Sqrt[-(Sqrt[c ]/Sqrt[a])]*c^4*x^4 + 15*a^2*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d*x^4 - 15*a^2 *A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2*x^4 + 5*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^4 *x^6 - 5*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*d*x^6 + 9*A*Sqrt[-(Sqrt[c]/Sqrt[ a])]*c^5*x^8 - 15*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*d*x^8 + 15*a*A*Sqrt[-(S qrt[c]/Sqrt[a])]*c^2*d^2*x^8 - (3*I)*Sqrt[a]*c^(3/2)*(-3*A*c^3 + 5*a*B*c*d - 5*a*A*d^2)*x^5*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/S qrt[a])]*x], -1] + I*Sqrt[a]*c^(3/2)*(-9*A*c^3 + 15*a*d*(B*c - A*d) + 5*Sq rt[a]*c^(3/2)*(-(B*c) + A*d))*x^5*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[ Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (15*I)*a^2*B*c*d^2*x^5*Sqrt[1 - (c*x^4) /a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]* x], -1] + (15*I)*a^2*A*d^3*x^5*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d )/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(15*a^2*Sqrt[-(Sqr t[c]/Sqrt[a])]*c^4*x^5*Sqrt[a - c*x^4])
Time = 0.77 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.40, 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.
| \(\displaystyle \int \frac {A+B x^2}{x^6 \sqrt {a-c x^4} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (\frac {d^2 (B c-A d)}{c^3 \sqrt {a-c x^4} \left (c+d x^2\right )}-\frac {d (B c-A d)}{c^3 x^2 \sqrt {a-c x^4}}+\frac {B c-A d}{c^2 x^4 \sqrt {a-c x^4}}+\frac {A}{c x^6 \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 a^{3/4} c^{5/4} \sqrt {a-c x^4}}+\frac {3 A \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 a^{5/4} \sqrt {a-c x^4}}-\frac {3 A \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt {a-c x^4}}-\frac {3 A \sqrt {a-c x^4}}{5 a^2 x}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{17/4} \sqrt {a-c x^4}}-\frac {d \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} c^{11/4} \sqrt {a-c x^4}}+\frac {d \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} c^{11/4} \sqrt {a-c x^4}}+\frac {d \sqrt {a-c x^4} (B c-A d)}{a c^3 x}-\frac {\sqrt {a-c x^4} (B c-A d)}{3 a c^2 x^3}-\frac {A \sqrt {a-c x^4}}{5 a c x^5}\) |
Input:
Int[(A + B*x^2)/(x^6*(c + d*x^2)*Sqrt[a - c*x^4]),x]
Output:
-1/5*(A*Sqrt[a - c*x^4])/(a*c*x^5) - ((B*c - A*d)*Sqrt[a - c*x^4])/(3*a*c^ 2*x^3) - (3*A*Sqrt[a - c*x^4])/(5*a^2*x) + (d*(B*c - A*d)*Sqrt[a - c*x^4]) /(a*c^3*x) - (3*A*c^(1/4)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x) /a^(1/4)], -1])/(5*a^(5/4)*Sqrt[a - c*x^4]) + (d*(B*c - A*d)*Sqrt[1 - (c*x ^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*c^(11/4)*Sqrt[ a - c*x^4]) + (3*A*c^(1/4)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x )/a^(1/4)], -1])/(5*a^(5/4)*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])/(3*a^(3/4)*c^(5/4)*Sqrt[ a - c*x^4]) - (d*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4) *x)/a^(1/4)], -1])/(a^(1/4)*c^(11/4)*Sqrt[a - c*x^4]) + (a^(1/4)*d^2*(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^(17/4)*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 = 3.82 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(-\frac {\sqrt {-c \,x^{4}+a}\, \left (15 A a \,d^{2} x^{4}+9 A \,c^{3} x^{4}-15 B a c d \,x^{4}-5 A a c d \,x^{2}+5 B a \,c^{2} x^{2}+3 A a \,c^{2}\right )}{15 c^{3} a^{2} x^{5}}-\frac {-\frac {3 \sqrt {c}\, \left (5 A a \,d^{2}+3 A \,c^{3}-5 a B c d \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {5 B \,c^{3} a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {5 A a \,c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {15 a^{2} d^{2} \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{15 a^{2} c^{3}}\) | \(441\) |
| default | \(\frac {A \left (-\frac {\sqrt {-c \,x^{4}+a}}{5 a \,x^{5}}-\frac {3 c \sqrt {-c \,x^{4}+a}}{5 a^{2} x}+\frac {3 c^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c}-\frac {\left (A d -B c \right ) \left (-\frac {\sqrt {-c \,x^{4}+a}}{3 a \,x^{3}}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c^{2}}+\frac {\left (A d -B c \right ) d \left (-\frac {\sqrt {-c \,x^{4}+a}}{a x}+\frac {\sqrt {c}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c^{3}}-\frac {d^{2} \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{c^{4} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(455\) |
