3.139 \(\int \frac{x^2 \left (a+b x^2+c x^4\right )}{\sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=216 \[ \frac{d^2 \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \sqrt{d-e x} \sqrt{d+e x} \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^6}-\frac{x^3 \sqrt{d-e x} \sqrt{d+e x} \left (6 b e^2+5 c d^2\right )}{24 e^4}+\frac{c x^5 (e x-d) \sqrt{d+e x}}{6 e^2 \sqrt{d-e x}} \]

[Out]

-((5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*x*Sqrt[d - e*x]*Sqrt[d + e*x])/(16*e^6) - ((
5*c*d^2 + 6*b*e^2)*x^3*Sqrt[d - e*x]*Sqrt[d + e*x])/(24*e^4) + (c*x^5*(-d + e*x)
*Sqrt[d + e*x])/(6*e^2*Sqrt[d - e*x]) + (d^2*(5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*S
qrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e^7*Sqrt[d - e*x]*Sqrt
[d + e*x])

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Rubi [A]  time = 0.601384, antiderivative size = 245, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ \frac{d^2 \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x^3 \left (d^2-e^2 x^2\right ) \left (6 b e^2+5 c d^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-((5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*x*(d^2 - e^2*x^2))/(16*e^6*Sqrt[d - e*x]*Sqr
t[d + e*x]) - ((5*c*d^2 + 6*b*e^2)*x^3*(d^2 - e^2*x^2))/(24*e^4*Sqrt[d - e*x]*Sq
rt[d + e*x]) - (c*x^5*(d^2 - e^2*x^2))/(6*e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + (d^
2*(5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 -
e^2*x^2]])/(16*e^7*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 27.8546, size = 194, normalized size = 0.9 \[ - \frac{c x^{5} \sqrt{d - e x} \sqrt{d + e x}}{6 e^{2}} + \frac{d^{2} \sqrt{d - e x} \sqrt{d + e x} \left (8 a e^{4} + 6 b d^{2} e^{2} + 5 c d^{4}\right ) \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{16 e^{7} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{x^{3} \sqrt{d - e x} \sqrt{d + e x} \left (6 b e^{2} + 5 c d^{2}\right )}{24 e^{4}} - \frac{x \sqrt{d - e x} \sqrt{d + e x} \left (8 a e^{4} + 6 b d^{2} e^{2} + 5 c d^{4}\right )}{16 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-c*x**5*sqrt(d - e*x)*sqrt(d + e*x)/(6*e**2) + d**2*sqrt(d - e*x)*sqrt(d + e*x)*
(8*a*e**4 + 6*b*d**2*e**2 + 5*c*d**4)*atan(e*x/sqrt(d**2 - e**2*x**2))/(16*e**7*
sqrt(d**2 - e**2*x**2)) - x**3*sqrt(d - e*x)*sqrt(d + e*x)*(6*b*e**2 + 5*c*d**2)
/(24*e**4) - x*sqrt(d - e*x)*sqrt(d + e*x)*(8*a*e**4 + 6*b*d**2*e**2 + 5*c*d**4)
/(16*e**6)

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Mathematica [A]  time = 0.229209, size = 136, normalized size = 0.63 \[ \frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d-e x} \sqrt{d+e x}}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )-e x \sqrt{d-e x} \sqrt{d+e x} \left (6 \left (4 a e^4+3 b d^2 e^2+2 b e^4 x^2\right )+c \left (15 d^4+10 d^2 e^2 x^2+8 e^4 x^4\right )\right )}{48 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

(-(e*x*Sqrt[d - e*x]*Sqrt[d + e*x]*(6*(3*b*d^2*e^2 + 4*a*e^4 + 2*b*e^4*x^2) + c*
(15*d^4 + 10*d^2*e^2*x^2 + 8*e^4*x^4))) + 3*d^2*(5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4
)*ArcTan[(e*x)/(Sqrt[d - e*x]*Sqrt[d + e*x])])/(48*e^7)

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Maple [C]  time = 0.046, size = 273, normalized size = 1.3 \[ -{\frac{{\it csgn} \left ( e \right ) }{48\,{e}^{7}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 8\,{\it csgn} \left ( e \right ){x}^{5}c{e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+12\,{\it csgn} \left ( e \right ){x}^{3}b{e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+10\,{\it csgn} \left ( e \right ){x}^{3}c{d}^{2}{e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+24\,ax\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{e}^{5}{\it csgn} \left ( e \right ) +18\,{d}^{2}x\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}b{e}^{3}{\it csgn} \left ( e \right ) +15\,c{d}^{4}x\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( e \right ) e-24\,a{d}^{2}\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ){e}^{4}-18\,{d}^{4}\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) b{e}^{2}-15\,c{d}^{6}\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/48*(-e*x+d)^(1/2)*(e*x+d)^(1/2)*(8*csgn(e)*x^5*c*e^5*(-e^2*x^2+d^2)^(1/2)+12*
csgn(e)*x^3*b*e^5*(-e^2*x^2+d^2)^(1/2)+10*csgn(e)*x^3*c*d^2*e^3*(-e^2*x^2+d^2)^(
1/2)+24*a*x*(-e^2*x^2+d^2)^(1/2)*e^5*csgn(e)+18*d^2*x*(-e^2*x^2+d^2)^(1/2)*b*e^3
*csgn(e)+15*c*d^4*x*(-e^2*x^2+d^2)^(1/2)*csgn(e)*e-24*a*d^2*arctan(csgn(e)*e*x/(
-e^2*x^2+d^2)^(1/2))*e^4-18*d^4*arctan(csgn(e)*e*x/(-e^2*x^2+d^2)^(1/2))*b*e^2-1
5*c*d^6*arctan(csgn(e)*e*x/(-e^2*x^2+d^2)^(1/2)))*csgn(e)/(-e^2*x^2+d^2)^(1/2)/e
^7

