Integrand size = 32, antiderivative size = 225 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=-\frac {(C d+B e) x \sqrt {a-c x^4}}{3 c}-\frac {C e x^3 \sqrt {a-c x^4}}{5 c}+\frac {a^{3/4} (5 B c d+5 A c e+3 a C e) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (5 \sqrt {c} (3 A c d+a C d+a B e)-3 \sqrt {a} (5 B c d+5 A c e+3 a C e)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15 c^{7/4} \sqrt {a-c x^4}} \] Output:
-1/3*(B*e+C*d)*x*(-c*x^4+a)^(1/2)/c-1/5*C*e*x^3*(-c*x^4+a)^(1/2)/c+1/5*a^( 3/4)*(5*A*c*e+5*B*c*d+3*C*a*e)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/ 4),I)/c^(7/4)/(-c*x^4+a)^(1/2)+1/15*a^(1/4)*(5*c^(1/2)*(3*A*c*d+B*a*e+C*a* d)-3*a^(1/2)*(5*A*c*e+5*B*c*d+3*C*a*e))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4 )*x/a^(1/4),I)/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 = 10.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\frac {x \left (5 C d+5 B e+3 C e x^2\right ) \left (-a+c x^4\right )+5 (3 A c d+a C d+a B e) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+(5 B c d+5 A c e+3 a C e) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )}{15 c \sqrt {a-c x^4}} \] Input:
Integrate[((d + e*x^2)*(A + B*x^2 + C*x^4))/Sqrt[a - c*x^4],x]
Output:
(x*(5*C*d + 5*B*e + 3*C*e*x^2)*(-a + c*x^4) + 5*(3*A*c*d + a*C*d + a*B*e)* x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + (5*B*c *d + 5*A*c*e + 3*a*C*e)*x^3*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4 , 7/4, (c*x^4)/a])/(15*c*Sqrt[a - c*x^4])
Time = 0.58 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.81, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {x^2 (A e+B d)}{\sqrt {a-c x^4}}+\frac {A d}{\sqrt {a-c x^4}}+\frac {x^4 (B e+C d)}{\sqrt {a-c x^4}}+\frac {C e x^6}{\sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} (A e+B d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} (A e+B d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {a^{5/4} \sqrt {1-\frac {c x^4}{a}} (B e+C d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{5/4} \sqrt {a-c x^4}}-\frac {3 a^{7/4} C e \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {3 a^{7/4} C e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} A d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {x \sqrt {a-c x^4} (B e+C d)}{3 c}-\frac {C e x^3 \sqrt {a-c x^4}}{5 c}\) |
Input:
Int[((d + e*x^2)*(A + B*x^2 + C*x^4))/Sqrt[a - c*x^4],x]
Output:
-1/3*((C*d + B*e)*x*Sqrt[a - c*x^4])/c - (C*e*x^3*Sqrt[a - c*x^4])/(5*c) + (3*a^(7/4)*C*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(7/4)*Sqrt[a - c*x^4]) + (a^(3/4)*(B*d + A*e)*Sqrt[1 - (c*x^4)/ a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*A*d*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], - 1])/(c^(1/4)*Sqrt[a - c*x^4]) - (3*a^(7/4)*C*e*Sqrt[1 - (c*x^4)/a]*Ellipti cF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(7/4)*Sqrt[a - c*x^4]) - (a^(3/4 )*(B*d + A*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], - 1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(5/4)*(C*d + B*e)*Sqrt[1 - (c*x^4)/a]*E llipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(5/4)*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]
Time = 1.73 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.01
| method | result | size |
| elliptic | \(-\frac {C e \,x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {\left (B e +C d \right ) x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (A d +\frac {a \left (B e +C d \right )}{3 c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (A e +B d +\frac {3 a e C}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(227\) |
| risch | \(-\frac {x \left (3 C \,x^{2} e +5 B e +5 C d \right ) \sqrt {-c \,x^{4}+a}}{15 c}+\frac {-\frac {\left (15 A c e +15 B c d +9 C a e \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {15 A c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {5 B a e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {5 C a d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{15 c}\) | \(344\) |
| default | \(\frac {A d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (A e +B d \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\left (B e +C d \right ) \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e C \left (-\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(363\) |
Input:
int((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/5*C*e*x^3*(-c*x^4+a)^(1/2)/c-1/3*(B*e+C*d)*x*(-c*x^4+a)^(1/2)/c+(A*d+1/ 3*a/c*(B*e+C*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)-(A*e+B*d+3/5*a/c*e*C)*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)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2 ),I))
Time = 0.09 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=-\frac {3 \, {\left (5 \, B a c d + {\left (3 \, C a^{2} + 5 \, A a c\right )} e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (5 \, {\left ({\left (3 \, B + C\right )} a c + 3 \, A c^{2}\right )} d + {\left (9 \, C a^{2} + 5 \, {\left (3 \, A + B\right )} a c\right )} e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3 \, C a c e x^{4} + 15 \, B a c d + 5 \, {\left (C a c d + B a c e\right )} x^{2} + 3 \, {\left (3 \, C a^{2} + 5 \, A a c\right )} e\right )} \sqrt {-c x^{4} + a}}{15 \, a c^{2} x} \] Input:
integrate((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x, algorithm="fricas" )
Output:
-1/15*(3*(5*B*a*c*d + (3*C*a^2 + 5*A*a*c)*e)*sqrt(-c)*x*(a/c)^(3/4)*ellipt ic_e(arcsin((a/c)^(1/4)/x), -1) - (5*((3*B + C)*a*c + 3*A*c^2)*d + (9*C*a^ 2 + 5*(3*A + B)*a*c)*e)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^(1/ 4)/x), -1) + (3*C*a*c*e*x^4 + 15*B*a*c*d + 5*(C*a*c*d + B*a*c*e)*x^2 + 3*( 3*C*a^2 + 5*A*a*c)*e)*sqrt(-c*x^4 + a))/(a*c^2*x)
Time = 3.04 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.16 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\frac {A d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {A e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {B d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {B e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {C d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {C e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((e*x**2+d)*(C*x**4+B*x**2+A)/(-c*x**4+a)**(1/2),x)
Output:
A*d*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4* sqrt(a)*gamma(5/4)) + A*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**4 *exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(7/4)) + B*d*x**3*gamma(3/4)*hyper(( 1/2, 3/4), (7/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(7/4)) + B* e*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4 *sqrt(a)*gamma(9/4)) + C*d*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x** 4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(9/4)) + C*e*x**7*gamma(7/4)*hyper( (1/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(11/4))
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)/sqrt(-c*x^4 + a), x)
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)/sqrt(-c*x^4 + a), x)
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (C\,x^4+B\,x^2+A\right )}{\sqrt {a-c\,x^4}} \,d x \] Input:
int(((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(1/2),x)
Output:
int(((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(1/2), x)
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\frac {-5 \sqrt {-c \,x^{4}+a}\, b e x -5 \sqrt {-c \,x^{4}+a}\, c d x -3 \sqrt {-c \,x^{4}+a}\, c e \,x^{3}+5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a b e +20 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a c d +24 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a c e +15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) b c d}{15 c} \] Input:
int((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x)
Output:
( - 5*sqrt(a - c*x**4)*b*e*x - 5*sqrt(a - c*x**4)*c*d*x - 3*sqrt(a - c*x** 4)*c*e*x**3 + 5*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a*b*e + 20*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a*c*d + 24*int((sqrt(a - c*x**4)*x**2)/(a - c*x **4),x)*a*c*e + 15*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*b*c*d)/(15* c)