3.12 \(\int \frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.170625, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d}+\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{8 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 19.7645, size = 88, normalized size = 0.81 \[ \frac{3 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{2}} - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{4 x^{4}} - \frac{3 e^{5} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 d x^{5}} \]

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**6,x)

[Out]

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

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Mathematica [A]  time = 0.0998187, size = 106, normalized size = 0.98 \[ \frac{-15 e^5 x^5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (-8 d^4-10 d^3 e x+16 d^2 e^2 x^2+25 d e^3 x^3-8 e^4 x^4\right )+15 e^5 x^5 \log (x)}{40 d x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^6,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-8*d^4 - 10*d^3*e*x + 16*d^2*e^2*x^2 + 25*d*e^3*x^3 - 8*e^
4*x^4) + 15*e^5*x^5*Log[x] - 15*e^5*x^5*Log[d + Sqrt[d^2 - e^2*x^2]])/(40*d*x^5)

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Maple [A]  time = 0.03, size = 158, normalized size = 1.5 \[ -{\frac{1}{5\,d{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{e}{4\,{d}^{2}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{3}}{8\,{d}^{4}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{5}}{8\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{5}}{8\,{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{5}}{8}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^6,x)

[Out]

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

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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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292921, size = 531, normalized size = 4.92 \[ -\frac{8 \, e^{10} x^{10} - 25 \, d e^{9} x^{9} - 120 \, d^{2} e^{8} x^{8} + 335 \, d^{3} e^{7} x^{7} + 440 \, d^{4} e^{6} x^{6} - 830 \, d^{5} e^{5} x^{5} - 680 \, d^{6} e^{4} x^{4} + 680 \, d^{7} e^{3} x^{3} + 480 \, d^{8} e^{2} x^{2} - 160 \, d^{9} e x - 128 \, d^{10} - 15 \,{\left (5 \, d e^{9} x^{9} - 20 \, d^{3} e^{7} x^{7} + 16 \, d^{5} e^{5} x^{5} -{\left (e^{9} x^{9} - 12 \, d^{2} e^{7} x^{7} + 16 \, d^{4} e^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (40 \, d e^{8} x^{8} - 125 \, d^{2} e^{7} x^{7} - 240 \, d^{3} e^{6} x^{6} + 550 \, d^{4} e^{5} x^{5} + 488 \, d^{5} e^{4} x^{4} - 600 \, d^{6} e^{3} x^{3} - 416 \, d^{7} e^{2} x^{2} + 160 \, d^{8} e x + 128 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{40 \,{\left (5 \, d^{2} e^{4} x^{9} - 20 \, d^{4} e^{2} x^{7} + 16 \, d^{6} x^{5} -{\left (d e^{4} x^{9} - 12 \, d^{3} e^{2} x^{7} + 16 \, d^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^6,x, algorithm="fricas")

[Out]

-1/40*(8*e^10*x^10 - 25*d*e^9*x^9 - 120*d^2*e^8*x^8 + 335*d^3*e^7*x^7 + 440*d^4*
e^6*x^6 - 830*d^5*e^5*x^5 - 680*d^6*e^4*x^4 + 680*d^7*e^3*x^3 + 480*d^8*e^2*x^2
- 160*d^9*e*x - 128*d^10 - 15*(5*d*e^9*x^9 - 20*d^3*e^7*x^7 + 16*d^5*e^5*x^5 - (
e^9*x^9 - 12*d^2*e^7*x^7 + 16*d^4*e^5*x^5)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(
-e^2*x^2 + d^2))/x) + (40*d*e^8*x^8 - 125*d^2*e^7*x^7 - 240*d^3*e^6*x^6 + 550*d^
4*e^5*x^5 + 488*d^5*e^4*x^4 - 600*d^6*e^3*x^3 - 416*d^7*e^2*x^2 + 160*d^8*e*x +
128*d^9)*sqrt(-e^2*x^2 + d^2))/(5*d^2*e^4*x^9 - 20*d^4*e^2*x^7 + 16*d^6*x^5 - (d
*e^4*x^9 - 12*d^3*e^2*x^7 + 16*d^5*x^5)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 26.174, size = 774, normalized size = 7.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**6,x)

[Out]

d**3*Piecewise((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7
) - 4*I*d*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2
*I*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e
**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x*
*2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) -
4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*
x**6*sqrt(1 - e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sq
rt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), True)) + d**2*e*Piecewi
se((-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*sqrt(-d**2/(e*
*2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) - d*e**2*Piecewise((-e*sq
rt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Ab
s(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sq
rt(-d**2/(e**2*x**2) + 1)/(3*d**2), True)) - e**3*Piecewise((-d**2/(2*e*x**3*sqr
t(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))

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GIAC/XCAS [A]  time = 0.295298, size = 497, normalized size = 4.6 \[ \frac{x^{5}{\left (\frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{10}}{x} - \frac{10 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{8}}{x^{2}} - \frac{40 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{6}}{x^{3}} + \frac{20 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{4}}{x^{4}} + 2 \, e^{12}\right )} e^{3}}{320 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d} - \frac{3 \, e^{5}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{8 \, d} - \frac{{\left (\frac{20 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{38}}{x} - \frac{40 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{36}}{x^{2}} - \frac{10 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{34}}{x^{3}} + \frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} e^{32}}{x^{4}} + \frac{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{4} e^{30}}{x^{5}}\right )} e^{\left (-35\right )}}{320 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^6,x, algorithm="giac")

[Out]

1/320*x^5*(5*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^10/x - 10*(d*e + sqrt(-x^2*e^2 + d
^2)*e)^2*e^8/x^2 - 40*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^6/x^3 + 20*(d*e + sqrt(
-x^2*e^2 + d^2)*e)^4*e^4/x^4 + 2*e^12)*e^3/((d*e + sqrt(-x^2*e^2 + d^2)*e)^5*d)
- 3/8*e^5*ln(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d - 1/320
*(20*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^4*e^38/x - 40*(d*e + sqrt(-x^2*e^2 + d^2)*
e)^2*d^4*e^36/x^2 - 10*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^4*e^34/x^3 + 5*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^4*d^4*e^32/x^4 + 2*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*d^4*
e^30/x^5)*e^(-35)/d^5