3.822 \(\int \frac{1}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.120878, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.2549, size = 82, normalized size = 0.82 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d e \left (d + e x\right )^{3}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{2} e \left (d + e x\right )^{2}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{3} e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0393493, size = 52, normalized size = 0.52 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (7 d^2+6 d e x+2 e^2 x^2\right )}{15 d^3 e (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{2}{x}^{2}+6\,dex+7\,{d}^{2} \right ) }{15\,e{d}^{3} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(-e*x+d)*(2*e^2*x^2+6*d*e*x+7*d^2)/(e*x+d)^2/d^3/e/(-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(1/(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 = 0.225133, size = 273, normalized size = 2.73 \[ -\frac{9 \, e^{4} x^{5} + 35 \, d e^{3} x^{4} + 20 \, d^{2} e^{2} x^{3} - 60 \, d^{3} e x^{2} - 60 \, d^{4} x - 5 \,{\left (e^{3} x^{4} - 2 \, d e^{2} x^{3} - 12 \, d^{2} e x^{2} - 12 \, d^{3} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{5} x^{5} + 5 \, d^{4} e^{4} x^{4} + 5 \, d^{5} e^{3} x^{3} - 5 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x - 4 \, d^{8} -{\left (d^{3} e^{4} x^{4} - 7 \, d^{5} e^{2} x^{2} - 10 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

-1/15*(9*e^4*x^5 + 35*d*e^3*x^4 + 20*d^2*e^2*x^3 - 60*d^3*e*x^2 - 60*d^4*x - 5*(
e^3*x^4 - 2*d*e^2*x^3 - 12*d^2*e*x^2 - 12*d^3*x)*sqrt(-e^2*x^2 + d^2))/(d^3*e^5*
x^5 + 5*d^4*e^4*x^4 + 5*d^5*e^3*x^3 - 5*d^6*e^2*x^2 - 10*d^7*e*x - 4*d^8 - (d^3*
e^4*x^4 - 7*d^5*e^2*x^2 - 10*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. \[ \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(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(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError