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

Optimal. Leaf size=73 \[ \frac{d (d+e x)}{e^3 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.133938, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{d (d+e x)}{e^3 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.712, size = 63, normalized size = 0.86 \[ - \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{3}} + \frac{d \sqrt{d^{2} - e^{2} x^{2}}}{e^{3} \left (d - e x\right )} + \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.146113, size = 63, normalized size = 0.86 \[ \sqrt{d^2-e^2 x^2} \left (\frac{1}{e^3}-\frac{d}{e^3 (e x-d)}\right )-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 99, normalized size = 1.4 \[{\frac{dx}{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{d}{{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{x}^{2}}{e}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+2\,{\frac{{d}^{2}}{{e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.802032, size = 123, normalized size = 1.68 \[ -\frac{x^{2}}{\sqrt{-e^{2} x^{2} + d^{2}} e} + \frac{d x}{\sqrt{-e^{2} x^{2} + d^{2}} e^{2}} - \frac{d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e^{2}} + \frac{2 \, d^{2}}{\sqrt{-e^{2} x^{2} + d^{2}} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.277382, size = 243, normalized size = 3.33 \[ \frac{e^{3} x^{3} - d e^{2} x^{2} + 2 \, d^{2} e x + 2 \,{\left (d e^{2} x^{2} + d^{2} e x - 2 \, d^{3} - \sqrt{-e^{2} x^{2} + d^{2}}{\left (d e x - 2 \, d^{2}\right )}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (e^{2} x^{2} - 2 \, d e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{e^{5} x^{2} + d e^{4} x - 2 \, d^{2} e^{3} -{\left (e^{4} x - 2 \, d e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.7418, size = 163, normalized size = 2.23 \[ d \left (\begin{cases} \frac{i \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{e^{3}} - \frac{i x}{d e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{\operatorname{asin}{\left (\frac{e x}{d} \right )}}{e^{3}} + \frac{x}{d e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} \tilde{\infty } x^{4} & \text{for}\: d = 0 \wedge e = 0 \\\frac{x^{4}}{4 \left (d^{2}\right )^{\frac{3}{2}}} & \text{for}\: e = 0 \\\tilde{\infty } x^{4} & \text{for}\: d = - \sqrt{e^{2} x^{2}} \vee d = \sqrt{e^{2} x^{2}} \\\frac{2 d^{2}}{e^{4} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{x^{2}}{e^{2} \sqrt{d^{2} - e^{2} x^{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((I*acosh(e*x/d)/e**3 - I*x/(d*e**2*sqrt(-1 + e**2*x**2/d**2)), Abs(e
**2*x**2/d**2) > 1), (-asin(e*x/d)/e**3 + x/(d*e**2*sqrt(1 - e**2*x**2/d**2)), T
rue)) + e*Piecewise((zoo*x**4, Eq(d, 0) & Eq(e, 0)), (x**4/(4*(d**2)**(3/2)), Eq
(e, 0)), (zoo*x**4, Eq(d, sqrt(e**2*x**2)) | Eq(d, -sqrt(e**2*x**2))), (2*d**2/(
e**4*sqrt(d**2 - e**2*x**2)) - x**2/(e**2*sqrt(d**2 - e**2*x**2)), True))

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GIAC/XCAS [A]  time = 0.296545, size = 89, normalized size = 1.22 \[ -d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )}{\rm sign}\left (d\right ) - \frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (2 \, d^{2} e^{\left (-3\right )} -{\left (x e^{\left (-1\right )} - d e^{\left (-2\right )}\right )} x\right )}}{x^{2} e^{2} - d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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