Integrand size = 32, antiderivative size = 275 \[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {A \sqrt {a-c x^4}}{3 a c x^3}-\frac {(B c-A d) \sqrt {a-c x^4}}{a c^2 x}-\frac {(B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}+\frac {\left (A c^{3/2}+3 \sqrt {a} (B c-A d)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 a^{3/4} c^{7/4} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} d (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{13/4} \sqrt {a-c x^4}} \] Output:
-1/3*A*(-c*x^4+a)^(1/2)/a/c/x^3-(-A*d+B*c)*(-c*x^4+a)^(1/2)/a/c^2/x-(-A*d+ B*c)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4)/c^(7/4)/(-c* x^4+a)^(1/2)+1/3*(A*c^(3/2)+3*a^(1/2)*(-A*d+B*c))*(1-c*x^4/a)^(1/2)*Ellipt icF(c^(1/4)*x/a^(1/4),I)/a^(3/4)/c^(7/4)/(-c*x^4+a)^(1/2)-a^(1/4)*d*(-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)/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.96 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.67 \[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2-3 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 x^2+3 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d x^2+A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 x^4+3 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 x^6-3 A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d x^6+3 i \sqrt {a} c^{3/2} (B c-A d) x^3 \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i c^{3/2} \left (A c^{3/2}+3 \sqrt {a} (B c-A d)\right ) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i a B c d x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i a A d^2 x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{3 a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 x^3 \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2)/(x^4*(c + d*x^2)*Sqrt[a - c*x^4]),x]
Output:
(-(a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2) - 3*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2* x^2 + 3*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*x^2 + A*Sqrt[-(Sqrt[c]/Sqrt[a])]* c^3*x^4 + 3*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*x^6 - 3*A*Sqrt[-(Sqrt[c]/Sqrt[a ])]*c^2*d*x^6 + (3*I)*Sqrt[a]*c^(3/2)*(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*c^(3/2 ) + 3*Sqrt[a]*(B*c - A*d))*x^3*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqr t[-(Sqrt[c]/Sqrt[a])]*x], -1] + (3*I)*a*B*c*d*x^3*Sqrt[1 - (c*x^4)/a]*Elli pticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*a*A*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*a*Sqrt[-(Sqrt[c]/Sqrt[a])] *c^3*x^3*Sqrt[a - c*x^4])
Time = 0.58 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.14, 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.
| \(\displaystyle \int \frac {A+B x^2}{x^4 \sqrt {a-c x^4} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (-\frac {d (B c-A d)}{c^2 \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {B c-A d}{c^2 x^2 \sqrt {a-c x^4}}+\frac {A}{c x^4 \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 a^{3/4} \sqrt [4]{c} \sqrt {a-c x^4}}+\frac {\sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}-\frac {\sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{13/4} \sqrt {a-c x^4}}-\frac {\sqrt {a-c x^4} (B c-A d)}{a c^2 x}-\frac {A \sqrt {a-c x^4}}{3 a c x^3}\) |
Input:
Int[(A + B*x^2)/(x^4*(c + d*x^2)*Sqrt[a - c*x^4]),x]
Output:
-1/3*(A*Sqrt[a - c*x^4])/(a*c*x^3) - ((B*c - A*d)*Sqrt[a - c*x^4])/(a*c^2* x) - ((B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4) ], -1])/(a^(1/4)*c^(7/4)*Sqrt[a - c*x^4]) + (A*Sqrt[1 - (c*x^4)/a]*Ellipti cF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*a^(3/4)*c^(1/4)*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])/(a^(1/4)*c^(7/4)*Sqrt[a - c*x^4]) - (a^(1/4)*d*(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)*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 = 2.09 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(\frac {A \left (-\frac {\sqrt {-c \,x^{4}+a}}{3 a \,x^{3}}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c}-\frac {\left (A d -B c \right ) \left (-\frac {\sqrt {-c \,x^{4}+a}}{a x}+\frac {\sqrt {c}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{c^{2}}+\frac {d \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{c^{3} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(314\) |
| risch | \(-\frac {\sqrt {-c \,x^{4}+a}\, \left (-3 A d \,x^{2}+3 B c \,x^{2}+A c \right )}{3 c^{2} a \,x^{3}}+\frac {\frac {A \,c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 B \,c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 A \sqrt {c}\, d \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 a d \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 a \,c^{2}}\) | \(398\) |
| elliptic | \(-\frac {A \sqrt {-c \,x^{4}+a}}{3 a c \,x^{3}}+\frac {\left (A d -B c \right ) \sqrt {-c \,x^{4}+a}}{a \,c^{2} x}+\frac {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 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}{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 {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{\sqrt {c}\, \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}{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 ) B}{\sqrt {c}\, \sqrt {a}\, \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}{c^{3} \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}{c^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(597\) |
Input:
int((B*x^2+A)/x^4/(d*x^2+c)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
A/c*(-1/3/a*(-c*x^4+a)^(1/2)/x^3+1/3*c/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)*Ellipt icF(x*(c^(1/2)/a^(1/2))^(1/2),I))-(A*d-B*c)/c^2*(-1/a*(-c*x^4+a)^(1/2)/x+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)))+d*(A*d-B*c)/c^3/(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. \[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)/x^4/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2}}{x^{4} \sqrt {a - c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)/x**4/(d*x**2+c)/(-c*x**4+a)**(1/2),x)
Output:
Integral((A + B*x**2)/(x**4*sqrt(a - c*x**4)*(c + d*x**2)), x)
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((B*x^2+A)/x^4/(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^4), x)
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((B*x^2+A)/x^4/(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^4), x)
Timed out. \[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {B\,x^2+A}{x^4\,\sqrt {a-c\,x^4}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((A + B*x^2)/(x^4*(a - c*x^4)^(1/2)*(c + d*x^2)),x)
Output:
int((A + B*x^2)/(x^4*(a - c*x^4)^(1/2)*(c + d*x^2)), x)
\[ \int \frac {A+B x^2}{x^4 \left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-\sqrt {-c \,x^{4}+a}-3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{8}-c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a d \,x^{3}+3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{8}-c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) b c \,x^{3}+\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) c^{2} x^{3}+\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) c d \,x^{3}}{3 c \,x^{3}} \] Input:
int((B*x^2+A)/x^4/(d*x^2+c)/(-c*x^4+a)^(1/2),x)
Output:
( - sqrt(a - c*x**4) - 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*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)*b*c*x**3 + int(sqrt(a - c*x**4)/(a*c + a*d*x** 2 - c**2*x**4 - c*d*x**6),x)*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)*c*d*x**3)/(3*c*x**3)