3.191 \(\int \frac{x^2 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=115 \[ \frac{3 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^3 (d+e x)^3}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{5 e^3 (d+e x)^4}-\frac{2 \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]

[Out]

(-2*Sqrt[d^2 - e^2*x^2])/(e^3*(d + e*x)) - (d*(d^2 - e^2*x^2)^(3/2))/(5*e^3*(d +
e*x)^4) + (3*(d^2 - e^2*x^2)^(3/2))/(5*e^3*(d + e*x)^3) - ArcTan[(e*x)/Sqrt[d^2
- e^2*x^2]]/e^3

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Rubi [A]  time = 0.278848, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{3 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^3 (d+e x)^3}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{5 e^3 (d+e x)^4}-\frac{2 \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d^2 - e^2*x^2])/(e^3*(d + e*x)) - (d*(d^2 - e^2*x^2)^(3/2))/(5*e^3*(d +
e*x)^4) + (3*(d^2 - e^2*x^2)^(3/2))/(5*e^3*(d + e*x)^3) - ArcTan[(e*x)/Sqrt[d^2
- e^2*x^2]]/e^3

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Rubi in Sympy [A]  time = 31.4054, size = 99, normalized size = 0.86 \[ - \frac{d \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 e^{3} \left (d + e x\right )^{4}} - \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{3}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{e^{3} \left (d + e x\right )} + \frac{3 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 e^{3} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

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

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Mathematica [A]  time = 0.135356, size = 73, normalized size = 0.63 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (8 d^2+19 d e x+13 e^2 x^2\right )}{(d+e x)^3}+5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{5 e^3} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[d^2 - e^2*x^2]*(8*d^2 + 19*d*e*x + 13*e^2*x^2))/(d + e*x)^3 + 5*ArcTan[(
e*x)/Sqrt[d^2 - e^2*x^2]])/(5*e^3)

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Maple [B]  time = 0.016, size = 214, normalized size = 1.9 \[ -{\frac{1}{{e}^{5}d} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}}-{\frac{1}{{e}^{3}d}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{1}{{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d}{5\,{e}^{7}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}}+{\frac{3}{5\,{e}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x)

[Out]

-1/e^5/d/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-1/e^3/d*(-(x+d/e)^2*e^2+
2*d*e*(x+d/e))^(1/2)-1/e^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))-1/5*d/e^7/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+3/5/e
^6/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)

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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(sqrt(-e^2*x^2 + d^2)*x^2/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294436, size = 455, normalized size = 3.96 \[ -\frac{21 \, e^{5} x^{5} + 20 \, d e^{4} x^{4} - 35 \, d^{2} e^{3} x^{3} - 50 \, d^{3} e^{2} x^{2} - 20 \, d^{4} e x - 10 \,{\left (e^{5} x^{5} + 5 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} - 5 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x - 4 \, d^{5} -{\left (e^{4} x^{4} - 7 \, d^{2} e^{2} x^{2} - 10 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (e^{4} x^{4} + 9 \, d e^{3} x^{3} + 10 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 5 \, d^{2} e^{6} x^{3} - 5 \, d^{3} e^{5} x^{2} - 10 \, d^{4} e^{4} x - 4 \, d^{5} e^{3} -{\left (e^{7} x^{4} - 7 \, d^{2} e^{5} x^{2} - 10 \, d^{3} e^{4} x - 4 \, d^{4} e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(21*e^5*x^5 + 20*d*e^4*x^4 - 35*d^2*e^3*x^3 - 50*d^3*e^2*x^2 - 20*d^4*e*x -
10*(e^5*x^5 + 5*d*e^4*x^4 + 5*d^2*e^3*x^3 - 5*d^3*e^2*x^2 - 10*d^4*e*x - 4*d^5
- (e^4*x^4 - 7*d^2*e^2*x^2 - 10*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)) + 5*(e^4*x^4 + 9*d*e^3*x^3 + 10*d^2*e^2*x^2 + 4
*d^3*e*x)*sqrt(-e^2*x^2 + d^2))/(e^8*x^5 + 5*d*e^7*x^4 + 5*d^2*e^6*x^3 - 5*d^3*e
^5*x^2 - 10*d^4*e^4*x - 4*d^5*e^3 - (e^7*x^4 - 7*d^2*e^5*x^2 - 10*d^3*e^4*x - 4*
d^4*e^3)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.295544, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done