3.144 \(\int \frac{a+b x^2+c x^4}{x^8 \sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=226 \[ -\frac{2 e^2 \left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^8 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^6 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (6 a e^2+7 b d^2\right )}{35 d^4 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}} \]

[Out]

-(a*(d^2 - e^2*x^2))/(7*d^2*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((7*b*d^2 + 6*a*e
^2)*(d^2 - e^2*x^2))/(35*d^4*x^5*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((35*c*d^4 + 28*
b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^3*Sqrt[d - e*x]*Sqrt[d + e*x])
- (2*e^2*(35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^8*x*Sqrt[
d - e*x]*Sqrt[d + e*x])

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Rubi [A]  time = 0.54504, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{2 e^2 \left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^8 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^6 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (6 a e^2+7 b d^2\right )}{35 d^4 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(a*(d^2 - e^2*x^2))/(7*d^2*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((7*b*d^2 + 6*a*e
^2)*(d^2 - e^2*x^2))/(35*d^4*x^5*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((35*c*d^4 + 28*
b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^3*Sqrt[d - e*x]*Sqrt[d + e*x])
- (2*e^2*(35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^8*x*Sqrt[
d - e*x]*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 26.3829, size = 207, normalized size = 0.92 \[ - \frac{a \sqrt{d - e x} \sqrt{d + e x}}{7 d^{2} x^{7}} + \frac{c \sqrt{d - e x} \sqrt{d + e x}}{4 e^{2} x^{5}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (24 a e^{4} + 28 b d^{2} e^{2} + 35 c d^{4}\right )}{140 d^{4} e^{2} x^{5}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (24 a e^{4} + 28 b d^{2} e^{2} + 35 c d^{4}\right )}{105 d^{6} x^{3}} - \frac{2 e^{2} \sqrt{d - e x} \sqrt{d + e x} \left (24 a e^{4} + 28 b d^{2} e^{2} + 35 c d^{4}\right )}{105 d^{8} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**8/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-a*sqrt(d - e*x)*sqrt(d + e*x)/(7*d**2*x**7) + c*sqrt(d - e*x)*sqrt(d + e*x)/(4*
e**2*x**5) - sqrt(d - e*x)*sqrt(d + e*x)*(24*a*e**4 + 28*b*d**2*e**2 + 35*c*d**4
)/(140*d**4*e**2*x**5) - sqrt(d - e*x)*sqrt(d + e*x)*(24*a*e**4 + 28*b*d**2*e**2
+ 35*c*d**4)/(105*d**6*x**3) - 2*e**2*sqrt(d - e*x)*sqrt(d + e*x)*(24*a*e**4 +
28*b*d**2*e**2 + 35*c*d**4)/(105*d**8*x)

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Mathematica [A]  time = 0.150053, size = 122, normalized size = 0.54 \[ \sqrt{d-e x} \sqrt{d+e x} \left (-\frac{2 e^2 \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^8 x}+\frac{-24 a e^4-28 b d^2 e^2-35 c d^4}{105 d^6 x^3}+\frac{-6 a e^2-7 b d^2}{35 d^4 x^5}-\frac{a}{7 d^2 x^7}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

(-a/(7*d^2*x^7) + (-7*b*d^2 - 6*a*e^2)/(35*d^4*x^5) + (-35*c*d^4 - 28*b*d^2*e^2
- 24*a*e^4)/(105*d^6*x^3) - (2*e^2*(35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4))/(105*d^
8*x))*Sqrt[d - e*x]*Sqrt[d + e*x]

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Maple [A]  time = 0.009, size = 118, normalized size = 0.5 \[ -{\frac{48\,a{e}^{6}{x}^{6}+56\,b{d}^{2}{e}^{4}{x}^{6}+70\,c{d}^{4}{e}^{2}{x}^{6}+24\,a{d}^{2}{e}^{4}{x}^{4}+28\,b{d}^{4}{e}^{2}{x}^{4}+35\,c{d}^{6}{x}^{4}+18\,a{d}^{4}{e}^{2}{x}^{2}+21\,b{d}^{6}{x}^{2}+15\,a{d}^{6}}{105\,{x}^{7}{d}^{8}}\sqrt{ex+d}\sqrt{-ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/105*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(48*a*e^6*x^6+56*b*d^2*e^4*x^6+70*c*d^4*e^2*
x^6+24*a*d^2*e^4*x^4+28*b*d^4*e^2*x^4+35*c*d^6*x^4+18*a*d^4*e^2*x^2+21*b*d^6*x^2
+15*a*d^6)/x^7/d^8

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)*x^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.556946, size = 713, normalized size = 3.15 \[ -\frac{960 \, a d^{14} + 2 \,{\left (35 \, c d^{4} e^{10} + 28 \, b d^{2} e^{12} + 24 \, a e^{14}\right )} x^{14} - 49 \,{\left (35 \, c d^{6} e^{8} + 28 \, b d^{4} e^{10} + 24 \, a d^{2} e^{12}\right )} x^{12} + 105 \,{\left (61 \, c d^{8} e^{6} + 49 \, b d^{6} e^{8} + 42 \, a d^{4} e^{10}\right )} x^{10} - 7 \,{\left (920 \, c d^{10} e^{4} + 811 \, b d^{8} e^{6} + 693 \, a d^{6} e^{8}\right )} x^{8} - 7 \,{\left (80 \, c d^{12} e^{2} - 248 \, b d^{10} e^{4} - 159 \, a d^{8} e^{6}\right )} x^{6} + 56 \,{\left (40 \, c d^{14} - 22 \, b d^{12} e^{2} + 9 \, a d^{10} e^{4}\right )} x^{4} + 336 \,{\left (4 \, b d^{14} - 3 \, a d^{12} e^{2}\right )} x^{2} -{\left (960 \, a d^{13} - 14 \,{\left (35 \, c d^{5} e^{8} + 28 \, b d^{3} e^{10} + 24 \, a d e^{12}\right )} x^{12} + 105 \,{\left (35 \, c d^{7} e^{6} + 28 \, b d^{5} e^{8} + 24 \, a d^{3} e^{10}\right )} x^{10} - 21 \,{\left (280 \, c d^{9} e^{4} + 231 \, b d^{7} e^{6} + 198 \, a d^{5} e^{8}\right )} x^{8} +{\left (560 \, c d^{11} e^{2} + 1624 \, b d^{9} e^{4} + 1287 \, a d^{7} e^{6}\right )} x^{6} + 40 \,{\left (56 \, c d^{13} - 14 \, b d^{11} e^{2} + 9 \, a d^{9} e^{4}\right )} x^{4} + 48 \,{\left (28 \, b d^{13} - 11 \, a d^{11} e^{2}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{105 \,{\left (7 \, d^{9} e^{6} x^{13} - 56 \, d^{11} e^{4} x^{11} + 112 \, d^{13} e^{2} x^{9} - 64 \, d^{15} x^{7} -{\left (d^{8} e^{6} x^{13} - 24 \, d^{10} e^{4} x^{11} + 80 \, d^{12} e^{2} x^{9} - 64 \, d^{14} x^{7}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)*x^8),x, algorithm="fricas")

[Out]

-1/105*(960*a*d^14 + 2*(35*c*d^4*e^10 + 28*b*d^2*e^12 + 24*a*e^14)*x^14 - 49*(35
*c*d^6*e^8 + 28*b*d^4*e^10 + 24*a*d^2*e^12)*x^12 + 105*(61*c*d^8*e^6 + 49*b*d^6*
e^8 + 42*a*d^4*e^10)*x^10 - 7*(920*c*d^10*e^4 + 811*b*d^8*e^6 + 693*a*d^6*e^8)*x
^8 - 7*(80*c*d^12*e^2 - 248*b*d^10*e^4 - 159*a*d^8*e^6)*x^6 + 56*(40*c*d^14 - 22
*b*d^12*e^2 + 9*a*d^10*e^4)*x^4 + 336*(4*b*d^14 - 3*a*d^12*e^2)*x^2 - (960*a*d^1
3 - 14*(35*c*d^5*e^8 + 28*b*d^3*e^10 + 24*a*d*e^12)*x^12 + 105*(35*c*d^7*e^6 + 2
8*b*d^5*e^8 + 24*a*d^3*e^10)*x^10 - 21*(280*c*d^9*e^4 + 231*b*d^7*e^6 + 198*a*d^
5*e^8)*x^8 + (560*c*d^11*e^2 + 1624*b*d^9*e^4 + 1287*a*d^7*e^6)*x^6 + 40*(56*c*d
^13 - 14*b*d^11*e^2 + 9*a*d^9*e^4)*x^4 + 48*(28*b*d^13 - 11*a*d^11*e^2)*x^2)*sqr
t(e*x + d)*sqrt(-e*x + d))/(7*d^9*e^6*x^13 - 56*d^11*e^4*x^11 + 112*d^13*e^2*x^9
- 64*d^15*x^7 - (d^8*e^6*x^13 - 24*d^10*e^4*x^11 + 80*d^12*e^2*x^9 - 64*d^14*x^
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((c*x**4+b*x**2+a)/x**8/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.864267, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)*x^8),x, algorithm="giac")

[Out]

Done