Integrand size = 32, antiderivative size = 280 \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \left (5 (7 A c d+a C d+a B e)+7 (3 B c d+3 A c e+a C e) x^2\right ) \sqrt {a-c x^4}}{105 c}-\frac {(C d+B e) x \left (a-c x^4\right )^{3/2}}{7 c}-\frac {C e x^3 \left (a-c x^4\right )^{3/2}}{9 c}+\frac {2 a^{7/4} (3 B c d+3 A c e+a C e) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 c^{7/4} \sqrt {a-c x^4}}+\frac {2 a^{5/4} \left (5 \sqrt {c} (7 A c d+a C d+a B e)-7 \sqrt {a} (3 B c d+3 A c e+a C e)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{105 c^{7/4} \sqrt {a-c x^4}} \] Output:
1/105*x*(35*A*c*d+5*B*a*e+5*C*a*d+7*(3*A*c*e+3*B*c*d+C*a*e)*x^2)*(-c*x^4+a )^(1/2)/c-1/7*(B*e+C*d)*x*(-c*x^4+a)^(3/2)/c-1/9*C*e*x^3*(-c*x^4+a)^(3/2)/ c+2/15*a^(7/4)*(3*A*c*e+3*B*c*d+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)+2/105*a^(5/4)*(5*c^(1/2)*(7*A*c*d+B *a*e+C*a*d)-7*a^(1/2)*(3*A*c*e+3*B*c*d+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 = 10.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.53 \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \sqrt {a-c x^4} \left (\left (9 C d+9 B e+7 C e x^2\right ) \left (-a+c x^4\right ) \sqrt {1-\frac {c x^4}{a}}+9 (7 A c d+a C d+a B e) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )+7 (3 B c d+3 A c e+a C e) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{63 c \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)*Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4),x]
Output:
(x*Sqrt[a - c*x^4]*((9*C*d + 9*B*e + 7*C*e*x^2)*(-a + c*x^4)*Sqrt[1 - (c*x ^4)/a] + 9*(7*A*c*d + a*C*d + a*B*e)*Hypergeometric2F1[-1/2, 1/4, 5/4, (c* x^4)/a] + 7*(3*B*c*d + 3*A*c*e + a*C*e)*x^2*Hypergeometric2F1[-1/2, 3/4, 7 /4, (c*x^4)/a]))/(63*c*Sqrt[1 - (c*x^4)/a])
Leaf count is larger than twice the leaf count of optimal. \(738\) vs. \(2(280)=560\).
Time = 0.97 (sec) , antiderivative size = 738, normalized size of antiderivative = 2.64, 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.
| \(\displaystyle \int \sqrt {a-c x^4} \left (d+e x^2\right ) \left (A+B x^2+C x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (-\frac {x^4 (-a B e-a C d+A c d)}{\sqrt {a-c x^4}}-\frac {x^6 (-a C e+A c e+B c d)}{\sqrt {a-c x^4}}+\frac {a x^2 (A e+B d)}{\sqrt {a-c x^4}}+\frac {a A d}{\sqrt {a-c x^4}}-\frac {c x^8 (B e+C d)}{\sqrt {a-c x^4}}-\frac {c C e x^{10}}{\sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{5/4} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) (-a B e-a C d+A c d)}{3 c^{5/4} \sqrt {a-c x^4}}+\frac {3 a^{7/4} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) (-a C e+A c e+B c d)}{5 c^{7/4} \sqrt {a-c x^4}}-\frac {3 a^{7/4} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) (-a C e+A c e+B c d)}{5 c^{7/4} \sqrt {a-c x^4}}-\frac {a^{7/4} \sqrt {1-\frac {c x^4}{a}} (A e+B d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {a^{7/4} \sqrt {1-\frac {c x^4}{a}} (A e+B d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {a^{5/4} A d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {5 a^{9/4} \sqrt {1-\frac {c x^4}{a}} (B e+C d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{21 c^{5/4} \sqrt {a-c x^4}}+\frac {7 a^{11/4} C e \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15 c^{7/4} \sqrt {a-c x^4}}-\frac {7 a^{11/4} C e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 c^{7/4} \sqrt {a-c x^4}}+\frac {x \sqrt {a-c x^4} (-a B e-a C d+A c d)}{3 c}+\frac {x^3 \sqrt {a-c x^4} (-a C e+A c e+B c d)}{5 c}+\frac {5 a x \sqrt {a-c x^4} (B e+C d)}{21 c}+\frac {1}{7} x^5 \sqrt {a-c x^4} (B e+C d)+\frac {1}{9} C e x^7 \sqrt {a-c x^4}+\frac {7 a C e x^3 \sqrt {a-c x^4}}{45 c}\) |
Input:
Int[(d + e*x^2)*Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4),x]
Output:
