3.101 \(\int \frac{\sqrt{d^2-e^2 x^2}}{x^5 (d+e x)} \, dx\)

Optimal. Leaf size=143 \[ -\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2} \]

[Out]

-Sqrt[d^2 - e^2*x^2]/(4*d*x^4) + (e*Sqrt[d^2 - e^2*x^2])/(3*d^2*x^3) - (3*e^2*Sq
rt[d^2 - e^2*x^2])/(8*d^3*x^2) + (2*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^4*x) - (3*e^4*
ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^4)

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Rubi [A]  time = 0.455178, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d^2 - e^2*x^2]/(x^5*(d + e*x)),x]

[Out]

-Sqrt[d^2 - e^2*x^2]/(4*d*x^4) + (e*Sqrt[d^2 - e^2*x^2])/(3*d^2*x^3) - (3*e^2*Sq
rt[d^2 - e^2*x^2])/(8*d^3*x^2) + (2*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^4*x) - (3*e^4*
ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^4)

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Rubi in Sympy [A]  time = 50.6556, size = 122, normalized size = 0.85 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{4 d x^{4}} + \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{2} x^{3}} - \frac{3 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{8 d^{3} x^{2}} - \frac{3 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d^{4}} + \frac{2 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(1/2)/x**5/(e*x+d),x)

[Out]

-sqrt(d**2 - e**2*x**2)/(4*d*x**4) + e*sqrt(d**2 - e**2*x**2)/(3*d**2*x**3) - 3*
e**2*sqrt(d**2 - e**2*x**2)/(8*d**3*x**2) - 3*e**4*atanh(sqrt(d**2 - e**2*x**2)/
d)/(8*d**4) + 2*e**3*sqrt(d**2 - e**2*x**2)/(3*d**4*x)

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Mathematica [A]  time = 0.0947802, size = 95, normalized size = 0.66 \[ \frac{-9 e^4 x^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (-6 d^3+8 d^2 e x-9 d e^2 x^2+16 e^3 x^3\right )+9 e^4 x^4 \log (x)}{24 d^4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d^2 - e^2*x^2]/(x^5*(d + e*x)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-6*d^3 + 8*d^2*e*x - 9*d*e^2*x^2 + 16*e^3*x^3) + 9*e^4*x^4
*Log[x] - 9*e^4*x^4*Log[d + Sqrt[d^2 - e^2*x^2]])/(24*d^4*x^4)

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Maple [B]  time = 0.018, size = 304, normalized size = 2.1 \[ -{\frac{1}{4\,{d}^{3}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{2}}{8\,{d}^{5}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{4}}{8\,{d}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{4}}{8\,{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{{e}^{4}}{{d}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{{e}^{5}}{{d}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{{e}^{3}}{{d}^{6}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{5}x}{{d}^{6}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{e}^{5}}{{d}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{3\,{d}^{4}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(1/2)/x^5/(e*x+d),x)

[Out]

-1/4/d^3/x^4*(-e^2*x^2+d^2)^(3/2)-5/8/d^5*e^2/x^2*(-e^2*x^2+d^2)^(3/2)+3/8/d^5*e
^4*(-e^2*x^2+d^2)^(1/2)-3/8/d^3*e^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^
2+d^2)^(1/2))/x)-1/d^5*e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-1/d^4*e^5/(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^6*e^3/x*(-e
^2*x^2+d^2)^(3/2)+1/d^6*e^5*x*(-e^2*x^2+d^2)^(1/2)+1/d^4*e^5/(e^2)^(1/2)*arctan(
(e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/3*e/d^4/x^3*(-e^2*x^2+d^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^5),x, algorithm="maxima")

[Out]

integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^5), x)

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Fricas [A]  time = 0.289201, size = 428, normalized size = 2.99 \[ -\frac{64 \, d e^{7} x^{7} - 36 \, d^{2} e^{6} x^{6} - 160 \, d^{3} e^{5} x^{5} + 84 \, d^{4} e^{4} x^{4} + 32 \, d^{5} e^{3} x^{3} + 64 \, d^{7} e x - 48 \, d^{8} - 9 \,{\left (e^{8} x^{8} - 8 \, d^{2} e^{6} x^{6} + 8 \, d^{4} e^{4} x^{4} + 4 \,{\left (d e^{6} x^{6} - 2 \, d^{3} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (16 \, e^{7} x^{7} - 9 \, d e^{6} x^{6} - 120 \, d^{2} e^{5} x^{5} + 66 \, d^{3} e^{4} x^{4} + 64 \, d^{4} e^{3} x^{3} - 24 \, d^{5} e^{2} x^{2} + 64 \, d^{6} e x - 48 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (d^{4} e^{4} x^{8} - 8 \, d^{6} e^{2} x^{6} + 8 \, d^{8} x^{4} + 4 \,{\left (d^{5} e^{2} x^{6} - 2 \, d^{7} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^5),x, algorithm="fricas")

[Out]

-1/24*(64*d*e^7*x^7 - 36*d^2*e^6*x^6 - 160*d^3*e^5*x^5 + 84*d^4*e^4*x^4 + 32*d^5
*e^3*x^3 + 64*d^7*e*x - 48*d^8 - 9*(e^8*x^8 - 8*d^2*e^6*x^6 + 8*d^4*e^4*x^4 + 4*
(d*e^6*x^6 - 2*d^3*e^4*x^4)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2)
)/x) - (16*e^7*x^7 - 9*d*e^6*x^6 - 120*d^2*e^5*x^5 + 66*d^3*e^4*x^4 + 64*d^4*e^3
*x^3 - 24*d^5*e^2*x^2 + 64*d^6*e*x - 48*d^7)*sqrt(-e^2*x^2 + d^2))/(d^4*e^4*x^8
- 8*d^6*e^2*x^6 + 8*d^8*x^4 + 4*(d^5*e^2*x^6 - 2*d^7*x^4)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{5} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(1/2)/x**5/(e*x+d),x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x**5*(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^5),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError