3.133 \(\int \frac{1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]

[Out]

(3*d - 2*e*x)/(3*d^4*Sqrt[d^2 - e^2*x^2]) + 1/(3*d^2*(d + e*x)*Sqrt[d^2 - e^2*x^
2]) - ArcTanh[Sqrt[d^2 - e^2*x^2]/d]/d^4

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Rubi [A]  time = 0.239111, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]

Antiderivative was successfully verified.

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

[Out]

(3*d - 2*e*x)/(3*d^4*Sqrt[d^2 - e^2*x^2]) + 1/(3*d^2*(d + e*x)*Sqrt[d^2 - e^2*x^
2]) - ArcTanh[Sqrt[d^2 - e^2*x^2]/d]/d^4

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Rubi in Sympy [A]  time = 27.8675, size = 70, normalized size = 0.8 \[ \frac{d - e x}{3 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{3 d - 2 e x}{3 d^{4} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.132128, size = 83, normalized size = 0.94 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (4 d^2+d e x-2 e^2 x^2\right )}{(d-e x) (d+e x)^2}-3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+3 \log (x)}{3 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(((4*d^2 + d*e*x - 2*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/((d - e*x)*(d + e*x)^2) + 3*L
og[x] - 3*Log[d + Sqrt[d^2 - e^2*x^2]])/(3*d^4)

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Maple [A]  time = 0.017, size = 142, normalized size = 1.6 \[{\frac{1}{{d}^{3}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{1}{3\,e{d}^{2}} \left ( x+{\frac{d}{e}} \right ) ^{-1}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{2\,ex}{3\,{d}^{4}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d^3/(-e^2*x^2+d^2)^(1/2)-1/d^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d
^2)^(1/2))/x)+1/3/d^2/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-2/3/d^4*e/(
-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)*x), x)

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Fricas [A]  time = 0.288523, size = 335, normalized size = 3.81 \[ \frac{2 \, e^{4} x^{4} + 7 \, d e^{3} x^{3} - 6 \, d^{3} e x + 3 \,{\left (2 \, d e^{3} x^{3} + 2 \, d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 2 \, d^{4} -{\left (e^{3} x^{3} + d e^{2} x^{2} - 2 \, d^{2} e x - 2 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) - 2 \,{\left (2 \, e^{3} x^{3} - 3 \, d^{2} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (2 \, d^{5} e^{3} x^{3} + 2 \, d^{6} e^{2} x^{2} - 2 \, d^{7} e x - 2 \, d^{8} -{\left (d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2} - 2 \, d^{6} e x - 2 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*e^4*x^4 + 7*d*e^3*x^3 - 6*d^3*e*x + 3*(2*d*e^3*x^3 + 2*d^2*e^2*x^2 - 2*d^
3*e*x - 2*d^4 - (e^3*x^3 + d*e^2*x^2 - 2*d^2*e*x - 2*d^3)*sqrt(-e^2*x^2 + d^2))*
log(-(d - sqrt(-e^2*x^2 + d^2))/x) - 2*(2*e^3*x^3 - 3*d^2*e*x)*sqrt(-e^2*x^2 + d
^2))/(2*d^5*e^3*x^3 + 2*d^6*e^2*x^2 - 2*d^7*e*x - 2*d^8 - (d^4*e^3*x^3 + d^5*e^2
*x^2 - 2*d^6*e*x - 2*d^7)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[undef, undef, undef, 1]