(5*a*(C*d + B*e)*x*Sqrt[a - c*x^4])/(21*c) + ((A*c*d - a*C*d - a*B*e)*x*Sq rt[a - c*x^4])/(3*c) + (7*a*C*e*x^3*Sqrt[a - c*x^4])/(45*c) + ((B*c*d + A* c*e - a*C*e)*x^3*Sqrt[a - c*x^4])/(5*c) + ((C*d + B*e)*x^5*Sqrt[a - c*x^4] )/7 + (C*e*x^7*Sqrt[a - c*x^4])/9 - (7*a^(11/4)*C*e*Sqrt[1 - (c*x^4)/a]*El lipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(15*c^(7/4)*Sqrt[a - c*x^4]) + ( a^(7/4)*(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]) - (3*a^(7/4)*(B*c*d + A*c*e - a*C*e)*S qrt[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^(5/4)*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]) + (7*a^(11/4)*C*e*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(15*c^(7/4)*Sqrt[a - c*x^4]) - (a^(7/4)*(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]) - (5*a^(9/4)*(C*d + B*e) *Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(21*c^(5/ 4)*Sqrt[a - c*x^4]) - (a^(5/4)*(A*c*d - a*C*d - a*B*e)*Sqrt[1 - (c*x^4)/a] *EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(5/4)*Sqrt[a - c*x^4]) + (3*a^(7/4)*(B*c*d + A*c*e - a*C*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[( c^(1/4)*x)/a^(1/4)], -1])/(5*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]
Time = 1.47 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.26
| method | result | size |
| elliptic | \(\frac {e C \,x^{7} \sqrt {-c \,x^{4}+a}}{9}-\frac {\left (-B c e -C c d \right ) x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {\left (-A c e -B c d +\frac {2}{9} C a e \right ) x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {\left (-A c d +B a e +C a d +\frac {5 a \left (-B c e -C c d \right )}{7 c}\right ) x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (a d A +\frac {a \left (-A c d +B a e +C a d +\frac {5 a \left (-B c e -C c d \right )}{7 c}\right )}{3 c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (A a e +B a d +\frac {3 a \left (-A c e -B c d +\frac {2}{9} C a e \right )}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(352\) |
| risch | \(\frac {x \left (35 e C \,x^{6} c +45 B c e \,x^{4}+45 C c d \,x^{4}+63 A c e \,x^{2}+63 B c d \,x^{2}-14 C a e \,x^{2}+105 A c d -30 B a e -30 C a d \right ) \sqrt {-c \,x^{4}+a}}{315 c}+\frac {2 a \left (-\frac {\left (21 A c e +21 B c d +7 C a e \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {35 A c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {5 B a e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {5 C a d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{105 c}\) | \(393\) |
| default | \(A d \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+\left (A e +B d \right ) \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5}-\frac {2 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )+\left (B e +C d \right ) \left (\frac {x^{5} \sqrt {-c \,x^{4}+a}}{7}-\frac {2 a x \sqrt {-c \,x^{4}+a}}{21 c}+\frac {2 a^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{21 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e C \left (\frac {x^{7} \sqrt {-c \,x^{4}+a}}{9}-\frac {2 a \,x^{3} \sqrt {-c \,x^{4}+a}}{45 c}-\frac {2 a^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{15 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(431\) |
Input:
int((e*x^2+d)*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/9*e*C*x^7*(-c*x^4+a)^(1/2)-1/7*(-B*c*e-C*c*d)/c*x^5*(-c*x^4+a)^(1/2)-1/5 *(-A*c*e-B*c*d+2/9*C*a*e)/c*x^3*(-c*x^4+a)^(1/2)-1/3*(-A*c*d+B*a*e+C*a*d+5 /7*a/c*(-B*c*e-C*c*d))/c*x*(-c*x^4+a)^(1/2)+(a*d*A+1/3*a/c*(-A*c*d+B*a*e+C *a*d+5/7*a/c*(-B*c*e-C*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*a*e+B*a*d+3/5*a/c*(-A*c*e-B*c*d+2/9*C*a*e))*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 = 0.08 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.86 \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=-\frac {42 \, {\left (3 \, B a c d + {\left (C a^{2} + 3 \, A a c\right )} e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 6 \, {\left ({\left ({\left (21 \, B + 5 \, C\right )} a c + 35 \, A c^{2}\right )} d + {\left (7 \, C a^{2} + {\left (21 \, A + 5 \, B\right )} a c\right )} e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (35 \, C c^{2} e x^{8} + 45 \, {\left (C c^{2} d + B c^{2} e\right )} x^{6} + 7 \, {\left (9 \, B c^{2} d - {\left (2 \, C a c - 9 \, A c^{2}\right )} e\right )} x^{4} - 126 \, B a c d - 15 \, {\left (2 \, B a c e + {\left (2 \, C a c - 7 \, A c^{2}\right )} d\right )} x^{2} - 42 \, {\left (C a^{2} + 3 \, A a c\right )} e\right )} \sqrt {-c x^{4} + a}}{315 \, c^{2} x} \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="fricas" )
Output:
-1/315*(42*(3*B*a*c*d + (C*a^2 + 3*A*a*c)*e)*sqrt(-c)*x*(a/c)^(3/4)*ellipt ic_e(arcsin((a/c)^(1/4)/x), -1) - 6*(((21*B + 5*C)*a*c + 35*A*c^2)*d + (7* C*a^2 + (21*A + 5*B)*a*c)*e)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/c )^(1/4)/x), -1) - (35*C*c^2*e*x^8 + 45*(C*c^2*d + B*c^2*e)*x^6 + 7*(9*B*c^ 2*d - (2*C*a*c - 9*A*c^2)*e)*x^4 - 126*B*a*c*d - 15*(2*B*a*c*e + (2*C*a*c - 7*A*c^2)*d)*x^2 - 42*(C*a^2 + 3*A*a*c)*e)*sqrt(-c*x^4 + a))/(c^2*x)
Time = 3.23 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.97 \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {A \sqrt {a} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {A \sqrt {a} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {a} d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {a} e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {C \sqrt {a} d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {C \sqrt {a} e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((e*x**2+d)*(-c*x**4+a)**(1/2)*(C*x**4+B*x**2+A),x)
Output:
A*sqrt(a)*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(2*I*p i)/a)/(4*gamma(5/4)) + A*sqrt(a)*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4 ,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(7/4)) + B*sqrt(a)*d*x**3*gamma(3/ 4)*hyper((-1/2, 3/4), (7/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(7/4)) + B*sqrt(a)*e*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(2 *I*pi)/a)/(4*gamma(9/4)) + C*sqrt(a)*d*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(9/4)) + C*sqrt(a)*e*x**7*gamm a(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(11 /4))
\[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)*(e*x^2 + d), x)
\[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)*(e*x^2 + d), x)
Timed out. \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int \sqrt {a-c\,x^4}\,\left (e\,x^2+d\right )\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:
int((a - c*x^4)^(1/2)*(d + e*x^2)*(A + B*x^2 + C*x^4),x)
Output:
int((a - c*x^4)^(1/2)*(d + e*x^2)*(A + B*x^2 + C*x^4), x)
\[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {-30 \sqrt {-c \,x^{4}+a}\, a b e x +75 \sqrt {-c \,x^{4}+a}\, a c d x +49 \sqrt {-c \,x^{4}+a}\, a c e \,x^{3}+63 \sqrt {-c \,x^{4}+a}\, b c d \,x^{3}+45 \sqrt {-c \,x^{4}+a}\, b c e \,x^{5}+45 \sqrt {-c \,x^{4}+a}\, c^{2} d \,x^{5}+35 \sqrt {-c \,x^{4}+a}\, c^{2} e \,x^{7}+30 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} b e +240 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} c d +168 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} c e +126 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a b c d}{315 c} \] Input:
int((e*x^2+d)*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x)
Output:
( - 30*sqrt(a - c*x**4)*a*b*e*x + 75*sqrt(a - c*x**4)*a*c*d*x + 49*sqrt(a - c*x**4)*a*c*e*x**3 + 63*sqrt(a - c*x**4)*b*c*d*x**3 + 45*sqrt(a - c*x**4 )*b*c*e*x**5 + 45*sqrt(a - c*x**4)*c**2*d*x**5 + 35*sqrt(a - c*x**4)*c**2* e*x**7 + 30*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**2*b*e + 240*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**2*c*d + 168*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**2*c*e + 126*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a*b *c*d)/(315*c)