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

Optimal. Leaf size=113 \[ \frac{1}{8} d^2 (8 d-3 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{12} (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{3}{8} d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

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

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Rubi [A]  time = 0.357984, antiderivative size = 113, 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 \[ \frac{1}{8} d^2 (8 d-3 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{12} (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{3}{8} d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.9324, size = 95, normalized size = 0.84 \[ - \frac{3 d^{4} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8} - d^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} + \frac{d^{2} \left (8 d - 3 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{8} + \frac{\left (4 d - 3 e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.101536, size = 108, normalized size = 0.96 \[ d^4 \log (x)-d^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-\frac{3}{8} d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{1}{24} \sqrt{d^2-e^2 x^2} \left (32 d^3-15 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 245, normalized size = 2.2 \[{\frac{1}{5\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{d}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{d}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}-{{d}^{5}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{1}{5\,d} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{ex}{4} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{2}ex}{8}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{3\,{d}^{4}e}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5/d*(-e^2*x^2+d^2)^(5/2)+1/3*d*(-e^2*x^2+d^2)^(3/2)+d^3*(-e^2*x^2+d^2)^(1/2)-d
^5/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/5/d*(-(x+d/e)^
2*e^2+2*d*e*(x+d/e))^(5/2)-1/4*e*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x-3/8*d^2*
e*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-3/8*d^4*e/(e^2)^(1/2)*arctan((e^2)^(1/2
)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(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)^(5/2)/((e*x + d)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291915, size = 524, normalized size = 4.64 \[ -\frac{24 \, d e^{7} x^{7} - 32 \, d^{2} e^{6} x^{6} - 132 \, d^{3} e^{5} x^{5} + 192 \, d^{4} e^{4} x^{4} + 228 \, d^{5} e^{3} x^{3} - 192 \, d^{6} e^{2} x^{2} - 120 \, d^{7} e x - 18 \,{\left (d^{4} e^{4} x^{4} - 8 \, d^{6} e^{2} x^{2} + 8 \, d^{8} + 4 \,{\left (d^{5} e^{2} x^{2} - 2 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 24 \,{\left (d^{4} e^{4} x^{4} - 8 \, d^{6} e^{2} x^{2} + 8 \, d^{8} + 4 \,{\left (d^{5} e^{2} x^{2} - 2 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (6 \, e^{7} x^{7} - 8 \, d e^{6} x^{6} - 63 \, d^{2} e^{5} x^{5} + 96 \, d^{3} e^{4} x^{4} + 168 \, d^{4} e^{3} x^{3} - 192 \, d^{5} e^{2} x^{2} - 120 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (e^{4} x^{4} - 8 \, d^{2} e^{2} x^{2} + 8 \, d^{4} + 4 \,{\left (d e^{2} x^{2} - 2 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(24*d*e^7*x^7 - 32*d^2*e^6*x^6 - 132*d^3*e^5*x^5 + 192*d^4*e^4*x^4 + 228*d
^5*e^3*x^3 - 192*d^6*e^2*x^2 - 120*d^7*e*x - 18*(d^4*e^4*x^4 - 8*d^6*e^2*x^2 + 8
*d^8 + 4*(d^5*e^2*x^2 - 2*d^7)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2
+ d^2))/(e*x)) - 24*(d^4*e^4*x^4 - 8*d^6*e^2*x^2 + 8*d^8 + 4*(d^5*e^2*x^2 - 2*d^
7)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (6*e^7*x^7 - 8*d*e
^6*x^6 - 63*d^2*e^5*x^5 + 96*d^3*e^4*x^4 + 168*d^4*e^3*x^3 - 192*d^5*e^2*x^2 - 1
20*d^6*e*x)*sqrt(-e^2*x^2 + d^2))/(e^4*x^4 - 8*d^2*e^2*x^2 + 8*d^4 + 4*(d*e^2*x^
2 - 2*d^3)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 26.7394, size = 469, normalized size = 4.15 \[ d^{3} \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) - d^{2} e \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x)) - e*x/s
qrt(d**2/(e**2*x**2) - 1), Abs(d**2/(e**2*x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/
(e**2*x**2) + 1)) + I*d*asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1), True)
) - d**2*e*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/
d**2)) + I*e**2*x**3/(2*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1),
(d**2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True)) - d*e**2*Piecew
ise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True
)) + e**3*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 +
e**2*x**2/d**2)) - 3*I*d*x**3/(8*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*s
qrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**4*asin(e*x/d)/(8*e**3)
- d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2
)) - e**2*x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError