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

Optimal. Leaf size=97 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.144726, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.6547, size = 78, normalized size = 0.8 \[ \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 e^{2} \left (d + e x\right )^{3}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d e^{2} \left (d + e x\right )^{2}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d^{2} e^{2} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0390686, size = 49, normalized size = 0.51 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (d^2+3 d e x+e^2 x^2\right )}{5 d^2 e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 52, normalized size = 0.5 \[ -{\frac{ \left ( -ex+d \right ) \left ({e}^{2}{x}^{2}+3\,dex+{d}^{2} \right ) }{5\,{e}^{2}{d}^{2} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(-e*x+d)*(e^2*x^2+3*d*e*x+d^2)/(e*x+d)^2/d^2/e^2/(-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/(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 = 0.281395, size = 238, normalized size = 2.45 \[ -\frac{2 \, e^{3} x^{5} + 5 \, d e^{2} x^{4} - 5 \, d^{2} e x^{3} - 10 \, d^{3} x^{2} + 5 \,{\left (d e x^{3} + 2 \, d^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{2} e^{5} x^{5} + 5 \, d^{3} e^{4} x^{4} + 5 \, d^{4} e^{3} x^{3} - 5 \, d^{5} e^{2} x^{2} - 10 \, d^{6} e x - 4 \, d^{7} -{\left (d^{2} e^{4} x^{4} - 7 \, d^{4} e^{2} x^{2} - 10 \, d^{5} e x - 4 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(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/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError