3.829 \(\int \frac{(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0511627, antiderivative size = 53, 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 \[ \frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.03719, size = 44, normalized size = 0.83 \[ \frac{2 \left (d + e x\right )}{3 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{x}{3 d^{2} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0348407, size = 42, normalized size = 0.79 \[ \frac{(2 d-e x) \sqrt{d^2-e^2 x^2}}{3 d^2 e (d-e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(d^2 - e^2*x^2)^(5/2),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 = 44, normalized size = 0.8 \[{\frac{ \left ( ex+d \right ) ^{3} \left ( -ex+d \right ) \left ( -ex+2\,d \right ) }{3\,{d}^{2}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.714753, size = 78, normalized size = 1.47 \[ \frac{2 \, x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{2 \, d}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} + \frac{x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218861, size = 153, normalized size = 2.89 \[ \frac{e^{2} x^{3} + 3 \, d e x^{2} - 6 \, d^{2} x - 3 \, \sqrt{-e^{2} x^{2} + d^{2}}{\left (e x^{2} - 2 \, d x\right )}}{3 \,{\left (d^{2} e^{3} x^{3} - 3 \, d^{4} e x + 2 \, d^{5} -{\left (d^{2} e^{2} x^{2} - 3 \, d^{3} e x + 2 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(-e^2*x^2 + d^2)^(5/2),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. \[ \int \frac{\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**2/(-(-d + e*x)*(d + e*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.228155, size = 65, normalized size = 1.23 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (x{\left (\frac{x^{2} e^{2}}{d^{2}} - 3\right )} - 2 \, d e^{\left (-1\right )}\right )}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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