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

Optimal. Leaf size=67 \[ -\frac{\sqrt{d^2-e^2 x^2}}{3 d^2 e (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{3 d e (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0795455, 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 \[ -\frac{\sqrt{d^2-e^2 x^2}}{3 d^2 e (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{3 d e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.69196, size = 51, normalized size = 0.76 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{3 d e \left (d + e x\right )^{2}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{3 d^{2} e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0332331, size = 40, normalized size = 0.6 \[ -\frac{(2 d+e x) \sqrt{d^2-e^2 x^2}}{3 d^2 e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 43, normalized size = 0.6 \[ -{\frac{ \left ( -ex+d \right ) \left ( ex+2\,d \right ) }{ \left ( 3\,ex+3\,d \right ){d}^{2}e}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-e*x+d)*(e*x+2*d)/(e*x+d)/d^2/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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226065, size = 151, normalized size = 2.25 \[ -\frac{e^{2} x^{3} - 3 \, d e x^{2} - 6 \, d^{2} x + 3 \, \sqrt{-e^{2} x^{2} + d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{3 \,{\left (d^{2} e^{3} x^{3} - 3 \, d^{4} e x - 2 \, d^{5} +{\left (d^{2} e^{2} x^{2} + 3 \, d^{3} e x + 2 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(e^2*x^3 - 3*d*e*x^2 - 6*d^2*x + 3*sqrt(-e^2*x^2 + d^2)*(e*x^2 + 2*d*x))/(d
^2*e^3*x^3 - 3*d^4*e*x - 2*d^5 + (d^2*e^2*x^2 + 3*d^3*e*x + 2*d^4)*sqrt(-e^2*x^2
+ d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**2), x)

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GIAC/XCAS [A]  time = 0.616004, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x