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Maxima [A]  time = 0.787224, size = 309, normalized size = 1.43 \[ -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{5}}{6 \, e^{2}} - \frac{5 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{3}}{24 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{3}}{4 \, e^{2}} + \frac{5 \, c d^{6} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e^{6}} + \frac{3 \, b d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{4}} + \frac{a d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{5 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x}{16 \, e^{6}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x}{8 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x}{2 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^2/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="maxima")

[Out]

-1/6*sqrt(-e^2*x^2 + d^2)*c*x^5/e^2 - 5/24*sqrt(-e^2*x^2 + d^2)*c*d^2*x^3/e^4 -
1/4*sqrt(-e^2*x^2 + d^2)*b*x^3/e^2 + 5/16*c*d^6*arcsin(e^2*x/sqrt(d^2*e^2))/(sqr
t(e^2)*e^6) + 3/8*b*d^4*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^4) + 1/2*a*d^2*
arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^2) - 5/16*sqrt(-e^2*x^2 + d^2)*c*d^4*x/
e^6 - 3/8*sqrt(-e^2*x^2 + d^2)*b*d^2*x/e^4 - 1/2*sqrt(-e^2*x^2 + d^2)*a*x/e^2

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Fricas [A]  time = 0.386573, size = 921, normalized size = 4.26 \[ \frac{48 \, c d e^{11} x^{11} - 4 \,{\left (61 \, c d^{3} e^{9} - 18 \, b d e^{11}\right )} x^{9} + 6 \,{\left (37 \, c d^{5} e^{7} - 58 \, b d^{3} e^{9} + 24 \, a d e^{11}\right )} x^{7} - 6 \,{\left (31 \, c d^{7} e^{5} - 14 \, b d^{5} e^{7} + 152 \, a d^{3} e^{9}\right )} x^{5} + 128 \,{\left (5 \, c d^{9} e^{3} + 6 \, b d^{7} e^{5} + 12 \, a d^{5} e^{7}\right )} x^{3} -{\left (8 \, c e^{11} x^{11} - 2 \,{\left (67 \, c d^{2} e^{9} - 6 \, b e^{11}\right )} x^{9} + 3 \,{\left (73 \, c d^{4} e^{7} - 66 \, b d^{2} e^{9} + 8 \, a e^{11}\right )} x^{7} - 2 \,{\left (23 \, c d^{6} e^{5} - 126 \, b d^{4} e^{7} + 216 \, a d^{2} e^{9}\right )} x^{5} + 16 \,{\left (25 \, c d^{8} e^{3} + 30 \, b d^{6} e^{5} + 72 \, a d^{4} e^{7}\right )} x^{3} - 96 \,{\left (5 \, c d^{10} e + 6 \, b d^{8} e^{3} + 8 \, a d^{6} e^{5}\right )} x\right )} \sqrt{e x + d} \sqrt{-e x + d} - 96 \,{\left (5 \, c d^{11} e + 6 \, b d^{9} e^{3} + 8 \, a d^{7} e^{5}\right )} x + 6 \,{\left (160 \, c d^{12} + 192 \, b d^{10} e^{2} + 256 \, a d^{8} e^{4} -{\left (5 \, c d^{6} e^{6} + 6 \, b d^{4} e^{8} + 8 \, a d^{2} e^{10}\right )} x^{6} + 18 \,{\left (5 \, c d^{8} e^{4} + 6 \, b d^{6} e^{6} + 8 \, a d^{4} e^{8}\right )} x^{4} - 48 \,{\left (5 \, c d^{10} e^{2} + 6 \, b d^{8} e^{4} + 8 \, a d^{6} e^{6}\right )} x^{2} - 2 \,{\left (80 \, c d^{11} + 96 \, b d^{9} e^{2} + 128 \, a d^{7} e^{4} + 3 \,{\left (5 \, c d^{7} e^{4} + 6 \, b d^{5} e^{6} + 8 \, a d^{3} e^{8}\right )} x^{4} - 16 \,{\left (5 \, c d^{9} e^{2} + 6 \, b d^{7} e^{4} + 8 \, a d^{5} e^{6}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right )}{48 \,{\left (e^{13} x^{6} - 18 \, d^{2} e^{11} x^{4} + 48 \, d^{4} e^{9} x^{2} - 32 \, d^{6} e^{7} + 2 \,{\left (3 \, d e^{11} x^{4} - 16 \, d^{3} e^{9} x^{2} + 16 \, d^{5} e^{7}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^2/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="fricas")

[Out]

1/48*(48*c*d*e^11*x^11 - 4*(61*c*d^3*e^9 - 18*b*d*e^11)*x^9 + 6*(37*c*d^5*e^7 -
58*b*d^3*e^9 + 24*a*d*e^11)*x^7 - 6*(31*c*d^7*e^5 - 14*b*d^5*e^7 + 152*a*d^3*e^9
)*x^5 + 128*(5*c*d^9*e^3 + 6*b*d^7*e^5 + 12*a*d^5*e^7)*x^3 - (8*c*e^11*x^11 - 2*
(67*c*d^2*e^9 - 6*b*e^11)*x^9 + 3*(73*c*d^4*e^7 - 66*b*d^2*e^9 + 8*a*e^11)*x^7 -
2*(23*c*d^6*e^5 - 126*b*d^4*e^7 + 216*a*d^2*e^9)*x^5 + 16*(25*c*d^8*e^3 + 30*b*
d^6*e^5 + 72*a*d^4*e^7)*x^3 - 96*(5*c*d^10*e + 6*b*d^8*e^3 + 8*a*d^6*e^5)*x)*sqr
t(e*x + d)*sqrt(-e*x + d) - 96*(5*c*d^11*e + 6*b*d^9*e^3 + 8*a*d^7*e^5)*x + 6*(1
60*c*d^12 + 192*b*d^10*e^2 + 256*a*d^8*e^4 - (5*c*d^6*e^6 + 6*b*d^4*e^8 + 8*a*d^
2*e^10)*x^6 + 18*(5*c*d^8*e^4 + 6*b*d^6*e^6 + 8*a*d^4*e^8)*x^4 - 48*(5*c*d^10*e^
2 + 6*b*d^8*e^4 + 8*a*d^6*e^6)*x^2 - 2*(80*c*d^11 + 96*b*d^9*e^2 + 128*a*d^7*e^4
+ 3*(5*c*d^7*e^4 + 6*b*d^5*e^6 + 8*a*d^3*e^8)*x^4 - 16*(5*c*d^9*e^2 + 6*b*d^7*e
^4 + 8*a*d^5*e^6)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d))*arctan((sqrt(e*x + d)*sqrt(
-e*x + d) - d)/(e*x)))/(e^13*x^6 - 18*d^2*e^11*x^4 + 48*d^4*e^9*x^2 - 32*d^6*e^7
+ 2*(3*d*e^11*x^4 - 16*d^3*e^9*x^2 + 16*d^5*e^7)*sqrt(e*x + d)*sqrt(-e*x + d))

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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**2*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.315731, size = 257, normalized size = 1.19 \[ \frac{1}{34603008} \,{\left ({\left (33 \, c d^{5} e^{36} + 30 \, b d^{3} e^{38} + 24 \, a d e^{40} -{\left (85 \, c d^{4} e^{36} + 54 \, b d^{2} e^{38} - 2 \,{\left (55 \, c d^{3} e^{36} + 18 \, b d e^{38} -{\left (45 \, c d^{2} e^{36} + 4 \,{\left ({\left (x e + d\right )} c e^{36} - 5 \, c d e^{36}\right )}{\left (x e + d\right )} + 6 \, b e^{38}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 24 \, a e^{40}\right )}{\left (x e + d\right )}\right )} \sqrt{x e + d} \sqrt{-x e + d} + 6 \,{\left (5 \, c d^{6} e^{36} + 6 \, b d^{4} e^{38} + 8 \, a d^{2} e^{40}\right )} \arcsin \left (\frac{\sqrt{2} \sqrt{x e + d}}{2 \, \sqrt{d}}\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^2/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="giac")

[Out]

1/34603008*((33*c*d^5*e^36 + 30*b*d^3*e^38 + 24*a*d*e^40 - (85*c*d^4*e^36 + 54*b
*d^2*e^38 - 2*(55*c*d^3*e^36 + 18*b*d*e^38 - (45*c*d^2*e^36 + 4*((x*e + d)*c*e^3
6 - 5*c*d*e^36)*(x*e + d) + 6*b*e^38)*(x*e + d))*(x*e + d) + 24*a*e^40)*(x*e + d
))*sqrt(x*e + d)*sqrt(-x*e + d) + 6*(5*c*d^6*e^36 + 6*b*d^4*e^38 + 8*a*d^2*e^40)
*arcsin(1/2*sqrt(2)*sqrt(x*e + d)/sqrt(d)))*e^(-1)