3.828 \(\int \frac{(d+e x)^3}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac{(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0409985, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.29009, size = 24, normalized size = 0.73 \[ \frac{\left (d + e x\right )^{3}}{3 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(-e**2*x**2+d**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0304966, size = 39, normalized size = 1.18 \[ \frac{(d+e x) \sqrt{d^2-e^2 x^2}}{3 d e (d-e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 36, normalized size = 1.1 \[{\frac{ \left ( ex+d \right ) ^{4} \left ( -ex+d \right ) }{3\,de} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.723042, size = 108, normalized size = 3.27 \[ \frac{e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{4 \, d x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{d^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} - \frac{x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(-e^2*x^2 + d^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.222858, size = 127, normalized size = 3.85 \[ \frac{2 \,{\left (e^{2} x^{3} - 3 \, d^{2} x + 3 \, \sqrt{-e^{2} x^{2} + d^{2}} d x\right )}}{3 \,{\left (d e^{3} x^{3} - 3 \, d^{3} e x + 2 \, d^{4} -{\left (d e^{2} x^{2} - 3 \, d^{2} e x + 2 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(-e^2*x^2 + d^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral((d + e*x)**3/(-(-d + e*x)*(d + e*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.228062, size = 76, normalized size = 2.3 \[ \frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (d^{2} e^{\left (-1\right )} +{\left (x{\left (\frac{x e^{2}}{d} + 3 \, e\right )} + 3 \, d\right )} x\right )}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(-e^2*x^2 + d^2)^(5/2),x, algorithm="giac")

[Out]

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