3.64 \(\int \frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{x^5} \, dx\)

Optimal. Leaf size=134 \[ -\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}+e^4 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+\frac{13}{8} e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

-(e^2*(13*d + 8*e*x)*Sqrt[d^2 - e^2*x^2])/(8*x^2) - (d*(d^2 - e^2*x^2)^(3/2))/(4
*x^4) - (e*(d^2 - e^2*x^2)^(3/2))/x^3 - e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] +
(13*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/8

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Rubi [A]  time = 0.409761, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ -\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}+e^4 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+\frac{13}{8} e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(e^2*(13*d + 8*e*x)*Sqrt[d^2 - e^2*x^2])/(8*x^2) - (d*(d^2 - e^2*x^2)^(3/2))/(4
*x^4) - (e*(d^2 - e^2*x^2)^(3/2))/x^3 - e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] +
(13*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/8

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Rubi in Sympy [A]  time = 46.7529, size = 129, normalized size = 0.96 \[ - \frac{d^{3} \sqrt{d^{2} - e^{2} x^{2}}}{4 x^{4}} - \frac{11 d e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{2}} - e^{4} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} + \frac{13 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8} - \frac{e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{x} - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(-e**2*x**2+d**2)**(1/2)/x**5,x)

[Out]

-d**3*sqrt(d**2 - e**2*x**2)/(4*x**4) - 11*d*e**2*sqrt(d**2 - e**2*x**2)/(8*x**2
) - e**4*atan(e*x/sqrt(d**2 - e**2*x**2)) + 13*e**4*atanh(sqrt(d**2 - e**2*x**2)
/d)/8 - e**3*sqrt(d**2 - e**2*x**2)/x - e*(d**2 - e**2*x**2)**(3/2)/x**3

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Mathematica [A]  time = 0.168626, size = 102, normalized size = 0.76 \[ \frac{1}{8} \left (-\frac{d \sqrt{d^2-e^2 x^2} \left (2 d^2+8 d e x+11 e^2 x^2\right )}{x^4}+13 e^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-8 e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-13 e^4 \log (x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((d*Sqrt[d^2 - e^2*x^2]*(2*d^2 + 8*d*e*x + 11*e^2*x^2))/x^4) - 8*e^4*ArcTan[(e
*x)/Sqrt[d^2 - e^2*x^2]] - 13*e^4*Log[x] + 13*e^4*Log[d + Sqrt[d^2 - e^2*x^2]])/
8

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Maple [A]  time = 0.03, size = 212, normalized size = 1.6 \[ -{\frac{d}{4\,{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{13\,{e}^{2}}{8\,d{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{13\,{e}^{4}}{8\,d}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{13\,d{e}^{4}}{8}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{{e}^{3}}{{d}^{2}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{5}x}{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{{e}^{5}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{e}{{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(-e^2*x^2+d^2)^(1/2)/x^5,x)

[Out]

-1/4*d*(-e^2*x^2+d^2)^(3/2)/x^4-13/8/d*e^2/x^2*(-e^2*x^2+d^2)^(3/2)-13/8*e^4/d*(
-e^2*x^2+d^2)^(1/2)+13/8*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)-e^3/d^2/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))-e*(-e^2*x^2+d^2)^(3/2)/x^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287954, size = 541, normalized size = 4.04 \[ \frac{44 \, d^{2} e^{6} x^{6} + 32 \, d^{3} e^{5} x^{5} - 124 \, d^{4} e^{4} x^{4} - 96 \, d^{5} e^{3} x^{3} + 64 \, d^{6} e^{2} x^{2} + 64 \, d^{7} e x + 16 \, d^{8} + 16 \,{\left (e^{8} x^{8} - 8 \, d^{2} e^{6} x^{6} + 8 \, d^{4} e^{4} x^{4} + 4 \,{\left (d e^{6} x^{6} - 2 \, d^{3} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 13 \,{\left (e^{8} x^{8} - 8 \, d^{2} e^{6} x^{6} + 8 \, d^{4} e^{4} x^{4} + 4 \,{\left (d e^{6} x^{6} - 2 \, d^{3} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (11 \, d e^{6} x^{6} + 8 \, d^{2} e^{5} x^{5} - 86 \, d^{3} e^{4} x^{4} - 64 \, d^{4} e^{3} x^{3} + 72 \, d^{5} e^{2} x^{2} + 64 \, d^{6} e x + 16 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{8 \,{\left (e^{4} x^{8} - 8 \, d^{2} e^{2} x^{6} + 8 \, d^{4} x^{4} + 4 \,{\left (d e^{2} x^{6} - 2 \, d^{3} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(44*d^2*e^6*x^6 + 32*d^3*e^5*x^5 - 124*d^4*e^4*x^4 - 96*d^5*e^3*x^3 + 64*d^6
*e^2*x^2 + 64*d^7*e*x + 16*d^8 + 16*(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)) - 13*(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) - (11*d*e^6*
x^6 + 8*d^2*e^5*x^5 - 86*d^3*e^4*x^4 - 64*d^4*e^3*x^3 + 72*d^5*e^2*x^2 + 64*d^6*
e*x + 16*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 = 26.0766, size = 544, normalized size = 4.06 \[ d^{3} \left (\begin{cases} - \frac{d^{2}}{4 e x^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{3 e}{8 x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{e^{3}}{8 d^{2} x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e^{4} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{8 d^{3}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i d^{2}}{4 e x^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{3 i e}{8 x^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{i e^{3}}{8 d^{2} x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{i e^{4} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{8 d^{3}} & \text{otherwise} \end{cases}\right ) + 3 d^{2} e \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac{e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac{i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text{otherwise} \end{cases}\right ) + 3 d e^{2} \left (\begin{cases} - \frac{d^{2}}{2 e x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e}{2 x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e^{2} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{2 d} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac{i e^{2} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{2 d} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(-e**2*x**2+d**2)**(1/2)/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)) + 3*d**2*e*Pi
ecewise((-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)) + 3*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*s
qrt(-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 = 0.30513, size = 398, normalized size = 2.97 \[ -\arcsin \left (\frac{x e}{d}\right ) e^{4}{\rm sign}\left (d\right ) + \frac{x^{4}{\left (\frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{8}}{x} + \frac{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{6}}{x^{2}} + \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{4}}{x^{3}} + e^{10}\right )} e^{2}}{64 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4}} - \frac{1}{64} \,{\left (\frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{26}}{x} + \frac{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{24}}{x^{2}} + \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{22}}{x^{3}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{20}}{x^{4}}\right )} e^{\left (-24\right )} + \frac{13}{8} \, e^{4}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(x*e/d)*e^4*sign(d) + 1/64*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 + 8*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3
*e^4/x^3 + e^10)*e^2/(d*e + sqrt(-x^2*e^2 + d^2)*e)^4 - 1/64*(8*(d*e + sqrt(-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 + (d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^20/x^4)*
e^(-24) + 13/8*e^4*ln(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))