3.206 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx\)

Optimal. Leaf size=110 \[ \frac{8 e^2 (d-e x)}{d \sqrt{d^2-e^2 x^2}}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d x}-\frac{\sqrt{d^2-e^2 x^2}}{2 x^2}-\frac{15 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d} \]

[Out]

(8*e^2*(d - e*x))/(d*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(2*x^2) + (4*e*S
qrt[d^2 - e^2*x^2])/(d*x) - (15*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d)

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Rubi [A]  time = 0.412657, antiderivative size = 110, 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 \[ \frac{8 e^2 (d-e x)}{d \sqrt{d^2-e^2 x^2}}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d x}-\frac{\sqrt{d^2-e^2 x^2}}{2 x^2}-\frac{15 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)^4),x]

[Out]

(8*e^2*(d - e*x))/(d*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(2*x^2) + (4*e*S
qrt[d^2 - e^2*x^2])/(d*x) - (15*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d)

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Rubi in Sympy [A]  time = 33.3633, size = 88, normalized size = 0.8 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{2 x^{2}} - \frac{15 e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d} + \frac{8 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{d \left (d + e x\right )} + \frac{4 e \sqrt{d^{2} - e^{2} x^{2}}}{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(5/2)/x**3/(e*x+d)**4,x)

[Out]

-sqrt(d**2 - e**2*x**2)/(2*x**2) - 15*e**2*atanh(sqrt(d**2 - e**2*x**2)/d)/(2*d)
+ 8*e**2*sqrt(d**2 - e**2*x**2)/(d*(d + e*x)) + 4*e*sqrt(d**2 - e**2*x**2)/(d*x
)

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Mathematica [A]  time = 0.200272, size = 85, normalized size = 0.77 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-d^2+7 d e x+24 e^2 x^2\right )}{x^2 (d+e x)}-15 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+15 e^2 \log (x)}{2 d} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)^4),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(-d^2 + 7*d*e*x + 24*e^2*x^2))/(x^2*(d + e*x)) + 15*e^2*Lo
g[x] - 15*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(2*d)

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Maple [B]  time = 0.021, size = 504, normalized size = 4.6 \[ -{\frac{1}{2\,{d}^{6}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{e}^{2}}{2\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{2}}{2\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{e}^{2}}{2\,{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{15\,{e}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-4\,{\frac{{e}^{2}}{{d}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{5/2}}-5\,{\frac{{e}^{3}x}{{d}^{5}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2}}-{\frac{15\,{e}^{3}x}{2\,{d}^{3}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{15\,{e}^{3}}{2\,d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+4\,{\frac{e \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{7/2}}{{d}^{7}x}}+4\,{\frac{{e}^{3}x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}{{d}^{7}}}+5\,{\frac{{e}^{3}x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}{{d}^{5}}}+{\frac{15\,{e}^{3}x}{2\,{d}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{15\,{e}^{3}}{2\,d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-2\,{\frac{1}{{d}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{7/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{1}{{d}^{4}{e}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(5/2)/x^3/(e*x+d)^4,x)

[Out]

-1/2/d^6/x^2*(-e^2*x^2+d^2)^(7/2)+3/2/d^6*e^2*(-e^2*x^2+d^2)^(5/2)+5/2/d^4*e^2*(
-e^2*x^2+d^2)^(3/2)+15/2/d^2*e^2*(-e^2*x^2+d^2)^(1/2)-15/2*e^2/(d^2)^(1/2)*ln((2
*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-4/d^6*e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d
/e))^(5/2)-5/d^5*e^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x-15/2/d^3*e^3*(-(x+d/
e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-15/2/d*e^3/(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/d^7*e/x*(-e^2*x^2+d^2)^(7/2)+4/d^7*e^3*x*(-e
^2*x^2+d^2)^(5/2)+5/d^5*e^3*x*(-e^2*x^2+d^2)^(3/2)+15/2/d^3*e^3*x*(-e^2*x^2+d^2)
^(1/2)+15/2/d*e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-2/d^6/(
x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+1/d^4/e^2/(x+d/e)^4*(-(x+d/e)^2*e^
2+2*d*e*(x+d/e))^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{4} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^3),x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^3), x)

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Fricas [A]  time = 0.281926, size = 405, normalized size = 3.68 \[ \frac{40 \, e^{5} x^{5} - 17 \, d e^{4} x^{4} - 102 \, d^{2} e^{3} x^{3} + 21 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x - 4 \, d^{5} + 15 \,{\left (e^{5} x^{5} + 3 \, d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} - 4 \, d^{3} e^{2} x^{2} -{\left (e^{4} x^{4} - 2 \, d e^{3} x^{3} - 4 \, d^{2} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (8 \, e^{4} x^{4} + 87 \, d e^{3} x^{3} - 19 \, d^{2} e^{2} x^{2} - 30 \, d^{3} e x + 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (d e^{3} x^{5} + 3 \, d^{2} e^{2} x^{4} - 2 \, d^{3} e x^{3} - 4 \, d^{4} x^{2} -{\left (d e^{2} x^{4} - 2 \, d^{2} e x^{3} - 4 \, d^{3} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^3),x, algorithm="fricas")

[Out]

1/2*(40*e^5*x^5 - 17*d*e^4*x^4 - 102*d^2*e^3*x^3 + 21*d^3*e^2*x^2 + 30*d^4*e*x -
4*d^5 + 15*(e^5*x^5 + 3*d*e^4*x^4 - 2*d^2*e^3*x^3 - 4*d^3*e^2*x^2 - (e^4*x^4 -
2*d*e^3*x^3 - 4*d^2*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2
))/x) + (8*e^4*x^4 + 87*d*e^3*x^3 - 19*d^2*e^2*x^2 - 30*d^3*e*x + 4*d^4)*sqrt(-e
^2*x^2 + d^2))/(d*e^3*x^5 + 3*d^2*e^2*x^4 - 2*d^3*e*x^3 - 4*d^4*x^2 - (d*e^2*x^4
- 2*d^2*e*x^3 - 4*d^3*x^2)*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**3/(e*x+d)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.338371, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^3),x, algorithm="giac")

[Out]

Done