Integrand size = 28, antiderivative size = 379 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {x \left (A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x^2\right )}{2 a c \sqrt {a+c x^4}}-\frac {\left (B c d^2+2 A c d e-3 a B e^2\right ) x \sqrt {a+c x^4}}{2 a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\left (B c d^2+2 A c d e-3 a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt {a+c x^4}}+\frac {\left (A c^{3/2} d^2+3 a^{3/2} B e^2+a \sqrt {c} e (2 B d+A e)-\sqrt {a} c d (B d+2 A e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} c^{7/4} \sqrt {a+c x^4}} \] Output:
1/2*x*(A*c*d^2-2*a*B*d*e-A*a*e^2+(2*A*c*d*e-B*a*e^2+B*c*d^2)*x^2)/a/c/(c*x ^4+a)^(1/2)-1/2*(2*A*c*d*e-3*B*a*e^2+B*c*d^2)*x*(c*x^4+a)^(1/2)/a/c^(3/2)/ (a^(1/2)+c^(1/2)*x^2)+1/2*(2*A*c*d*e-3*B*a*e^2+B*c*d^2)*(a^(1/2)+c^(1/2)*x ^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/ 4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/c^(7/4)/(c*x^4+a)^(1/2)+1/4*(A*c^(3/2) *d^2+3*a^(3/2)*B*e^2+a*c^(1/2)*e*(A*e+2*B*d)-a^(1/2)*c*d*(2*A*e+B*d))*(a^( 1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiA M(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(5/4)/c^(7/4)/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.21 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.44 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {3 A \left (c d^2-a e^2\right ) x+6 a B e x \left (-d+e x^2\right )+3 \left (A c d^2+2 a B d e+a A e^2\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+2 \left (B c d^2+2 A c d e-3 a B e^2\right ) x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {c x^4}{a}\right )}{6 a c \sqrt {a+c x^4}} \] Input:
Integrate[((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2),x]
Output:
(3*A*(c*d^2 - a*e^2)*x + 6*a*B*e*x*(-d + e*x^2) + 3*(A*c*d^2 + 2*a*B*d*e + a*A*e^2)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4) /a)] + 2*(B*c*d^2 + 2*A*c*d*e - 3*a*B*e^2)*x^3*Sqrt[1 + (c*x^4)/a]*Hyperge ometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(6*a*c*Sqrt[a + c*x^4])
Time = 0.83 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2259, 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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {x^2 \left (-a B e^2+2 A c d e+B c d^2\right )-a A e^2-2 a B d e+A c d^2}{c \left (a+c x^4\right )^{3/2}}+\frac {e (A e+2 B d)}{c \sqrt {a+c x^4}}+\frac {B e^2 x^2}{c \sqrt {a+c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (-\frac {\sqrt {c} \left (-a A e^2-2 a B d e+A c d^2\right )}{\sqrt {a}}-a B e^2+2 A c d e+B c d^2\right )}{4 a^{3/4} c^{7/4} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) \left (-a B e^2+2 A c d e+B c d^2\right )}{2 a^{3/4} c^{7/4} \sqrt {a+c x^4}}+\frac {e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (A e+2 B d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4} \left (-a B e^2+2 A c d e+B c d^2\right )}{2 a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {x \left (x^2 \left (-a B e^2+2 A c d e+B c d^2\right )-a A e^2-2 a B d e+A c d^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 c^{7/4} \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {a+c x^4}}+\frac {B e^2 x \sqrt {a+c x^4}}{c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}\) |
Input:
Int[((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2),x]
Output:
(x*(A*c*d^2 - 2*a*B*d*e - a*A*e^2 + (B*c*d^2 + 2*A*c*d*e - a*B*e^2)*x^2))/ (2*a*c*Sqrt[a + c*x^4]) + (B*e^2*x*Sqrt[a + c*x^4])/(c^(3/2)*(Sqrt[a] + Sq rt[c]*x^2)) - ((B*c*d^2 + 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*c^( 3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*B*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqr t[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^ (1/4)], 1/2])/(c^(7/4)*Sqrt[a + c*x^4]) + ((B*c*d^2 + 2*A*c*d*e - a*B*e^2) *(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip ticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*c^(7/4)*Sqrt[a + c*x^ 4]) + (a^(1/4)*B*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + S qrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(7/4)*S qrt[a + c*x^4]) + (e*(2*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4 )/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2] )/(2*a^(1/4)*c^(5/4)*Sqrt[a + c*x^4]) - ((B*c*d^2 + 2*A*c*d*e - a*B*e^2 - (Sqrt[c]*(A*c*d^2 - 2*a*B*d*e - a*A*e^2))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2) *Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x )/a^(1/4)], 1/2])/(4*a^(3/4)*c^(7/4)*Sqrt[a + c*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 2] && IntegerQ[q]
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.85
| method | result | size |
| elliptic | \(-\frac {2 c \left (-\frac {\left (2 A c d e -B a \,e^{2}+B c \,d^{2}\right ) x^{3}}{4 a \,c^{2}}+\frac {\left (A a \,e^{2}-A c \,d^{2}+2 a B d e \right ) x}{4 c^{2} a}\right )}{\sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\left (\frac {e \left (A e +2 B d \right )}{c}-\frac {A a \,e^{2}-A c \,d^{2}+2 a B d e}{2 a c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (\frac {B \,e^{2}}{c}-\frac {2 A c d e -B a \,e^{2}+B c \,d^{2}}{2 a c}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(323\) |
