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

Optimal. Leaf size=89 \[ \frac{x}{15 d^2 e^2 \sqrt{d^2-e^2 x^2}}-\frac{d}{5 e^3 (d+e x)^2 \sqrt{d^2-e^2 x^2}}+\frac{7}{15 e^3 (d+e x) \sqrt{d^2-e^2 x^2}} \]

[Out]

x/(15*d^2*e^2*Sqrt[d^2 - e^2*x^2]) - d/(5*e^3*(d + e*x)^2*Sqrt[d^2 - e^2*x^2]) +
7/(15*e^3*(d + e*x)*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.311352, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{x}{15 d^2 e^2 \sqrt{d^2-e^2 x^2}}-\frac{d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 (d-e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(d - e*x)^2)/(5*e^3*(d^2 - e^2*x^2)^(5/2)) + (7*(d - e*x))/(15*e^3*(d^2 - e^
2*x^2)^(3/2)) + x/(15*d^2*e^2*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 25.727, size = 75, normalized size = 0.84 \[ - \frac{d}{5 e^{3} \left (d + e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{7}{15 e^{3} \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{x}{15 d^{2} e^{2} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(5*e**3*(d + e*x)**2*sqrt(d**2 - e**2*x**2)) + 7/(15*e**3*(d + e*x)*sqrt(d**2
- e**2*x**2)) + x/(15*d**2*e**2*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0483472, size = 70, normalized size = 0.79 \[ \frac{\sqrt{d^2-e^2 x^2} \left (4 d^3+8 d^2 e x+2 d e^2 x^2+e^3 x^3\right )}{15 d^2 e^3 (d-e x) (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(4*d^3 + 8*d^2*e*x + 2*d*e^2*x^2 + e^3*x^3))/(15*d^2*e^3*(d
- e*x)*(d + e*x)^3)

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Maple [A]  time = 0.012, size = 65, normalized size = 0.7 \[{\frac{ \left ( -ex+d \right ) \left ({e}^{3}{x}^{3}+2\,d{e}^{2}{x}^{2}+8\,x{d}^{2}e+4\,{d}^{3} \right ) }{ \left ( 15\,ex+15\,d \right ){d}^{2}{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-e*x+d)*(e^3*x^3+2*d*e^2*x^2+8*d^2*e*x+4*d^3)/(e*x+d)/d^2/e^3/(-e^2*x^2+d^
2)^(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(x^2/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282937, size = 278, normalized size = 3.12 \[ \frac{4 \, e^{3} x^{6} + 11 \, d e^{2} x^{5} - 10 \, d^{2} e x^{4} - 20 \, d^{3} x^{3} -{\left (e^{2} x^{5} - 10 \, d e x^{4} - 20 \, d^{2} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{2} e^{6} x^{6} + 2 \, d^{3} e^{5} x^{5} - 4 \, d^{4} e^{4} x^{4} - 10 \, d^{5} e^{3} x^{3} - d^{6} e^{2} x^{2} + 8 \, d^{7} e x + 4 \, d^{8} +{\left (3 \, d^{3} e^{4} x^{4} + 6 \, d^{4} e^{3} x^{3} - d^{5} e^{2} x^{2} - 8 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(4*e^3*x^6 + 11*d*e^2*x^5 - 10*d^2*e*x^4 - 20*d^3*x^3 - (e^2*x^5 - 10*d*e*x
^4 - 20*d^2*x^3)*sqrt(-e^2*x^2 + d^2))/(d^2*e^6*x^6 + 2*d^3*e^5*x^5 - 4*d^4*e^4*
x^4 - 10*d^5*e^3*x^3 - d^6*e^2*x^2 + 8*d^7*e*x + 4*d^8 + (3*d^3*e^4*x^4 + 6*d^4*
e^3*x^3 - d^5*e^2*x^2 - 8*d^6*e*x - 4*d^7)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.632812, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x