| elliptic | \(-\frac {A \sqrt {-c \,x^{4}+a}}{5 a c \,x^{5}}+\frac {\left (A d -B c \right ) \sqrt {-c \,x^{4}+a}}{3 c^{2} a \,x^{3}}-\frac {\left (5 A a \,d^{2}+3 A \,c^{3}-5 a B c d \right ) \sqrt {-c \,x^{4}+a}}{5 a^{2} c^{3} 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 ) A d}{3 c a \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 {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{3 a \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 {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) A \,d^{2}}{c^{\frac {5}{2}} \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 \sqrt {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}{5 a^{\frac {3}{2}} \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 {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B d}{c^{\frac {3}{2}} \sqrt {a}\, \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 {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) A \,d^{2}}{c^{\frac {5}{2}} \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {c}\, \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}{5 a^{\frac {3}{2}} \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 {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B d}{c^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {d^{3} \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^{4} \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 ) B}{c^{3} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(861\) |
Input:
int((B*x^2+A)/x^6/(d*x^2+c)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/15*(-c*x^4+a)^(1/2)*(15*A*a*d^2*x^4+9*A*c^3*x^4-15*B*a*c*d*x^4-5*A*a*c* d*x^2+5*B*a*c^2*x^2+3*A*a*c^2)/c^3/a^2/x^5-1/15/a^2/c^3*(-3*c^(1/2)*(5*A*a *d^2+3*A*c^3-5*B*a*c*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)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))-5*B*c^3*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)+5*A*a*c ^2*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)+15* a^2*d^2*(A*d-B*c)/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)))
\[ \int \frac {A+B x^2}{x^6 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )} x^{6}} \,d x } \] Input:
integrate((B*x^2+A)/x^6/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-c*x^4 + a)*(B*x^2 + A)/(c*d*x^12 + c^2*x^10 - a*d*x^8 - a* c*x^6), x)
\[ \int \frac {A+B x^2}{x^6 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2}}{x^{6} \sqrt {a - c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)/x**6/(d*x**2+c)/(-c*x**4+a)**(1/2),x)
Output:
Integral((A + B*x**2)/(x**6*sqrt(a - c*x**4)*(c + d*x**2)), x)
\[ \int \frac {A+B x^2}{x^6 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )} x^{6}} \,d x } \] Input:
integrate((B*x^2+A)/x^6/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*(d*x^2 + c)*x^6), x)
\[ \int \frac {A+B x^2}{x^6 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )} x^{6}} \,d x } \] Input:
integrate((B*x^2+A)/x^6/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*(d*x^2 + c)*x^6), x)
Timed out. \[ \int \frac {A+B x^2}{x^6 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {B\,x^2+A}{x^6\,\sqrt {a-c\,x^4}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((A + B*x^2)/(x^6*(a - c*x^4)^(1/2)*(c + d*x^2)),x)
Output:
int((A + B*x^2)/(x^6*(a - c*x^4)^(1/2)*(c + d*x^2)), x)
\[ \int \frac {A+B x^2}{x^6 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-3 \sqrt {-c \,x^{4}+a}\, a -5 \sqrt {-c \,x^{4}+a}\, b \,x^{2}-15 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{10}-c^{2} x^{8}+a d \,x^{6}+a c \,x^{4}}d x \right ) a^{2} d \,x^{5}-15 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{8}-c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a b d \,x^{5}+9 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{8}-c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a \,c^{2} x^{5}+9 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a c d \,x^{5}+5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{2} x^{5}+5 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b c d \,x^{5}}{15 a c \,x^{5}} \] Input:
int((B*x^2+A)/x^6/(d*x^2+c)/(-c*x^4+a)^(1/2),x)
Output:
( - 3*sqrt(a - c*x**4)*a - 5*sqrt(a - c*x**4)*b*x**2 - 15*int(sqrt(a - c*x **4)/(a*c*x**4 + a*d*x**6 - c**2*x**8 - c*d*x**10),x)*a**2*d*x**5 - 15*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*d*x* *5 + 9*int(sqrt(a - c*x**4)/(a*c*x**2 + a*d*x**4 - c**2*x**6 - c*d*x**8),x )*a*c**2*x**5 + 9*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x **6),x)*a*c*d*x**5 + 5*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*b*c**2*x**5 + 5*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*d*x**5)/(15*a*c*x**5)