| default | \(A \,d^{2} \left (\frac {x}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+e \left (A e +2 B d \right ) \left (-\frac {x}{2 c \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+d \left (2 A e +B d \right ) \left (\frac {x^{3}}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+B \,e^{2} \left (-\frac {x^{3}}{2 c \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )\) | \(454\) |
Input:
int((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*c*(-1/4*(2*A*c*d*e-B*a*e^2+B*c*d^2)/a/c^2*x^3+1/4*(A*a*e^2-A*c*d^2+2*B* a*d*e)/c^2/a*x)/(c*(a/c+x^4))^(1/2)+(e*(A*e+2*B*d)/c-1/2*(A*a*e^2-A*c*d^2+ 2*B*a*d*e)/a/c)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)* (1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^( 1/2))^(1/2),I)+I*(B*e^2/c-1/2*(2*A*c*d*e-B*a*e^2+B*c*d^2)/a/c)*a^(1/2)/(I* c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^ (1/2))^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2 ),I)-EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.88 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=-\frac {{\left ({\left (B a c^{2} d^{2} + 2 \, A a c^{2} d e - 3 \, B a^{2} c e^{2}\right )} x^{5} + {\left (B a^{2} c d^{2} + 2 \, A a^{2} c d e - 3 \, B a^{3} e^{2}\right )} x\right )} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (2 \, {\left (A + B\right )} a c^{2} d e + {\left (B a c^{2} + A c^{3}\right )} d^{2} - {\left (3 \, B a^{2} c - A a c^{2}\right )} e^{2}\right )} x^{5} + {\left (2 \, {\left (A + B\right )} a^{2} c d e + {\left (B a^{2} c + A a c^{2}\right )} d^{2} - {\left (3 \, B a^{3} - A a^{2} c\right )} e^{2}\right )} x\right )} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (2 \, B a^{2} c e^{2} x^{4} - B a^{2} c d^{2} - 2 \, A a^{2} c d e + 3 \, B a^{3} e^{2} + {\left (A a c^{2} d^{2} - 2 \, B a^{2} c d e - A a^{2} c e^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + a}}{2 \, {\left (a^{2} c^{3} x^{5} + a^{3} c^{2} x\right )}} \] Input:
integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/2*(((B*a*c^2*d^2 + 2*A*a*c^2*d*e - 3*B*a^2*c*e^2)*x^5 + (B*a^2*c*d^2 + 2*A*a^2*c*d*e - 3*B*a^3*e^2)*x)*sqrt(c)*(-a/c)^(3/4)*elliptic_e(arcsin((-a /c)^(1/4)/x), -1) - ((2*(A + B)*a*c^2*d*e + (B*a*c^2 + A*c^3)*d^2 - (3*B*a ^2*c - A*a*c^2)*e^2)*x^5 + (2*(A + B)*a^2*c*d*e + (B*a^2*c + A*a*c^2)*d^2 - (3*B*a^3 - A*a^2*c)*e^2)*x)*sqrt(c)*(-a/c)^(3/4)*elliptic_f(arcsin((-a/c )^(1/4)/x), -1) - (2*B*a^2*c*e^2*x^4 - B*a^2*c*d^2 - 2*A*a^2*c*d*e + 3*B*a ^3*e^2 + (A*a*c^2*d^2 - 2*B*a^2*c*d*e - A*a^2*c*e^2)*x^2)*sqrt(c*x^4 + a)) /(a^2*c^3*x^5 + a^3*c^2*x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{2}}{\left (a + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**2/(c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)*(d + e*x**2)**2/(a + c*x**4)**(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^2/(c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^2/(c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^2}{{\left (c\,x^4+a\right )}^{3/2}} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2),x)
Output:
int(((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {-\sqrt {c \,x^{4}+a}\, a \,e^{2} x -2 \sqrt {c \,x^{4}+a}\, b d e x +\sqrt {c \,x^{4}+a}\, b \,e^{2} x^{3}+\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{3} e^{2}+2 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} b d e +\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c \,d^{2}+\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c \,e^{2} x^{4}+2 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a b c d e \,x^{4}+\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{2} d^{2} x^{4}-3 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} b \,e^{2}+2 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c d e +\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a b c \,d^{2}-3 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a b c \,e^{2} x^{4}+2 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{2} d e \,x^{4}+\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) b \,c^{2} d^{2} x^{4}}{c \left (c \,x^{4}+a \right )} \] Input:
int((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x)
Output:
( - sqrt(a + c*x**4)*a*e**2*x - 2*sqrt(a + c*x**4)*b*d*e*x + sqrt(a + c*x* *4)*b*e**2*x**3 + int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)* a**3*e**2 + 2*int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2 *b*d*e + int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*c*d* *2 + int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*c*e**2*x **4 + 2*int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*b*c*d*e* x**4 + int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c**2*d**2 *x**4 - 3*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a **2*b*e**2 + 2*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8) ,x)*a**2*c*d*e + int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x** 8),x)*a*b*c*d**2 - 3*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2 *x**8),x)*a*b*c*e**2*x**4 + 2*int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x* *4 + c**2*x**8),x)*a*c**2*d*e*x**4 + int((sqrt(a + c*x**4)*x**2)/(a**2 + 2 *a*c*x**4 + c**2*x**8),x)*b*c**2*d**2*x**4)/(c*(a + c*x**4))