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

Optimal. Leaf size=118 \[ \frac{e^2 (3 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{3}{8} e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

(e^2*(3*d + 8*e*x)*Sqrt[d^2 - e^2*x^2])/(8*x^2) - ((3*d + 4*e*x)*(d^2 - e^2*x^2)
^(3/2))/(12*x^4) + e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - (3*e^4*ArcTanh[Sqrt[d
^2 - e^2*x^2]/d])/8

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Rubi [A]  time = 0.282199, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{e^2 (3 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{3}{8} e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(3*d + 8*e*x)*Sqrt[d^2 - e^2*x^2])/(8*x^2) - ((3*d + 4*e*x)*(d^2 - e^2*x^2)
^(3/2))/(12*x^4) + e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - (3*e^4*ArcTanh[Sqrt[d
^2 - e^2*x^2]/d])/8

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Rubi in Sympy [A]  time = 43.0902, size = 102, normalized size = 0.86 \[ e^{4} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - \frac{3 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8} + \frac{e^{2} \left (6 d + 16 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{16 x^{2}} - \frac{\left (3 d + 4 e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**4*atan(e*x/sqrt(d**2 - e**2*x**2)) - 3*e**4*atanh(sqrt(d**2 - e**2*x**2)/d)/8
+ e**2*(6*d + 16*e*x)*sqrt(d**2 - e**2*x**2)/(16*x**2) - (3*d + 4*e*x)*(d**2 -
e**2*x**2)**(3/2)/(12*x**4)

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Mathematica [A]  time = 0.225609, size = 111, normalized size = 0.94 \[ \frac{1}{24} \left (-9 e^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+24 e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (-6 d^3-8 d^2 e x+15 d e^2 x^2+32 e^3 x^3\right )}{x^4}+9 e^4 \log (x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(-6*d^3 - 8*d^2*e*x + 15*d*e^2*x^2 + 32*e^3*x^3))/x^4 + 24
*e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + 9*e^4*Log[x] - 9*e^4*Log[d + Sqrt[d^2 -
e^2*x^2]])/24

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Maple [B]  time = 0.026, size = 260, normalized size = 2.2 \[ -{\frac{1}{4\,d{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{2}}{8\,{d}^{3}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{4}}{8\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{4}}{8\,d}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,d{e}^{4}}{8}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{e}{3\,{d}^{2}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{e}^{3}}{3\,{d}^{4}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{e}^{5}x}{3\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{5}x}{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{{e}^{5}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^(3/2)*(e*x + d)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303354, size = 564, normalized size = 4.78 \[ -\frac{128 \, d e^{7} x^{7} + 60 \, d^{2} e^{6} x^{6} - 416 \, d^{3} e^{5} x^{5} - 204 \, d^{4} e^{4} x^{4} + 352 \, d^{5} e^{3} x^{3} + 192 \, d^{6} e^{2} x^{2} - 64 \, d^{7} e x - 48 \, d^{8} + 48 \,{\left (e^{8} x^{8} - 8 \, d^{2} e^{6} x^{6} + 8 \, d^{4} e^{4} x^{4} + 4 \,{\left (d e^{6} x^{6} - 2 \, d^{3} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 9 \,{\left (e^{8} x^{8} - 8 \, d^{2} e^{6} x^{6} + 8 \, d^{4} e^{4} x^{4} + 4 \,{\left (d e^{6} x^{6} - 2 \, d^{3} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (32 \, e^{7} x^{7} + 15 \, d e^{6} x^{6} - 264 \, d^{2} e^{5} x^{5} - 126 \, d^{3} e^{4} x^{4} + 320 \, d^{4} e^{3} x^{3} + 168 \, d^{5} e^{2} x^{2} - 64 \, d^{6} e x - 48 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (e^{4} x^{8} - 8 \, d^{2} e^{2} x^{6} + 8 \, d^{4} x^{4} + 4 \,{\left (d e^{2} x^{6} - 2 \, d^{3} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(128*d*e^7*x^7 + 60*d^2*e^6*x^6 - 416*d^3*e^5*x^5 - 204*d^4*e^4*x^4 + 352*
d^5*e^3*x^3 + 192*d^6*e^2*x^2 - 64*d^7*e*x - 48*d^8 + 48*(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))*arctan(-
(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 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) - (32*e^7*x^7 + 15*d*e^6*x^6 - 264*d^2*e^5*x^5 - 126*d^3*e^4*x^4 + 320*d
^4*e^3*x^3 + 168*d^5*e^2*x^2 - 64*d^6*e*x - 48*d^7)*sqrt(-e^2*x^2 + d^2))/(e^4*x
^8 - 8*d^2*e^2*x^6 + 8*d^4*x^4 + 4*(d*e^2*x^6 - 2*d^3*x^4)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 27.8105, size = 541, normalized size = 4.58 \[ d^{3} \left (\begin{cases} - \frac{d^{2}}{4 e x^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{3 e}{8 x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{e^{3}}{8 d^{2} x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e^{4} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{8 d^{3}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i d^{2}}{4 e x^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{3 i e}{8 x^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{i e^{3}}{8 d^{2} x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{i e^{4} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{8 d^{3}} & \text{otherwise} \end{cases}\right ) + d^{2} e \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac{e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac{i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} - \frac{d^{2}}{2 e x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e}{2 x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e^{2} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{2 d} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac{i e^{2} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{2 d} & \text{otherwise} \end{cases}\right ) - e^{3} \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3*sqrt(d
**2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(
d/(e*x))/(8*d**3), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**
2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**3/(8*d**2*x*sq
rt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) + d**2*e*Piec
ewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/
(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2)
+ I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True)) - d*e**2*Piecewise((-d**2
/(2*e*x**3*sqrt(d**2/(e**2*x**2) - 1)) + e/(2*x*sqrt(d**2/(e**2*x**2) - 1)) + e*
*2*acosh(d/(e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2
) + 1)/(2*x) - I*e**2*asin(d/(e*x))/(2*d), True)) - e**3*Piecewise((I*d/(x*sqrt(
-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)
), Abs(e**2*x**2/d**2) > 1), (-d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) +
e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True))

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GIAC/XCAS [A]  time = 0.31666, size = 401, normalized size = 3.4 \[ \arcsin \left (\frac{x e}{d}\right ) e^{4}{\rm sign}\left (d\right ) + \frac{x^{4}{\left (\frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{8}}{x} - \frac{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{6}}{x^{2}} - \frac{120 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{4}}{x^{3}} + 3 \, e^{10}\right )} e^{2}}{192 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4}} + \frac{1}{192} \,{\left (\frac{120 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{26}}{x} + \frac{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{24}}{x^{2}} - \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{22}}{x^{3}} - \frac{3 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{20}}{x^{4}}\right )} e^{\left (-24\right )} - \frac{3}{8} \, e^{4}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(x*e/d)*e^4*sign(d) + 1/192*x^4*(8*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^8/x -
24*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^6/x^2 - 120*(d*e + sqrt(-x^2*e^2 + d^2)*e)
^3*e^4/x^3 + 3*e^10)*e^2/(d*e + sqrt(-x^2*e^2 + d^2)*e)^4 + 1/192*(120*(d*e + sq
rt(-x^2*e^2 + d^2)*e)*e^26/x + 24*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^24/x^2 - 8*
(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^22/x^3 - 3*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e
^20/x^4)*e^(-24) - 3/8*e^4*ln(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/
abs(x))