3.41 \(\int \frac{(d+e x)^2}{x^4 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=107 \[ -\frac{5 e^2 \sqrt{d^2-e^2 x^2}}{3 d^2 x}-\frac{e \sqrt{d^2-e^2 x^2}}{d x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 x^3}-\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.315298, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{5 e^2 \sqrt{d^2-e^2 x^2}}{3 d^2 x}-\frac{e \sqrt{d^2-e^2 x^2}}{d x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 x^3}-\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.1774, size = 88, normalized size = 0.82 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{3 x^{3}} - \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{d x^{2}} - \frac{e^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{2}} - \frac{5 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/x**4/(-e**2*x**2+d**2)**(1/2),x)

[Out]

-sqrt(d**2 - e**2*x**2)/(3*x**3) - e*sqrt(d**2 - e**2*x**2)/(d*x**2) - e**3*atan
h(sqrt(d**2 - e**2*x**2)/d)/d**2 - 5*e**2*sqrt(d**2 - e**2*x**2)/(3*d**2*x)

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Mathematica [A]  time = 0.164305, size = 76, normalized size = 0.71 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (d^2+3 d e x+5 e^2 x^2\right )}{x^3}+3 e^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-3 e^3 \log (x)}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[d^2 - e^2*x^2]*(d^2 + 3*d*e*x + 5*e^2*x^2))/x^3 - 3*e^3*Log[x] + 3*e^3*L
og[d + Sqrt[d^2 - e^2*x^2]])/(3*d^2)

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Maple [A]  time = 0.018, size = 114, normalized size = 1.1 \[ -{\frac{1}{3\,{x}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{5\,{e}^{2}}{3\,{d}^{2}x}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{e}{d{x}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{3}}{d}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/x^4/(-e^2*x^2+d^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274398, size = 355, normalized size = 3.32 \[ -\frac{5 \, e^{6} x^{6} + 3 \, d e^{5} x^{5} - 24 \, d^{2} e^{4} x^{4} - 15 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 12 \, d^{5} e x + 4 \, d^{6} - 3 \,{\left (3 \, d e^{5} x^{5} - 4 \, d^{3} e^{3} x^{3} -{\left (e^{5} x^{5} - 4 \, d^{2} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (15 \, d e^{4} x^{4} + 9 \, d^{2} e^{3} x^{3} - 17 \, d^{3} e^{2} x^{2} - 12 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (3 \, d^{3} e^{2} x^{5} - 4 \, d^{5} x^{3} -{\left (d^{2} e^{2} x^{5} - 4 \, d^{4} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(5*e^6*x^6 + 3*d*e^5*x^5 - 24*d^2*e^4*x^4 - 15*d^3*e^3*x^3 + 15*d^4*e^2*x^2
+ 12*d^5*e*x + 4*d^6 - 3*(3*d*e^5*x^5 - 4*d^3*e^3*x^3 - (e^5*x^5 - 4*d^2*e^3*x^
3)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (15*d*e^4*x^4 + 9*
d^2*e^3*x^3 - 17*d^3*e^2*x^2 - 12*d^4*e*x - 4*d^5)*sqrt(-e^2*x^2 + d^2))/(3*d^3*
e^2*x^5 - 4*d^5*x^3 - (d^2*e^2*x^5 - 4*d^4*x^3)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 13.7215, size = 303, normalized size = 2.83 \[ d^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac{2 e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac{2 i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac{e^{2} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{2 d^{3}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i}{2 e x^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{i e}{2 d^{2} x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{i e^{2} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{2 d^{3}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{d^{2}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{d^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/x**4/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2*x**2) - 2*e**3*sqrt(d**2/(
e**2*x**2) - 1)/(3*d**4), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**
2) + 1)/(3*d**2*x**2) - 2*I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**4), True)) +
2*d*e*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(2*d**2*x) - e**2*acosh(d/(e*x))/
(2*d**3), Abs(d**2/(e**2*x**2)) > 1), (I/(2*e*x**3*sqrt(-d**2/(e**2*x**2) + 1))
- I*e/(2*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**2*asin(d/(e*x))/(2*d**3), Tr
ue)) + e**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/d**2, Abs(d**2/(e**2*x**2))
> 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/d**2, True))

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GIAC/XCAS [A]  time = 0.295299, size = 323, normalized size = 3.02 \[ \frac{x^{3}{\left (\frac{6 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{x} + \frac{21 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{4}}{x^{2}} + e^{8}\right )} e}{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2}} - \frac{e^{3}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{2}} - \frac{{\left (\frac{21 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{16}}{x} + \frac{6 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{14}}{x^{2}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{12}}{x^{3}}\right )} e^{\left (-15\right )}}{24 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*x^3*(6*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^6/x + 21*(d*e + sqrt(-x^2*e^2 + d^2
)*e)^2*e^4/x^2 + e^8)*e/((d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^2) - e^3*ln(1/2*abs(
-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^2 - 1/24*(21*(d*e + sqrt(-x^
2*e^2 + d^2)*e)*d^4*e^16/x + 6*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^4*e^14/x^2 + (
d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^4*e^12/x^3)*e^(-15)/d^6