3.182 \(\int \frac{x^2}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{d \sqrt{d^2-e^2 x^2}}{5 e^3 (d+e x)^3}+\frac{8 \sqrt{d^2-e^2 x^2}}{15 e^3 (d+e x)^2}-\frac{7 \sqrt{d^2-e^2 x^2}}{15 d e^3 (d+e x)} \]

[Out]

-(d*Sqrt[d^2 - e^2*x^2])/(5*e^3*(d + e*x)^3) + (8*Sqrt[d^2 - e^2*x^2])/(15*e^3*(
d + e*x)^2) - (7*Sqrt[d^2 - e^2*x^2])/(15*d*e^3*(d + e*x))

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Rubi [A]  time = 0.274871, antiderivative size = 95, 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 \[ -\frac{d \sqrt{d^2-e^2 x^2}}{5 e^3 (d+e x)^3}+\frac{8 \sqrt{d^2-e^2 x^2}}{15 e^3 (d+e x)^2}-\frac{7 \sqrt{d^2-e^2 x^2}}{15 d e^3 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*Sqrt[d^2 - e^2*x^2])/(5*e^3*(d + e*x)^3) + (8*Sqrt[d^2 - e^2*x^2])/(15*e^3*(
d + e*x)^2) - (7*Sqrt[d^2 - e^2*x^2])/(15*d*e^3*(d + e*x))

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Rubi in Sympy [A]  time = 30.1672, size = 80, normalized size = 0.84 \[ - \frac{d \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{3} \left (d + e x\right )^{3}} + \frac{8 \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{3} \left (d + e x\right )^{2}} - \frac{7 \sqrt{d^{2} - e^{2} x^{2}}}{15 d e^{3} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)

[Out]

-d*sqrt(d**2 - e**2*x**2)/(5*e**3*(d + e*x)**3) + 8*sqrt(d**2 - e**2*x**2)/(15*e
**3*(d + e*x)**2) - 7*sqrt(d**2 - e**2*x**2)/(15*d*e**3*(d + e*x))

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Mathematica [A]  time = 0.0436854, size = 52, normalized size = 0.55 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (2 d^2+6 d e x+7 e^2 x^2\right )}{15 d e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d^2 - e^2*x^2]*(2*d^2 + 6*d*e*x + 7*e^2*x^2))/(15*d*e^3*(d + e*x)^3)

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Maple [A]  time = 0.011, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( -ex+d \right ) \left ( 7\,{e}^{2}{x}^{2}+6\,dex+2\,{d}^{2} \right ) }{15\,{e}^{3}d \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/15*(-e*x+d)*(7*e^2*x^2+6*d*e*x+2*d^2)/(e*x+d)^2/d/e^3/(-e^2*x^2+d^2)^(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(x^2/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283272, size = 213, normalized size = 2.24 \[ -\frac{9 \, e^{2} x^{5} - 5 \, d e x^{4} - 20 \, d^{2} x^{3} + 5 \,{\left (e x^{4} + 4 \, d x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d e^{5} x^{5} + 5 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} - 5 \, d^{4} e^{2} x^{2} - 10 \, d^{5} e x - 4 \, d^{6} -{\left (d e^{4} x^{4} - 7 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(9*e^2*x^5 - 5*d*e*x^4 - 20*d^2*x^3 + 5*(e*x^4 + 4*d*x^3)*sqrt(-e^2*x^2 +
d^2))/(d*e^5*x^5 + 5*d^2*e^4*x^4 + 5*d^3*e^3*x^3 - 5*d^4*e^2*x^2 - 10*d^5*e*x -
4*d^6 - (d*e^4*x^4 - 7*d^3*e^2*x^2 - 10*d^4*e*x - 4*d^5)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError