Integrand size = 32, antiderivative size = 336 \[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=-\frac {(B c-A d) x \sqrt {a-c x^4}}{3 d^2}+\frac {B x^3 \sqrt {a-c x^4}}{5 d}-\frac {a^{3/4} \left (5 B c^3-5 A c^2 d-2 a B d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{3/4} d^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (5 \sqrt {c} (B c-A d) \left (3 c^3-2 a d^2\right )+3 \sqrt {a} d \left (5 B c^3-5 A c^2 d-2 a B d^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15 c^{3/4} d^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} (B c-A d) \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d^4 \sqrt {a-c x^4}} \] Output:
-1/3*(-A*d+B*c)*x*(-c*x^4+a)^(1/2)/d^2+1/5*B*x^3*(-c*x^4+a)^(1/2)/d-1/5*a^ (3/4)*(-5*A*c^2*d-2*B*a*d^2+5*B*c^3)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x /a^(1/4),I)/c^(3/4)/d^3/(-c*x^4+a)^(1/2)+1/15*a^(1/4)*(5*c^(1/2)*(-A*d+B*c )*(-2*a*d^2+3*c^3)+3*a^(1/2)*d*(-5*A*c^2*d-2*B*a*d^2+5*B*c^3))*(1-c*x^4/a) ^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(3/4)/d^4/(-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^(1/4)/d^4/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.62 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\frac {-3 i \sqrt {a} d \left (-5 B c^3+5 A c^2 d+2 a B d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (5 A \sqrt {c} d \left (3 c^3+3 \sqrt {a} c^{3/2} d-2 a d^2\right )+B \left (-15 c^{9/2}-15 \sqrt {a} c^3 d+10 a c^{3/2} d^2+6 a^{3/2} d^3\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {c} \left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^2 x \left (5 B c-5 A d-3 B d x^2\right ) \left (-a+c x^4\right )-15 i (B c-A d) \left (-c^3+a d^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{15 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \sqrt {c} d^4 \sqrt {a-c x^4}} \] Input:
Integrate[(x^2*(A + B*x^2)*Sqrt[a - c*x^4])/(c + d*x^2),x]
Output:
((-3*I)*Sqrt[a]*d*(-5*B*c^3 + 5*A*c^2*d + 2*a*B*d^2)*Sqrt[1 - (c*x^4)/a]*E llipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*(5*A*Sqrt[c]*d*(3* c^3 + 3*Sqrt[a]*c^(3/2)*d - 2*a*d^2) + B*(-15*c^(9/2) - 15*Sqrt[a]*c^3*d + 10*a*c^(3/2)*d^2 + 6*a^(3/2)*d^3))*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSin h[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + Sqrt[c]*(Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2 *x*(5*B*c - 5*A*d - 3*B*d*x^2)*(-a + c*x^4) - (15*I)*(B*c - A*d)*(-c^3 + a *d^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqr t[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(15*Sqrt[-(Sqrt[c]/Sqrt[a])]*Sqrt[c]*d^4*S qrt[a - c*x^4])
Time = 0.80 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.65, 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.
| \(\displaystyle \int \frac {x^2 \sqrt {a-c x^4} \left (A+B x^2\right )}{c+d x^2} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (\frac {\left (c^3-a d^2\right ) (B c-A d)}{d^4 \sqrt {a-c x^4}}-\frac {x^2 \left (-a B d^2-A c^2 d+B c^3\right )}{d^3 \sqrt {a-c x^4}}+\frac {-a A c d^3+a B c^2 d^2+A c^4 d-B c^5}{d^4 \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {c x^4 (B c-A d)}{d^2 \sqrt {a-c x^4}}-\frac {B c x^6}{d \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} \left (-a B d^2-A c^2 d+B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} d^3 \sqrt {a-c x^4}}-\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} \left (-a B d^2-A c^2 d+B c^3\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} d^3 \sqrt {a-c x^4}}+\frac {a^{5/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt [4]{c} d^2 \sqrt {a-c x^4}}+\frac {3 a^{7/4} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{3/4} d \sqrt {a-c x^4}}-\frac {3 a^{7/4} B \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{3/4} d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (c^3-a d^2\right ) \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d^4 \sqrt {a-c x^4}}-\frac {x \sqrt {a-c x^4} (B c-A d)}{3 d^2}+\frac {B x^3 \sqrt {a-c x^4}}{5 d}\) |
Input:
Int[(x^2*(A + B*x^2)*Sqrt[a - c*x^4])/(c + d*x^2),x]
Output:
-1/3*((B*c - A*d)*x*Sqrt[a - c*x^4])/d^2 + (B*x^3*Sqrt[a - c*x^4])/(5*d) - (3*a^(7/4)*B*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], - 1])/(5*c^(3/4)*d*Sqrt[a - c*x^4]) - (a^(3/4)*(B*c^3 - A*c^2*d - a*B*d^2)*S qrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*d^ 3*Sqrt[a - c*x^4]) + (3*a^(7/4)*B*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^ (1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*d*Sqrt[a - c*x^4]) + (a^(5/4)*(B*c - A* d)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(1 /4)*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]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d^4*Sqrt[a - c*x^4]) + (a^(3/4)*(B*c^3 - A*c^2*d - a*B*d^2)*Sqrt[1 - (c*x^4)/a]*Ellipti cF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*d^3*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^(1/4)*d^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 = 2.31 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(\frac {x \left (3 B \,x^{2} d +5 A d -5 B c \right ) \sqrt {-c \,x^{4}+a}}{15 d^{2}}+\frac {\frac {5 \left (2 A a \,d^{3}-3 A \,c^{3} d -2 B a c \,d^{2}+3 B \,c^{4}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \left (5 A \,c^{2} d +2 B a \,d^{2}-5 B \,c^{3}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {15 \left (A a \,d^{3}-A \,c^{3} d -B a c \,d^{2}+B \,c^{4}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{15 d^{2}}\) | \(368\) |
| default | \(\frac {A d \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 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 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+B d \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5}-\frac {2 a^{\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 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )-B c \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 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 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d^{2}}-\frac {c \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 )}{d^{2}}\) | \(698\) |
| elliptic | \(\text {Expression too large to display}\) | \(1154\) |
Input:
int(x^2*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
1/15*x*(3*B*d*x^2+5*A*d-5*B*c)*(-c*x^4+a)^(1/2)/d^2+1/15/d^2*(5*(2*A*a*d^3 -3*A*c^3*d-2*B*a*c*d^2+3*B*c^4)/d^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/d*(5*A*c^2*d+2*B*a*d^2-5*B*c^3)*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)-Ell ipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))-15*(A*a*d^3-A*c^3*d-B*a*c*d^2+B*c^4)/ d^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)*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. \[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\text {Timed out} \] Input:
integrate(x^2*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int \frac {x^{2} \left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{c + d x^{2}}\, dx \] Input:
integrate(x**2*(B*x**2+A)*(-c*x**4+a)**(1/2)/(d*x**2+c),x)
Output:
Integral(x**2*(A + B*x**2)*sqrt(a - c*x**4)/(c + d*x**2), x)
\[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} x^{2}}{d x^{2} + c} \,d x } \] Input:
integrate(x^2*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*x^2/(d*x^2 + c), x)
\[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} x^{2}}{d x^{2} + c} \,d x } \] Input:
integrate(x^2*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*x^2/(d*x^2 + c), x)
Timed out. \[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\int \frac {x^2\,\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{d\,x^2+c} \,d x \] Input:
int((x^2*(A + B*x^2)*(a - c*x^4)^(1/2))/(c + d*x^2),x)
Output:
int((x^2*(A + B*x^2)*(a - c*x^4)^(1/2))/(c + d*x^2), x)
\[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{c+d x^2} \, dx=\frac {5 \sqrt {-c \,x^{4}+a}\, a d x -5 \sqrt {-c \,x^{4}+a}\, b c x +3 \sqrt {-c \,x^{4}+a}\, b d \,x^{3}-5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} c d +5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b \,c^{2}+6 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b \,d^{2}+15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a \,c^{2} d -15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{3}+10 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a^{2} d^{2}-4 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c d \,x^{6}-c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b c d}{15 d^{2}} \] Input:
int(x^2*(B*x^2+A)*(-c*x^4+a)^(1/2)/(d*x^2+c),x)
Output:
(5*sqrt(a - c*x**4)*a*d*x - 5*sqrt(a - c*x**4)*b*c*x + 3*sqrt(a - c*x**4)* b*d*x**3 - 5*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6), x)*a**2*c*d + 5*int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x** 6),x)*a*b*c**2 + 6*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*b*d**2 + 15*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**2*d - 15*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**3 + 10*int((sqrt(a - c*x**4)*x* *2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a**2*d**2 - 4*int((sqrt(a - c*x**4)*x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*b*c*d)/(15*d** 2)