3.824 \(\int \frac{1}{(d+e x)^5 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3} \]

[Out]

-Sqrt[d^2 - e^2*x^2]/(9*d*e*(d + e*x)^5) - (4*Sqrt[d^2 - e^2*x^2])/(63*d^2*e*(d
+ e*x)^4) - (4*Sqrt[d^2 - e^2*x^2])/(105*d^3*e*(d + e*x)^3) - (8*Sqrt[d^2 - e^2*
x^2])/(315*d^4*e*(d + e*x)^2) - (8*Sqrt[d^2 - e^2*x^2])/(315*d^5*e*(d + e*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.225636, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^5*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-Sqrt[d^2 - e^2*x^2]/(9*d*e*(d + e*x)^5) - (4*Sqrt[d^2 - e^2*x^2])/(63*d^2*e*(d
+ e*x)^4) - (4*Sqrt[d^2 - e^2*x^2])/(105*d^3*e*(d + e*x)^3) - (8*Sqrt[d^2 - e^2*
x^2])/(315*d^4*e*(d + e*x)^2) - (8*Sqrt[d^2 - e^2*x^2])/(315*d^5*e*(d + e*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.8762, size = 139, normalized size = 0.84 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{9 d e \left (d + e x\right )^{5}} - \frac{4 \sqrt{d^{2} - e^{2} x^{2}}}{63 d^{2} e \left (d + e x\right )^{4}} - \frac{4 \sqrt{d^{2} - e^{2} x^{2}}}{105 d^{3} e \left (d + e x\right )^{3}} - \frac{8 \sqrt{d^{2} - e^{2} x^{2}}}{315 d^{4} e \left (d + e x\right )^{2}} - \frac{8 \sqrt{d^{2} - e^{2} x^{2}}}{315 d^{5} e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**5/(-e**2*x**2+d**2)**(1/2),x)

[Out]

-sqrt(d**2 - e**2*x**2)/(9*d*e*(d + e*x)**5) - 4*sqrt(d**2 - e**2*x**2)/(63*d**2
*e*(d + e*x)**4) - 4*sqrt(d**2 - e**2*x**2)/(105*d**3*e*(d + e*x)**3) - 8*sqrt(d
**2 - e**2*x**2)/(315*d**4*e*(d + e*x)**2) - 8*sqrt(d**2 - e**2*x**2)/(315*d**5*
e*(d + e*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0495135, size = 74, normalized size = 0.45 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (83 d^4+100 d^3 e x+84 d^2 e^2 x^2+40 d e^3 x^3+8 e^4 x^4\right )}{315 d^5 e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^5*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(83*d^4 + 100*d^3*e*x + 84*d^2*e^2*x^2 + 40*d*e^3*x^3 + 8*
e^4*x^4))/(315*d^5*e*(d + e*x)^5)

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 77, normalized size = 0.5 \[ -{\frac{ \left ( -ex+d \right ) \left ( 8\,{e}^{4}{x}^{4}+40\,{e}^{3}{x}^{3}d+84\,{e}^{2}{x}^{2}{d}^{2}+100\,x{d}^{3}e+83\,{d}^{4} \right ) }{315\,e{d}^{5} \left ( ex+d \right ) ^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/315*(-e*x+d)*(8*e^4*x^4+40*d*e^3*x^3+84*d^2*e^2*x^2+100*d^3*e*x+83*d^4)/(e*x+
d)^4/d^5/e/(-e^2*x^2+d^2)^(1/2)

_______________________________________________________________________________________

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(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.255071, size = 512, normalized size = 3.08 \[ -\frac{91 \, e^{8} x^{9} + 747 \, d e^{7} x^{8} + 1314 \, d^{2} e^{6} x^{7} - 1974 \, d^{3} e^{5} x^{6} - 8442 \, d^{4} e^{4} x^{5} - 7560 \, d^{5} e^{3} x^{4} + 3360 \, d^{6} e^{2} x^{3} + 10080 \, d^{7} e x^{2} + 5040 \, d^{8} x - 3 \,{\left (25 \, e^{7} x^{8} - 24 \, d e^{6} x^{7} - 658 \, d^{2} e^{5} x^{6} - 1624 \, d^{3} e^{4} x^{5} - 840 \, d^{4} e^{3} x^{4} + 1960 \, d^{5} e^{2} x^{3} + 3360 \, d^{6} e x^{2} + 1680 \, d^{7} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{315 \,{\left (d^{5} e^{9} x^{9} + 9 \, d^{6} e^{8} x^{8} + 18 \, d^{7} e^{7} x^{7} - 18 \, d^{8} e^{6} x^{6} - 99 \, d^{9} e^{5} x^{5} - 99 \, d^{10} e^{4} x^{4} + 24 \, d^{11} e^{3} x^{3} + 108 \, d^{12} e^{2} x^{2} + 72 \, d^{13} e x + 16 \, d^{14} -{\left (d^{5} e^{8} x^{8} - 22 \, d^{7} e^{6} x^{6} - 60 \, d^{8} e^{5} x^{5} - 39 \, d^{9} e^{4} x^{4} + 60 \, d^{10} e^{3} x^{3} + 116 \, d^{11} e^{2} x^{2} + 72 \, d^{12} e x + 16 \, d^{13}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^5),x, algorithm="fricas")

[Out]

-1/315*(91*e^8*x^9 + 747*d*e^7*x^8 + 1314*d^2*e^6*x^7 - 1974*d^3*e^5*x^6 - 8442*
d^4*e^4*x^5 - 7560*d^5*e^3*x^4 + 3360*d^6*e^2*x^3 + 10080*d^7*e*x^2 + 5040*d^8*x
- 3*(25*e^7*x^8 - 24*d*e^6*x^7 - 658*d^2*e^5*x^6 - 1624*d^3*e^4*x^5 - 840*d^4*e
^3*x^4 + 1960*d^5*e^2*x^3 + 3360*d^6*e*x^2 + 1680*d^7*x)*sqrt(-e^2*x^2 + d^2))/(
d^5*e^9*x^9 + 9*d^6*e^8*x^8 + 18*d^7*e^7*x^7 - 18*d^8*e^6*x^6 - 99*d^9*e^5*x^5 -
99*d^10*e^4*x^4 + 24*d^11*e^3*x^3 + 108*d^12*e^2*x^2 + 72*d^13*e*x + 16*d^14 -
(d^5*e^8*x^8 - 22*d^7*e^6*x^6 - 60*d^8*e^5*x^5 - 39*d^9*e^4*x^4 + 60*d^10*e^3*x^
3 + 116*d^11*e^2*x^2 + 72*d^12*e*x + 16*d^13)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**5/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**5), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.626974, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^5),x, algorithm="giac")

[Out]

sage0*x