Optimal. Leaf size=210 \[ -\frac{(d-e x)^{5/2} (d+e x)^{5/2} \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10}}+\frac{d^2 (d-e x)^{3/2} (d+e x)^{3/2} \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10}}-\frac{d^4 \sqrt{d-e x} \sqrt{d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10}}+\frac{(d-e x)^{7/2} (d+e x)^{7/2} \left (b e^2+4 c d^2\right )}{7 e^{10}}-\frac{c (d-e x)^{9/2} (d+e x)^{9/2}}{9 e^{10}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.818034, antiderivative size = 278, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ -\frac{\left (d^2-e^2 x^2\right )^3 \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \left (d^2-e^2 x^2\right )^2 \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^4 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^4 \left (b e^2+4 c d^2\right )}{7 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{c \sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right )^{4}}{9 e^{10}} + \frac{d^{2} \sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right ) \left (2 a e^{4} + 3 b d^{2} e^{2} + 4 c d^{4}\right )}{3 e^{10}} + \frac{\sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right )^{3} \left (b e^{2} + 4 c d^{2}\right )}{7 e^{10}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right )^{2} \left (a e^{4} + 3 b d^{2} e^{2} + 6 c d^{4}\right )}{5 e^{10}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (a e^{4} + b d^{2} e^{2} + c d^{4}\right ) \int ^{\sqrt{d^{2} - e^{2} x^{2}}} d^{4}\, dx}{e^{10} \sqrt{d^{2} - e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.163924, size = 149, normalized size = 0.71 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (21 a e^4 \left (8 d^4+4 d^2 e^2 x^2+3 e^4 x^4\right )+9 b \left (16 d^6 e^2+8 d^4 e^4 x^2+6 d^2 e^6 x^4+5 e^8 x^6\right )+c \left (128 d^8+64 d^6 e^2 x^2+48 d^4 e^4 x^4+40 d^2 e^6 x^6+35 e^8 x^8\right )\right )}{315 e^{10}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.011, size = 145, normalized size = 0.7 \[ -{\frac{35\,c{x}^{8}{e}^{8}+45\,b{e}^{8}{x}^{6}+40\,c{d}^{2}{e}^{6}{x}^{6}+63\,a{e}^{8}{x}^{4}+54\,b{d}^{2}{e}^{6}{x}^{4}+48\,c{d}^{4}{e}^{4}{x}^{4}+84\,a{d}^{2}{e}^{6}{x}^{2}+72\,b{d}^{4}{e}^{4}{x}^{2}+64\,c{d}^{6}{e}^{2}{x}^{2}+168\,a{d}^{4}{e}^{4}+144\,b{d}^{6}{e}^{2}+128\,c{d}^{8}}{315\,{e}^{10}}\sqrt{-ex+d}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.789336, size = 398, normalized size = 1.9 \[ -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{8}}{9 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{6}}{63 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{6}}{7 \, e^{2}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x^{4}}{105 \, e^{6}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x^{4}}{35 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x^{4}}{5 \, e^{2}} - \frac{64 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{6} x^{2}}{315 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{4} x^{2}}{35 \, e^{6}} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{2} x^{2}}{15 \, e^{4}} - \frac{128 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{8}}{315 \, e^{10}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{6}}{35 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{4}}{15 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)*x^5/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.283802, size = 554, normalized size = 2.64 \[ -\frac{35 \, c e^{8} x^{18} - 45 \,{\left (31 \, c d^{2} e^{6} - b e^{8}\right )} x^{16} + 13440 \, a d^{8} x^{6} + 9 \,{\left (912 \, c d^{4} e^{4} - 199 \, b d^{2} e^{6} + 7 \, a e^{8}\right )} x^{14} - 21 \,{\left (704 \, c d^{6} e^{2} - 498 \, b d^{4} e^{4} + 119 \, a d^{2} e^{6}\right )} x^{12} + 252 \,{\left (32 \, c d^{8} - 74 \, b d^{6} e^{2} + 57 \, a d^{4} e^{4}\right )} x^{10} + 5040 \,{\left (2 \, b d^{8} - 5 \, a d^{6} e^{2}\right )} x^{8} + 3 \,{\left (105 \, c d e^{6} x^{16} - 5 \,{\left (256 \, c d^{3} e^{4} - 27 \, b d e^{6}\right )} x^{14} - 4480 \, a d^{7} x^{6} + 7 \,{\left (512 \, c d^{5} e^{2} - 234 \, b d^{3} e^{4} + 27 \, a d e^{6}\right )} x^{12} - 84 \,{\left (32 \, c d^{7} - 54 \, b d^{5} e^{2} + 27 \, a d^{3} e^{4}\right )} x^{10} - 560 \,{\left (6 \, b d^{7} - 11 \, a d^{5} e^{2}\right )} x^{8}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{315 \,{\left (9 \, d e^{8} x^{8} - 120 \, d^{3} e^{6} x^{6} + 432 \, d^{5} e^{4} x^{4} - 576 \, d^{7} e^{2} x^{2} + 256 \, d^{9} -{\left (e^{8} x^{8} - 40 \, d^{2} e^{6} x^{6} + 240 \, d^{4} e^{4} x^{4} - 448 \, d^{6} e^{2} x^{2} + 256 \, d^{8}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)*x^5/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.300016, size = 328, normalized size = 1.56 \[ -\frac{1}{2807562240} \,{\left (315 \, c d^{8} e^{81} + 315 \, b d^{6} e^{83} + 315 \, a d^{4} e^{85} -{\left (840 \, c d^{7} e^{81} + 630 \, b d^{5} e^{83} + 420 \, a d^{3} e^{85} -{\left (1932 \, c d^{6} e^{81} + 1071 \, b d^{4} e^{83} + 462 \, a d^{2} e^{85} -{\left (2952 \, c d^{5} e^{81} + 1116 \, b d^{3} e^{83} + 252 \, a d e^{85} -{\left (3098 \, c d^{4} e^{81} + 729 \, b d^{2} e^{83} - 5 \,{\left (440 \, c d^{3} e^{81} + 54 \, b d e^{83} -{\left (204 \, c d^{2} e^{81} + 7 \,{\left ({\left (x e + d\right )} c e^{81} - 8 \, c d e^{81}\right )}{\left (x e + d\right )} + 9 \, b e^{83}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 63 \, a e^{85}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )} \sqrt{x e + d} \sqrt{-x e + d} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)*x^5/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="giac")
[Out]