3.800 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(9*d*e*(d + e*x)^9)

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Rubi [A]  time = 0.0366368, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^9,x]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(9*d*e*(d + e*x)^9)

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Rubi in Sympy [A]  time = 4.88217, size = 26, normalized size = 0.79 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{9 d e \left (d + e x\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**9,x)

[Out]

-(d**2 - e**2*x**2)**(9/2)/(9*d*e*(d + e*x)**9)

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Mathematica [A]  time = 0.0742725, size = 41, normalized size = 1.24 \[ -\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^9,x]

[Out]

-((d - e*x)^4*Sqrt[d^2 - e^2*x^2])/(9*d*e*(d + e*x)^5)

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Maple [A]  time = 0.01, size = 36, normalized size = 1.1 \[ -{\frac{-ex+d}{9\, \left ( ex+d \right ) ^{8}de} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^9,x)

[Out]

-1/9/(e*x+d)^8*(-e*x+d)/d/e*(-e^2*x^2+d^2)^(7/2)

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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((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236118, size = 386, normalized size = 11.7 \[ -\frac{2 \,{\left (e^{8} x^{9} + 18 \, d^{2} e^{6} x^{7} - 99 \, d^{4} e^{4} x^{5} + 24 \, d^{6} e^{2} x^{3} + 72 \, d^{8} x + 12 \,{\left (5 \, d^{3} e^{4} x^{5} - 5 \, d^{5} e^{2} x^{3} - 6 \, d^{7} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}}{9 \,{\left (d e^{9} x^{9} + 9 \, d^{2} e^{8} x^{8} + 18 \, d^{3} e^{7} x^{7} - 18 \, d^{4} e^{6} x^{6} - 99 \, d^{5} e^{5} x^{5} - 99 \, d^{6} e^{4} x^{4} + 24 \, d^{7} e^{3} x^{3} + 108 \, d^{8} e^{2} x^{2} + 72 \, d^{9} e x + 16 \, d^{10} -{\left (d e^{8} x^{8} - 22 \, d^{3} e^{6} x^{6} - 60 \, d^{4} e^{5} x^{5} - 39 \, d^{5} e^{4} x^{4} + 60 \, d^{6} e^{3} x^{3} + 116 \, d^{7} e^{2} x^{2} + 72 \, d^{8} e x + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^9,x, algorithm="fricas")

[Out]

-2/9*(e^8*x^9 + 18*d^2*e^6*x^7 - 99*d^4*e^4*x^5 + 24*d^6*e^2*x^3 + 72*d^8*x + 12
*(5*d^3*e^4*x^5 - 5*d^5*e^2*x^3 - 6*d^7*x)*sqrt(-e^2*x^2 + d^2))/(d*e^9*x^9 + 9*
d^2*e^8*x^8 + 18*d^3*e^7*x^7 - 18*d^4*e^6*x^6 - 99*d^5*e^5*x^5 - 99*d^6*e^4*x^4
+ 24*d^7*e^3*x^3 + 108*d^8*e^2*x^2 + 72*d^9*e*x + 16*d^10 - (d*e^8*x^8 - 22*d^3*
e^6*x^6 - 60*d^4*e^5*x^5 - 39*d^5*e^4*x^4 + 60*d^6*e^3*x^3 + 116*d^7*e^2*x^2 + 7
2*d^8*e*x + 16*d^9)*sqrt(-e^2*x^2 + d^2))

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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((-e**2*x**2+d**2)**(7/2)/(e*x+d)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.606707, size = 1, normalized size = 0.03 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^9,x, algorithm="giac")

[Out]

Done