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

Optimal. Leaf size=140 \[ -\frac{7 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}-\frac{2 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{7 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3}-\frac{4 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x} \]

[Out]

-Sqrt[d^2 - e^2*x^2]/(4*x^4) - (2*e*Sqrt[d^2 - e^2*x^2])/(3*d*x^3) - (7*e^2*Sqrt
[d^2 - e^2*x^2])/(8*d^2*x^2) - (4*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^3*x) - (7*e^4*Ar
cTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^3)

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Rubi [A]  time = 0.402565, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{7 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}-\frac{2 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{7 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3}-\frac{4 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[d^2 - e^2*x^2]/(4*x^4) - (2*e*Sqrt[d^2 - e^2*x^2])/(3*d*x^3) - (7*e^2*Sqrt
[d^2 - e^2*x^2])/(8*d^2*x^2) - (4*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^3*x) - (7*e^4*Ar
cTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^3)

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Rubi in Sympy [A]  time = 37.3071, size = 122, normalized size = 0.87 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{4 x^{4}} - \frac{2 e \sqrt{d^{2} - e^{2} x^{2}}}{3 d x^{3}} - \frac{7 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{8 d^{2} x^{2}} - \frac{7 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d^{3}} - \frac{4 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(d**2 - e**2*x**2)/(4*x**4) - 2*e*sqrt(d**2 - e**2*x**2)/(3*d*x**3) - 7*e**
2*sqrt(d**2 - e**2*x**2)/(8*d**2*x**2) - 7*e**4*atanh(sqrt(d**2 - e**2*x**2)/d)/
(8*d**3) - 4*e**3*sqrt(d**2 - e**2*x**2)/(3*d**3*x)

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Mathematica [A]  time = 0.133011, size = 95, normalized size = 0.68 \[ -\frac{21 e^4 x^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (6 d^3+16 d^2 e x+21 d e^2 x^2+32 e^3 x^3\right )-21 e^4 x^4 \log (x)}{24 d^3 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d^2 - e^2*x^2]*(6*d^3 + 16*d^2*e*x + 21*d*e^2*x^2 + 32*e^3*x^3) - 21*e^4*
x^4*Log[x] + 21*e^4*x^4*Log[d + Sqrt[d^2 - e^2*x^2]])/(24*d^3*x^4)

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Maple [A]  time = 0.019, size = 139, normalized size = 1. \[ -{\frac{1}{4\,{x}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{7\,{e}^{2}}{8\,{d}^{2}{x}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{7\,{e}^{4}}{8\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{2\,e}{3\,d{x}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{4\,{e}^{3}}{3\,{d}^{3}x}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289214, size = 443, normalized size = 3.16 \[ \frac{128 \, d e^{7} x^{7} + 84 \, d^{2} e^{6} x^{6} - 320 \, d^{3} e^{5} x^{5} - 228 \, d^{4} e^{4} x^{4} + 64 \, d^{5} e^{3} x^{3} + 96 \, d^{6} e^{2} x^{2} + 128 \, d^{7} e x + 48 \, d^{8} + 21 \,{\left (e^{8} x^{8} - 8 \, d^{2} e^{6} x^{6} + 8 \, d^{4} e^{4} x^{4} + 4 \,{\left (d e^{6} x^{6} - 2 \, d^{3} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (32 \, e^{7} x^{7} + 21 \, d e^{6} x^{6} - 240 \, d^{2} e^{5} x^{5} - 162 \, d^{3} e^{4} x^{4} + 128 \, d^{4} e^{3} x^{3} + 120 \, d^{5} e^{2} x^{2} + 128 \, d^{6} e x + 48 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (d^{3} e^{4} x^{8} - 8 \, d^{5} e^{2} x^{6} + 8 \, d^{7} x^{4} + 4 \,{\left (d^{4} e^{2} x^{6} - 2 \, d^{6} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(128*d*e^7*x^7 + 84*d^2*e^6*x^6 - 320*d^3*e^5*x^5 - 228*d^4*e^4*x^4 + 64*d^
5*e^3*x^3 + 96*d^6*e^2*x^2 + 128*d^7*e*x + 48*d^8 + 21*(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) - (32*e^7*x^7 + 21*d*e^6*x^6 - 240*d^2*e^5*x^5 - 162*d^
3*e^4*x^4 + 128*d^4*e^3*x^3 + 120*d^5*e^2*x^2 + 128*d^6*e*x + 48*d^7)*sqrt(-e^2*
x^2 + d^2))/(d^3*e^4*x^8 - 8*d^5*e^2*x^6 + 8*d^7*x^4 + 4*(d^4*e^2*x^6 - 2*d^6*x^
4)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 24.9604, size = 449, normalized size = 3.21 \[ d^{2} \left (\begin{cases} - \frac{1}{4 e x^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{e}{8 d^{2} x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{3 e^{3}}{8 d^{4} x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{3 e^{4} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{8 d^{5}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i}{4 e x^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{i e}{8 d^{2} x^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{3 i e^{3}}{8 d^{4} x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{3 i e^{4} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{8 d^{5}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac{2 e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac{2 i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac{e^{2} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{2 d^{3}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i}{2 e x^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{i e}{2 d^{2} x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{i e^{2} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{2 d^{3}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*Piecewise((-1/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) - e/(8*d**2*x**3*sqrt(d
**2/(e**2*x**2) - 1)) + 3*e**3/(8*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) - 3*e**4*ac
osh(d/(e*x))/(8*d**5), Abs(d**2/(e**2*x**2)) > 1), (I/(4*e*x**5*sqrt(-d**2/(e**2
*x**2) + 1)) + I*e/(8*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e**3/(8*d**4*
x*sqrt(-d**2/(e**2*x**2) + 1)) + 3*I*e**4*asin(d/(e*x))/(8*d**5), True)) + 2*d*e
*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)) + e**2
*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), True))

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GIAC/XCAS [A]  time = 0.293015, size = 412, normalized size = 2.94 \[ \frac{x^{4}{\left (\frac{16 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{8}}{x} + \frac{48 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{6}}{x^{2}} + \frac{144 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{4}}{x^{3}} + 3 \, e^{10}\right )} e^{2}}{192 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3}} - \frac{7 \, e^{4}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{8 \, d^{3}} - \frac{{\left (\frac{144 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{9} e^{26}}{x} + \frac{48 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{9} e^{24}}{x^{2}} + \frac{16 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{9} e^{22}}{x^{3}} + \frac{3 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{9} e^{20}}{x^{4}}\right )} e^{\left (-24\right )}}{192 \, d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*x^4*(16*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^8/x + 48*(d*e + sqrt(-x^2*e^2 + d
^2)*e)^2*e^6/x^2 + 144*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^4/x^3 + 3*e^10)*e^2/((
d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^3) - 7/8*e^4*ln(1/2*abs(-2*d*e - 2*sqrt(-x^2*e
^2 + d^2)*e)*e^(-2)/abs(x))/d^3 - 1/192*(144*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^9*
e^26/x + 48*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^9*e^24/x^2 + 16*(d*e + sqrt(-x^2*
e^2 + d^2)*e)^3*d^9*e^22/x^3 + 3*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^9*e^20/x^4)*
e^(-24)/d^12