3.100 \(\int \frac{\sqrt{d^2-e^2 x^2}}{x^4 (d+e x)} \, dx\)

Optimal. Leaf size=114 \[ \frac{e \sqrt{d^2-e^2 x^2}}{2 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{2 e^2 \sqrt{d^2-e^2 x^2}}{3 d^3 x}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^3} \]

[Out]

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

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Rubi [A]  time = 0.364858, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{e \sqrt{d^2-e^2 x^2}}{2 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{2 e^2 \sqrt{d^2-e^2 x^2}}{3 d^3 x}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d^2 - e^2*x^2]/(x^4*(d + e*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 38.3383, size = 94, normalized size = 0.82 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{3 d x^{3}} + \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{2 d^{2} x^{2}} + \frac{e^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{3}} - \frac{2 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(1/2)/x**4/(e*x+d),x)

[Out]

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

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Mathematica [A]  time = 0.0764567, size = 84, normalized size = 0.74 \[ \frac{\left (-2 d^2+3 d e x-4 e^2 x^2\right ) \sqrt{d^2-e^2 x^2}+3 e^3 x^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-3 e^3 x^3 \log (x)}{6 d^3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d^2 - e^2*x^2]/(x^4*(d + e*x)),x]

[Out]

((-2*d^2 + 3*d*e*x - 4*e^2*x^2)*Sqrt[d^2 - e^2*x^2] - 3*e^3*x^3*Log[x] + 3*e^3*x
^3*Log[d + Sqrt[d^2 - e^2*x^2]])/(6*d^3*x^3)

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Maple [B]  time = 0.017, size = 280, normalized size = 2.5 \[ -{\frac{1}{3\,{d}^{3}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{3}}{{d}^{4}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{{e}^{4}}{{d}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{2}}{{d}^{5}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{4}x}{{d}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{4}}{{d}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{3}}{2\,{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{e}^{3}}{2\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{e}{2\,{d}^{4}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(1/2)/x^4/(e*x+d),x)

[Out]

-1/3/d^3/x^3*(-e^2*x^2+d^2)^(3/2)+1/d^4*e^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)
+1/d^3*e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)
)-1/d^5*e^2/x*(-e^2*x^2+d^2)^(3/2)-1/d^5*e^4*x*(-e^2*x^2+d^2)^(1/2)-1/d^3*e^4/(e
^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/2/d^4*e^3*(-e^2*x^2+d^2)^
(1/2)+1/2/d^2*e^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+1
/2*e/d^4/x^2*(-e^2*x^2+d^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^4),x, algorithm="maxima")

[Out]

integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^4), x)

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Fricas [A]  time = 0.284932, size = 355, normalized size = 3.11 \[ -\frac{4 \, e^{6} x^{6} - 3 \, d e^{5} x^{5} - 18 \, d^{2} e^{4} x^{4} + 15 \, d^{3} e^{3} x^{3} + 6 \, d^{4} e^{2} x^{2} - 12 \, d^{5} e x + 8 \, d^{6} + 3 \,{\left (3 \, d e^{5} x^{5} - 4 \, d^{3} e^{3} x^{3} -{\left (e^{5} x^{5} - 4 \, d^{2} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (12 \, d e^{4} x^{4} - 9 \, d^{2} e^{3} x^{3} - 10 \, d^{3} e^{2} x^{2} + 12 \, d^{4} e x - 8 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (3 \, d^{4} e^{2} x^{5} - 4 \, d^{6} x^{3} -{\left (d^{3} e^{2} x^{5} - 4 \, d^{5} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^4),x, algorithm="fricas")

[Out]

-1/6*(4*e^6*x^6 - 3*d*e^5*x^5 - 18*d^2*e^4*x^4 + 15*d^3*e^3*x^3 + 6*d^4*e^2*x^2
- 12*d^5*e*x + 8*d^6 + 3*(3*d*e^5*x^5 - 4*d^3*e^3*x^3 - (e^5*x^5 - 4*d^2*e^3*x^3
)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (12*d*e^4*x^4 - 9*d
^2*e^3*x^3 - 10*d^3*e^2*x^2 + 12*d^4*e*x - 8*d^5)*sqrt(-e^2*x^2 + d^2))/(3*d^4*e
^2*x^5 - 4*d^6*x^3 - (d^3*e^2*x^5 - 4*d^5*x^3)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(1/2)/x**4/(e*x+d),x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x**4*(d + e*x)), 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(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError