3.801 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}} \]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(11*d*e*(d + e*x)^10) - (d^2 - e^2*x^2)^(9/2)/(99*d^2*e*(
d + e*x)^9)

_______________________________________________________________________________________

Rubi [A]  time = 0.0758616, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(11*d*e*(d + e*x)^10) - (d^2 - e^2*x^2)^(9/2)/(99*d^2*e*(
d + e*x)^9)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.19573, size = 53, normalized size = 0.79 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{11 d e \left (d + e x\right )^{10}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{99 d^{2} e \left (d + e x\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**10,x)

[Out]

-(d**2 - e**2*x**2)**(9/2)/(11*d*e*(d + e*x)**10) - (d**2 - e**2*x**2)**(9/2)/(9
9*d**2*e*(d + e*x)**9)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0700145, size = 48, normalized size = 0.72 \[ -\frac{(d-e x)^4 (10 d+e x) \sqrt{d^2-e^2 x^2}}{99 d^2 e (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]

[Out]

-((d - e*x)^4*(10*d + e*x)*Sqrt[d^2 - e^2*x^2])/(99*d^2*e*(d + e*x)^6)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 43, normalized size = 0.6 \[ -{\frac{ \left ( ex+10\,d \right ) \left ( -ex+d \right ) }{99\, \left ( ex+d \right ) ^{9}{d}^{2}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x)

[Out]

-1/99*(-e*x+d)*(e*x+10*d)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^9/d^2/e

_______________________________________________________________________________________

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)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.258034, size = 613, normalized size = 9.15 \[ -\frac{9 \, e^{10} x^{11} - 11 \, d e^{9} x^{10} - 308 \, d^{2} e^{8} x^{9} - 858 \, d^{3} e^{7} x^{8} - 1551 \, d^{4} e^{6} x^{7} + 2409 \, d^{5} e^{5} x^{6} + 6666 \, d^{6} e^{4} x^{5} - 132 \, d^{7} e^{3} x^{4} - 2112 \, d^{8} e^{2} x^{3} - 1584 \, d^{9} e x^{2} - 3168 \, d^{10} x + 11 \,{\left (e^{9} x^{10} + 10 \, d e^{8} x^{9} + 18 \, d^{2} e^{7} x^{8} - 159 \, d^{4} e^{5} x^{6} - 402 \, d^{5} e^{4} x^{5} + 84 \, d^{6} e^{3} x^{4} + 336 \, d^{7} e^{2} x^{3} + 144 \, d^{8} e x^{2} + 288 \, d^{9} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{99 \,{\left (d^{2} e^{11} x^{11} - 33 \, d^{4} e^{9} x^{9} - 110 \, d^{5} e^{8} x^{8} - 77 \, d^{6} e^{7} x^{7} + 220 \, d^{7} e^{6} x^{6} + 473 \, d^{8} e^{5} x^{5} + 242 \, d^{9} e^{4} x^{4} - 220 \, d^{10} e^{3} x^{3} - 352 \, d^{11} e^{2} x^{2} - 176 \, d^{12} e x - 32 \, d^{13} +{\left (d^{2} e^{10} x^{10} + 11 \, d^{3} e^{9} x^{9} + 28 \, d^{4} e^{8} x^{8} - 22 \, d^{5} e^{7} x^{7} - 199 \, d^{6} e^{6} x^{6} - 297 \, d^{7} e^{5} x^{5} - 54 \, d^{8} e^{4} x^{4} + 308 \, d^{9} e^{3} x^{3} + 368 \, d^{10} e^{2} x^{2} + 176 \, d^{11} e x + 32 \, d^{12}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^10,x, algorithm="fricas")

[Out]

-1/99*(9*e^10*x^11 - 11*d*e^9*x^10 - 308*d^2*e^8*x^9 - 858*d^3*e^7*x^8 - 1551*d^
4*e^6*x^7 + 2409*d^5*e^5*x^6 + 6666*d^6*e^4*x^5 - 132*d^7*e^3*x^4 - 2112*d^8*e^2
*x^3 - 1584*d^9*e*x^2 - 3168*d^10*x + 11*(e^9*x^10 + 10*d*e^8*x^9 + 18*d^2*e^7*x
^8 - 159*d^4*e^5*x^6 - 402*d^5*e^4*x^5 + 84*d^6*e^3*x^4 + 336*d^7*e^2*x^3 + 144*
d^8*e*x^2 + 288*d^9*x)*sqrt(-e^2*x^2 + d^2))/(d^2*e^11*x^11 - 33*d^4*e^9*x^9 - 1
10*d^5*e^8*x^8 - 77*d^6*e^7*x^7 + 220*d^7*e^6*x^6 + 473*d^8*e^5*x^5 + 242*d^9*e^
4*x^4 - 220*d^10*e^3*x^3 - 352*d^11*e^2*x^2 - 176*d^12*e*x - 32*d^13 + (d^2*e^10
*x^10 + 11*d^3*e^9*x^9 + 28*d^4*e^8*x^8 - 22*d^5*e^7*x^7 - 199*d^6*e^6*x^6 - 297
*d^7*e^5*x^5 - 54*d^8*e^4*x^4 + 308*d^9*e^3*x^3 + 368*d^10*e^2*x^2 + 176*d^11*e*
x + 32*d^12)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

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)**10,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 1.01868, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^10,x, algorithm="giac")

[Out]

Done