3.823 \(\int \frac{1}{(d+e x)^4 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2} \]

[Out]

-Sqrt[d^2 - e^2*x^2]/(7*d*e*(d + e*x)^4) - (3*Sqrt[d^2 - e^2*x^2])/(35*d^2*e*(d
+ e*x)^3) - (2*Sqrt[d^2 - e^2*x^2])/(35*d^3*e*(d + e*x)^2) - (2*Sqrt[d^2 - e^2*x
^2])/(35*d^4*e*(d + e*x))

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Rubi [A]  time = 0.168746, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-Sqrt[d^2 - e^2*x^2]/(7*d*e*(d + e*x)^4) - (3*Sqrt[d^2 - e^2*x^2])/(35*d^2*e*(d
+ e*x)^3) - (2*Sqrt[d^2 - e^2*x^2])/(35*d^3*e*(d + e*x)^2) - (2*Sqrt[d^2 - e^2*x
^2])/(35*d^4*e*(d + e*x))

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Rubi in Sympy [A]  time = 18.8941, size = 110, normalized size = 0.83 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{7 d e \left (d + e x\right )^{4}} - \frac{3 \sqrt{d^{2} - e^{2} x^{2}}}{35 d^{2} e \left (d + e x\right )^{3}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{35 d^{3} e \left (d + e x\right )^{2}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{35 d^{4} e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(d**2 - e**2*x**2)/(7*d*e*(d + e*x)**4) - 3*sqrt(d**2 - e**2*x**2)/(35*d**2
*e*(d + e*x)**3) - 2*sqrt(d**2 - e**2*x**2)/(35*d**3*e*(d + e*x)**2) - 2*sqrt(d*
*2 - e**2*x**2)/(35*d**4*e*(d + e*x))

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Mathematica [A]  time = 0.0447698, size = 63, normalized size = 0.47 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (12 d^3+13 d^2 e x+8 d e^2 x^2+2 e^3 x^3\right )}{35 d^4 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(12*d^3 + 13*d^2*e*x + 8*d*e^2*x^2 + 2*e^3*x^3))/(35*d^4*e
*(d + e*x)^4)

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Maple [A]  time = 0.011, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{3}{x}^{3}+8\,{e}^{2}{x}^{2}d+13\,x{d}^{2}e+12\,{d}^{3} \right ) }{35\,e{d}^{4} \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(-e*x+d)*(2*e^3*x^3+8*d*e^2*x^2+13*d^2*e*x+12*d^3)/(e*x+d)^3/d^4/e/(-e^2*x
^2+d^2)^(1/2)

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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(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228687, size = 377, normalized size = 2.83 \[ -\frac{10 \, e^{6} x^{7} - 14 \, d e^{5} x^{6} - 189 \, d^{2} e^{4} x^{5} - 315 \, d^{3} e^{3} x^{4} + 420 \, d^{5} e x^{2} + 280 \, d^{6} x + 7 \,{\left (2 \, e^{5} x^{6} + 12 \, d e^{4} x^{5} + 15 \, d^{2} e^{3} x^{4} - 20 \, d^{3} e^{2} x^{3} - 60 \, d^{4} e x^{2} - 40 \, d^{5} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{35 \,{\left (d^{4} e^{7} x^{7} - 14 \, d^{6} e^{5} x^{5} - 28 \, d^{7} e^{4} x^{4} - 7 \, d^{8} e^{3} x^{3} + 28 \, d^{9} e^{2} x^{2} + 28 \, d^{10} e x + 8 \, d^{11} +{\left (d^{4} e^{6} x^{6} + 7 \, d^{5} e^{5} x^{5} + 11 \, d^{6} e^{4} x^{4} - 7 \, d^{7} e^{3} x^{3} - 32 \, d^{8} e^{2} x^{2} - 28 \, d^{9} e x - 8 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

-1/35*(10*e^6*x^7 - 14*d*e^5*x^6 - 189*d^2*e^4*x^5 - 315*d^3*e^3*x^4 + 420*d^5*e
*x^2 + 280*d^6*x + 7*(2*e^5*x^6 + 12*d*e^4*x^5 + 15*d^2*e^3*x^4 - 20*d^3*e^2*x^3
- 60*d^4*e*x^2 - 40*d^5*x)*sqrt(-e^2*x^2 + d^2))/(d^4*e^7*x^7 - 14*d^6*e^5*x^5
- 28*d^7*e^4*x^4 - 7*d^8*e^3*x^3 + 28*d^9*e^2*x^2 + 28*d^10*e*x + 8*d^11 + (d^4*
e^6*x^6 + 7*d^5*e^5*x^5 + 11*d^6*e^4*x^4 - 7*d^7*e^3*x^3 - 32*d^8*e^2*x^2 - 28*d
^9*e*x - 8*d^10)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**4), x)

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GIAC/XCAS [A]  time = 0.239057, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^4),x, algorithm="giac")

[Out]

Done