3.203 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=113 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac{5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac{15 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{15 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

(-15*d*Sqrt[d^2 - e^2*x^2])/(2*e) - (5*(d^2 - e^2*x^2)^(3/2))/(2*e*(d + e*x)) -
(2*(d^2 - e^2*x^2)^(5/2))/(e*(d + e*x)^3) - (15*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*
x^2]])/(2*e)

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Rubi [A]  time = 0.115866, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac{5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac{15 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{15 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4,x]

[Out]

(-15*d*Sqrt[d^2 - e^2*x^2])/(2*e) - (5*(d^2 - e^2*x^2)^(3/2))/(2*e*(d + e*x)) -
(2*(d^2 - e^2*x^2)^(5/2))/(e*(d + e*x)^3) - (15*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*
x^2]])/(2*e)

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Rubi in Sympy [A]  time = 23.0227, size = 95, normalized size = 0.84 \[ - \frac{15 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} - \frac{15 d \sqrt{d^{2} - e^{2} x^{2}}}{2 e} - \frac{5 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{e \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(5/2)/(e*x+d)**4,x)

[Out]

-15*d**2*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e) - 15*d*sqrt(d**2 - e**2*x**2)/(2
*e) - 5*(d**2 - e**2*x**2)**(3/2)/(2*e*(d + e*x)) - 2*(d**2 - e**2*x**2)**(5/2)/
(e*(d + e*x)**3)

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Mathematica [A]  time = 0.0997608, size = 75, normalized size = 0.66 \[ \sqrt{d^2-e^2 x^2} \left (-\frac{8 d^2}{e (d+e x)}-\frac{4 d}{e}+\frac{x}{2}\right )-\frac{15 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4,x]

[Out]

Sqrt[d^2 - e^2*x^2]*((-4*d)/e + x/2 - (8*d^2)/(e*(d + e*x))) - (15*d^2*ArcTan[(e
*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Maple [B]  time = 0.008, size = 284, normalized size = 2.5 \[ -{\frac{1}{{e}^{5}d} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}}-3\,{\frac{1}{{e}^{4}{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{7/2} \left ( x+{\frac{d}{e}} \right ) ^{-3}}-4\,{\frac{1}{{e}^{3}{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{7/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}}-4\,{\frac{1}{e{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{5/2}}-5\,{\frac{x}{{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2}}-{\frac{15\,x}{2}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{15\,{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x)

[Out]

-1/e^5/d/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)-3/e^4/d^2/(x+d/e)^3*(-(x
+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)-4/e^3/d^3/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/
e))^(7/2)-4/e/d^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-5/d^2*(-(x+d/e)^2*e^2+2*d
*e*(x+d/e))^(3/2)*x-15/2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-15/2*d^2/(e^2)^(
1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(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((-e^2*x^2 + d^2)^(5/2)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289165, size = 366, normalized size = 3.24 \[ \frac{e^{5} x^{5} - 10 \, d e^{4} x^{4} - 29 \, d^{2} e^{3} x^{3} + 18 \, d^{3} e^{2} x^{2} + 68 \, d^{4} e x + 30 \,{\left (d^{2} e^{3} x^{3} + 3 \, d^{3} e^{2} x^{2} - 2 \, d^{4} e x - 4 \, d^{5} -{\left (d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (e^{4} x^{4} - 5 \, d e^{3} x^{3} - 18 \, d^{2} e^{2} x^{2} - 68 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} - 2 \, d^{2} e^{2} x - 4 \, d^{3} e -{\left (e^{3} x^{2} - 2 \, d e^{2} x - 4 \, d^{2} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/2*(e^5*x^5 - 10*d*e^4*x^4 - 29*d^2*e^3*x^3 + 18*d^3*e^2*x^2 + 68*d^4*e*x + 30*
(d^2*e^3*x^3 + 3*d^3*e^2*x^2 - 2*d^4*e*x - 4*d^5 - (d^2*e^2*x^2 - 2*d^3*e*x - 4*
d^4)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (e^4*x^4
- 5*d*e^3*x^3 - 18*d^2*e^2*x^2 - 68*d^3*e*x)*sqrt(-e^2*x^2 + d^2))/(e^4*x^3 + 3*
d*e^3*x^2 - 2*d^2*e^2*x - 4*d^3*e - (e^3*x^2 - 2*d*e^2*x - 4*d^2*e)*sqrt(-e^2*x^
2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(5/2)/(e*x+d)**4,x)

[Out]

Integral((-(-d + e*x)*(d + e*x))**(5/2)/(d + e*x)**4, x)

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GIAC/XCAS [A]  time = 0.301862, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/(e*x + d)^4,x, algorithm="giac")

[Out]

Done