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

Optimal. Leaf size=120 \[ \frac{e^2 (2 d+3 e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(2 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{3}{2} d e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

(e^2*(2*d + 3*e*x)*Sqrt[d^2 - e^2*x^2])/(2*x) - ((2*d - 3*e*x)*(d^2 - e^2*x^2)^(
3/2))/(6*x^3) + d*e^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - (3*d*e^3*ArcTanh[Sqrt[
d^2 - e^2*x^2]/d])/2

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Rubi [A]  time = 0.370038, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{e^2 (2 d+3 e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(2 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{3}{2} d e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(2*d + 3*e*x)*Sqrt[d^2 - e^2*x^2])/(2*x) - ((2*d - 3*e*x)*(d^2 - e^2*x^2)^(
3/2))/(6*x^3) + d*e^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - (3*d*e^3*ArcTanh[Sqrt[
d^2 - e^2*x^2]/d])/2

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Rubi in Sympy [A]  time = 51.1633, size = 104, normalized size = 0.87 \[ d e^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - \frac{3 d e^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2} + \frac{e^{2} \left (4 d + 6 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{4 x} - \frac{\left (2 d - 3 e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.162312, size = 116, normalized size = 0.97 \[ -\frac{3}{2} d e^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+d e^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\left (-\frac{d^3}{3 x^3}+\frac{d^2 e}{2 x^2}+\frac{4 d e^2}{3 x}+e^3\right ) \sqrt{d^2-e^2 x^2}+\frac{3}{2} d e^3 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

(e^3 - d^3/(3*x^3) + (d^2*e)/(2*x^2) + (4*d*e^2)/(3*x))*Sqrt[d^2 - e^2*x^2] + d*
e^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (3*d*e^3*Log[x])/2 - (3*d*e^3*Log[d + Sq
rt[d^2 - e^2*x^2]])/2

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Maple [B]  time = 0.018, size = 439, normalized size = 3.7 \[ -{\frac{1}{3\,{d}^{3}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{2}}{3\,{d}^{5}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{4}x}{3\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{4}x}{12\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{e}^{4}x}{8\,d}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{5\,d{e}^{4}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{{e}^{3}}{5\,{d}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{4}x}{4\,{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{4}x}{8\,d}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{3\,d{e}^{4}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{2\,{d}^{4}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{e}^{3}}{10\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{3}}{2\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{3}}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{3}{d}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/d^3/x^3*(-e^2*x^2+d^2)^(7/2)+1/3/d^5*e^2/x*(-e^2*x^2+d^2)^(7/2)+1/3/d^5*e^4
*x*(-e^2*x^2+d^2)^(5/2)+5/12*e^4/d^3*x*(-e^2*x^2+d^2)^(3/2)+5/8*e^4/d*x*(-e^2*x^
2+d^2)^(1/2)+5/8*d*e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/
5/d^4*e^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+1/4/d^3*e^4*(-(x+d/e)^2*e^2+2*d*e
*(x+d/e))^(3/2)*x+3/8/d*e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x+3/8*d*e^4/(e^
2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))+1/2*e/d^4/x^
2*(-e^2*x^2+d^2)^(7/2)+3/10*e^3/d^4*(-e^2*x^2+d^2)^(5/2)+1/2*e^3/d^2*(-e^2*x^2+d
^2)^(3/2)+3/2*e^3*(-e^2*x^2+d^2)^(1/2)-3/2*e^3*d^2/(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)^(5/2)/((e*x + d)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296722, size = 572, normalized size = 4.77 \[ -\frac{18 \, d e^{7} x^{7} + 32 \, d^{2} e^{6} x^{6} - 12 \, d^{3} e^{5} x^{5} - 104 \, d^{4} e^{4} x^{4} - 36 \, d^{5} e^{3} x^{3} + 88 \, d^{6} e^{2} x^{2} + 24 \, d^{7} e x - 16 \, d^{8} + 12 \,{\left (d e^{7} x^{7} - 8 \, d^{3} e^{5} x^{5} + 8 \, d^{5} e^{3} x^{3} + 4 \,{\left (d^{2} e^{5} x^{5} - 2 \, d^{4} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 9 \,{\left (d e^{7} x^{7} - 8 \, d^{3} e^{5} x^{5} + 8 \, d^{5} e^{3} x^{3} + 4 \,{\left (d^{2} e^{5} x^{5} - 2 \, d^{4} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (6 \, e^{7} x^{7} + 8 \, d e^{6} x^{6} - 21 \, d^{2} e^{5} x^{5} - 66 \, d^{3} e^{4} x^{4} - 24 \, d^{4} e^{3} x^{3} + 80 \, d^{5} e^{2} x^{2} + 24 \, d^{6} e x - 16 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{4} x^{7} - 8 \, d^{2} e^{2} x^{5} + 8 \, d^{4} x^{3} + 4 \,{\left (d e^{2} x^{5} - 2 \, d^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(18*d*e^7*x^7 + 32*d^2*e^6*x^6 - 12*d^3*e^5*x^5 - 104*d^4*e^4*x^4 - 36*d^5*
e^3*x^3 + 88*d^6*e^2*x^2 + 24*d^7*e*x - 16*d^8 + 12*(d*e^7*x^7 - 8*d^3*e^5*x^5 +
8*d^5*e^3*x^3 + 4*(d^2*e^5*x^5 - 2*d^4*e^3*x^3)*sqrt(-e^2*x^2 + d^2))*arctan(-(
d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 9*(d*e^7*x^7 - 8*d^3*e^5*x^5 + 8*d^5*e^3*x^3
+ 4*(d^2*e^5*x^5 - 2*d^4*e^3*x^3)*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 - 21*d^2*e^5*x^5 - 66*d^3*e^4*x^4 - 24*d^4
*e^3*x^3 + 80*d^5*e^2*x^2 + 24*d^6*e*x - 16*d^7)*sqrt(-e^2*x^2 + d^2))/(e^4*x^7
- 8*d^2*e^2*x^5 + 8*d^4*x^3 + 4*(d*e^2*x^5 - 2*d^3*x^3)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 22.5667, size = 457, normalized size = 3.81 \[ d^{3} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac{e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac{i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text{otherwise} \end{cases}\right ) - d^{2} e \left (\begin{cases} - \frac{d^{2}}{2 e x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e}{2 x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e^{2} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{2 d} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac{i e^{2} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{2 d} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*Piecewise((-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**2*e*Piecewi
se((-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)) - d*e**2*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*asi
n(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True)) + e**3*Piecewise((d**2/(e
*x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(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))

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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^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError