3.167 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^2} \, dx\)

Optimal. Leaf size=108 \[ -\frac{5 e^2 \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac{2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}+\frac{5 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d} \]

[Out]

(-5*e^2*Sqrt[d^2 - e^2*x^2])/(8*x^2) - (d^2 - e^2*x^2)^(3/2)/(4*x^4) + (2*e*(d^2
- e^2*x^2)^(3/2))/(3*d*x^3) + (5*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d)

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Rubi [A]  time = 0.328318, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{5 e^2 \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac{2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}+\frac{5 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d} \]

Antiderivative was successfully verified.

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

[Out]

(-5*e^2*Sqrt[d^2 - e^2*x^2])/(8*x^2) - (d^2 - e^2*x^2)^(3/2)/(4*x^4) + (2*e*(d^2
- e^2*x^2)^(3/2))/(3*d*x^3) + (5*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d)

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Rubi in Sympy [A]  time = 38.5662, size = 94, normalized size = 0.87 \[ - \frac{d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{4 x^{4}} - \frac{3 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{2}} + \frac{5 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d} + \frac{2 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 d x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.122108, size = 95, normalized size = 0.88 \[ -\frac{-15 e^4 x^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (6 d^3-16 d^2 e x+9 d e^2 x^2+16 e^3 x^3\right )+15 e^4 x^4 \log (x)}{24 d x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d^2 - e^2*x^2]*(6*d^3 - 16*d^2*e*x + 9*d*e^2*x^2 + 16*e^3*x^3) + 15*e^4*x
^4*Log[x] - 15*e^4*x^4*Log[d + Sqrt[d^2 - e^2*x^2]])/(24*d*x^4)

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Maple [B]  time = 0.02, size = 513, normalized size = 4.8 \[ -{\frac{1}{4\,{d}^{4}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{9\,{e}^{2}}{8\,{d}^{6}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{4}}{8\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{e}^{4}}{24\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{4}}{8\,{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{5\,{e}^{4}}{8}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{4\,{e}^{4}}{3\,{d}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{e}^{5}x}{3\,{d}^{5}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{5}x}{2\,{d}^{3}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{5\,{e}^{5}}{2\,d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{4\,{e}^{3}}{3\,{d}^{7}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{4\,{e}^{5}x}{3\,{d}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{5}x}{3\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{e}^{5}x}{2\,{d}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{5\,{e}^{5}}{2\,d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{2}}{3\,{d}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{2\,e}{3\,{d}^{5}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/d^4/x^4*(-e^2*x^2+d^2)^(7/2)-9/8/d^6*e^2/x^2*(-e^2*x^2+d^2)^(7/2)-1/8/d^6*e
^4*(-e^2*x^2+d^2)^(5/2)-5/24/d^4*e^4*(-e^2*x^2+d^2)^(3/2)-5/8/d^2*e^4*(-e^2*x^2+
d^2)^(1/2)+5/8*e^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-
4/3/d^6*e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-5/3/d^5*e^5*(-(x+d/e)^2*e^2+2*d
*e*(x+d/e))^(3/2)*x-5/2/d^3*e^5*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-5/2/d*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))+4/3/d^7*
e^3/x*(-e^2*x^2+d^2)^(7/2)+4/3/d^7*e^5*x*(-e^2*x^2+d^2)^(5/2)+5/3/d^5*e^5*x*(-e^
2*x^2+d^2)^(3/2)+5/2/d^3*e^5*x*(-e^2*x^2+d^2)^(1/2)+5/2/d*e^5/(e^2)^(1/2)*arctan
((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/3/d^6*e^2/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e
*(x+d/e))^(7/2)+2/3/d^5*e/x^3*(-e^2*x^2+d^2)^(7/2)

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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((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^2*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291319, size = 425, normalized size = 3.94 \[ \frac{64 \, d e^{7} x^{7} + 36 \, d^{2} e^{6} x^{6} - 256 \, d^{3} e^{5} x^{5} - 84 \, d^{4} e^{4} x^{4} + 320 \, d^{5} e^{3} x^{3} - 128 \, d^{7} e x + 48 \, d^{8} - 15 \,{\left (e^{8} x^{8} - 8 \, d^{2} e^{6} x^{6} + 8 \, d^{4} e^{4} x^{4} + 4 \,{\left (d e^{6} x^{6} - 2 \, d^{3} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (16 \, e^{7} x^{7} + 9 \, d e^{6} x^{6} - 144 \, d^{2} e^{5} x^{5} - 66 \, d^{3} e^{4} x^{4} + 256 \, d^{4} e^{3} x^{3} + 24 \, d^{5} e^{2} x^{2} - 128 \, d^{6} e x + 48 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (d e^{4} x^{8} - 8 \, d^{3} e^{2} x^{6} + 8 \, d^{5} x^{4} + 4 \,{\left (d^{2} e^{2} x^{6} - 2 \, d^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(64*d*e^7*x^7 + 36*d^2*e^6*x^6 - 256*d^3*e^5*x^5 - 84*d^4*e^4*x^4 + 320*d^5
*e^3*x^3 - 128*d^7*e*x + 48*d^8 - 15*(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 - 144*d^2*e^5*x^5 - 66*d^3*e^4*x^4 + 256*d^4*
e^3*x^3 + 24*d^5*e^2*x^2 - 128*d^6*e*x + 48*d^7)*sqrt(-e^2*x^2 + d^2))/(d*e^4*x^
8 - 8*d^3*e^2*x^6 + 8*d^5*x^4 + 4*(d^2*e^2*x^6 - 2*d^4*x^4)*sqrt(-e^2*x^2 + d^2)
)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError