3.145 \(\int \frac{a+b x^2+c x^4}{x^{10} \sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=292 \[ -\frac{8 e^4 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^{10} x \sqrt{d-e x} \sqrt{d+e x}}-\frac{4 e^2 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^8 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{105 d^6 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (8 a e^2+9 b d^2\right )}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}} \]

[Out]

-(a*(d^2 - e^2*x^2))/(9*d^2*x^9*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((9*b*d^2 + 8*a*e
^2)*(d^2 - e^2*x^2))/(63*d^4*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((21*c*d^4 + 18*
b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^5*Sqrt[d - e*x]*Sqrt[d + e*x])
- (4*e^2*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(315*d^8*x^3*Sqr
t[d - e*x]*Sqrt[d + e*x]) - (8*e^4*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e
^2*x^2))/(315*d^10*x*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi [A]  time = 0.684247, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{8 e^4 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^{10} x \sqrt{d-e x} \sqrt{d+e x}}-\frac{4 e^2 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^8 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{105 d^6 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (8 a e^2+9 b d^2\right )}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(a*(d^2 - e^2*x^2))/(9*d^2*x^9*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((9*b*d^2 + 8*a*e
^2)*(d^2 - e^2*x^2))/(63*d^4*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((21*c*d^4 + 18*
b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^5*Sqrt[d - e*x]*Sqrt[d + e*x])
- (4*e^2*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(315*d^8*x^3*Sqr
t[d - e*x]*Sqrt[d + e*x]) - (8*e^4*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e
^2*x^2))/(315*d^10*x*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 31.3793, size = 262, normalized size = 0.9 \[ - \frac{a \sqrt{d - e x} \sqrt{d + e x}}{9 d^{2} x^{9}} + \frac{c \sqrt{d - e x} \sqrt{d + e x}}{6 e^{2} x^{7}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (16 a e^{4} + 18 b d^{2} e^{2} + 21 c d^{4}\right )}{126 d^{4} e^{2} x^{7}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (16 a e^{4} + 18 b d^{2} e^{2} + 21 c d^{4}\right )}{105 d^{6} x^{5}} - \frac{4 e^{2} \sqrt{d - e x} \sqrt{d + e x} \left (16 a e^{4} + 18 b d^{2} e^{2} + 21 c d^{4}\right )}{315 d^{8} x^{3}} - \frac{8 e^{4} \sqrt{d - e x} \sqrt{d + e x} \left (16 a e^{4} + 18 b d^{2} e^{2} + 21 c d^{4}\right )}{315 d^{10} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**10/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-a*sqrt(d - e*x)*sqrt(d + e*x)/(9*d**2*x**9) + c*sqrt(d - e*x)*sqrt(d + e*x)/(6*
e**2*x**7) - sqrt(d - e*x)*sqrt(d + e*x)*(16*a*e**4 + 18*b*d**2*e**2 + 21*c*d**4
)/(126*d**4*e**2*x**7) - sqrt(d - e*x)*sqrt(d + e*x)*(16*a*e**4 + 18*b*d**2*e**2
+ 21*c*d**4)/(105*d**6*x**5) - 4*e**2*sqrt(d - e*x)*sqrt(d + e*x)*(16*a*e**4 +
18*b*d**2*e**2 + 21*c*d**4)/(315*d**8*x**3) - 8*e**4*sqrt(d - e*x)*sqrt(d + e*x)
*(16*a*e**4 + 18*b*d**2*e**2 + 21*c*d**4)/(315*d**10*x)

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Mathematica [A]  time = 0.186516, size = 157, normalized size = 0.54 \[ \sqrt{d-e x} \sqrt{d+e x} \left (-\frac{8 e^4 \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^{10} x}-\frac{4 e^2 \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^8 x^3}+\frac{-16 a e^4-18 b d^2 e^2-21 c d^4}{105 d^6 x^5}+\frac{-8 a e^2-9 b d^2}{63 d^4 x^7}-\frac{a}{9 d^2 x^9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

(-a/(9*d^2*x^9) + (-9*b*d^2 - 8*a*e^2)/(63*d^4*x^7) + (-21*c*d^4 - 18*b*d^2*e^2
- 16*a*e^4)/(105*d^6*x^5) - (4*e^2*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4))/(315*d^
8*x^3) - (8*e^4*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4))/(315*d^10*x))*Sqrt[d - e*x
]*Sqrt[d + e*x]

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Maple [A]  time = 0.014, size = 154, normalized size = 0.5 \[ -{\frac{128\,a{e}^{8}{x}^{8}+144\,b{d}^{2}{e}^{6}{x}^{8}+168\,c{d}^{4}{e}^{4}{x}^{8}+64\,a{d}^{2}{e}^{6}{x}^{6}+72\,b{d}^{4}{e}^{4}{x}^{6}+84\,c{d}^{6}{e}^{2}{x}^{6}+48\,a{d}^{4}{e}^{4}{x}^{4}+54\,b{d}^{6}{e}^{2}{x}^{4}+63\,c{d}^{8}{x}^{4}+40\,a{d}^{6}{e}^{2}{x}^{2}+45\,b{d}^{8}{x}^{2}+35\,a{d}^{8}}{315\,{x}^{9}{d}^{10}}\sqrt{ex+d}\sqrt{-ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/315*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(128*a*e^8*x^8+144*b*d^2*e^6*x^8+168*c*d^4*e
^4*x^8+64*a*d^2*e^6*x^6+72*b*d^4*e^4*x^6+84*c*d^6*e^2*x^6+48*a*d^4*e^4*x^4+54*b*
d^6*e^2*x^4+63*c*d^8*x^4+40*a*d^6*e^2*x^2+45*b*d^8*x^2+35*a*d^8)/x^9/d^10

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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^10),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.14287, size = 922, normalized size = 3.16 \[ \text{result too large to display} \]

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^10),x, algorithm="fricas")

[Out]

1/315*(8960*a*d^18 - 8*(21*c*d^4*e^14 + 18*b*d^2*e^16 + 16*a*e^18)*x^18 + 324*(2
1*c*d^6*e^12 + 18*b*d^4*e^14 + 16*a*d^2*e^16)*x^16 - 2079*(21*c*d^8*e^10 + 18*b*
d^6*e^12 + 16*a*d^4*e^14)*x^14 + 21*(4507*c*d^10*e^8 + 3861*b*d^8*e^10 + 3432*a*
d^6*e^12)*x^12 - 45*(1736*c*d^12*e^6 + 1447*b*d^10*e^8 + 1287*a*d^8*e^10)*x^10 +
9*(3024*c*d^14*e^4 + 1192*b*d^12*e^6 + 1219*a*d^10*e^8)*x^8 - 24*(952*c*d^16*e^
2 - 474*b*d^14*e^4 - 13*a*d^12*e^6)*x^6 + 144*(112*c*d^18 - 124*b*d^16*e^2 + 57*
a*d^14*e^4)*x^4 + 2880*(4*b*d^18 - 5*a*d^16*e^2)*x^2 - (8960*a*d^17 + 72*(21*c*d
^5*e^12 + 18*b*d^3*e^14 + 16*a*d*e^16)*x^16 - 924*(21*c*d^7*e^10 + 18*b*d^5*e^12
+ 16*a*d^3*e^14)*x^14 + 3003*(21*c*d^9*e^8 + 18*b*d^7*e^10 + 16*a*d^5*e^12)*x^1
2 - 45*(1512*c*d^11*e^6 + 1287*b*d^9*e^8 + 1144*a*d^7*e^10)*x^10 + 5*(4368*c*d^1
3*e^4 + 2664*b*d^11*e^6 + 2431*a*d^9*e^8)*x^8 - 8*(1848*c*d^15*e^2 - 846*b*d^13*
e^4 - 227*a*d^11*e^6)*x^6 + 336*(48*c*d^17 - 36*b*d^15*e^2 + 13*a*d^13*e^4)*x^4
+ 320*(36*b*d^17 - 31*a*d^15*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d))/(9*d^11*e^8
*x^17 - 120*d^13*e^6*x^15 + 432*d^15*e^4*x^13 - 576*d^17*e^2*x^11 + 256*d^19*x^9
- (d^10*e^8*x^17 - 40*d^12*e^6*x^15 + 240*d^14*e^4*x^13 - 448*d^16*e^2*x^11 + 2
56*d^18*x^9)*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**10/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.16691, 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^10),x, algorithm="giac")

[Out]

Done