3.43 \(\int \frac{(d+e x)^2}{x^6 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=169 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{4 d^4}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2} \]

[Out]

-Sqrt[d^2 - e^2*x^2]/(5*x^5) - (e*Sqrt[d^2 - e^2*x^2])/(2*d*x^4) - (3*e^2*Sqrt[d
^2 - e^2*x^2])/(5*d^2*x^3) - (3*e^3*Sqrt[d^2 - e^2*x^2])/(4*d^3*x^2) - (6*e^4*Sq
rt[d^2 - e^2*x^2])/(5*d^4*x) - (3*e^5*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(4*d^4)

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Rubi [A]  time = 0.499527, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{4 d^4}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[d^2 - e^2*x^2]/(5*x^5) - (e*Sqrt[d^2 - e^2*x^2])/(2*d*x^4) - (3*e^2*Sqrt[d
^2 - e^2*x^2])/(5*d^2*x^3) - (3*e^3*Sqrt[d^2 - e^2*x^2])/(4*d^3*x^2) - (6*e^4*Sq
rt[d^2 - e^2*x^2])/(5*d^4*x) - (3*e^5*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(4*d^4)

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Rubi in Sympy [A]  time = 42.136, size = 148, normalized size = 0.88 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 x^{5}} - \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{2 d x^{4}} - \frac{3 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{2} x^{3}} - \frac{3 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{4 d^{3} x^{2}} - \frac{3 e^{5} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{4 d^{4}} - \frac{6 e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/x**6/(-e**2*x**2+d**2)**(1/2),x)

[Out]

-sqrt(d**2 - e**2*x**2)/(5*x**5) - e*sqrt(d**2 - e**2*x**2)/(2*d*x**4) - 3*e**2*
sqrt(d**2 - e**2*x**2)/(5*d**2*x**3) - 3*e**3*sqrt(d**2 - e**2*x**2)/(4*d**3*x**
2) - 3*e**5*atanh(sqrt(d**2 - e**2*x**2)/d)/(4*d**4) - 6*e**4*sqrt(d**2 - e**2*x
**2)/(5*d**4*x)

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Mathematica [A]  time = 0.171842, size = 106, normalized size = 0.63 \[ -\frac{15 e^5 x^5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (4 d^4+10 d^3 e x+12 d^2 e^2 x^2+15 d e^3 x^3+24 e^4 x^4\right )-15 e^5 x^5 \log (x)}{20 d^4 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d^2 - e^2*x^2]*(4*d^4 + 10*d^3*e*x + 12*d^2*e^2*x^2 + 15*d*e^3*x^3 + 24*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]])/(20*d^4*x
^5)

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Maple [A]  time = 0.022, size = 164, normalized size = 1. \[ -{\frac{1}{5\,{x}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{2}}{5\,{d}^{2}{x}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{6\,{e}^{4}}{5\,{d}^{4}x}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{e}{2\,d{x}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{3}}{4\,{d}^{3}{x}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{5}}{4\,{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/x^6/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/5*(-e^2*x^2+d^2)^(1/2)/x^5-3/5*e^2*(-e^2*x^2+d^2)^(1/2)/d^2/x^3-6/5*e^4*(-e^2
*x^2+d^2)^(1/2)/d^4/x-1/2*e*(-e^2*x^2+d^2)^(1/2)/d/x^4-3/4*e^3*(-e^2*x^2+d^2)^(1
/2)/d^3/x^2-3/4/d^3*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*x + d)^2/(sqrt(-e^2*x^2 + d^2)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295339, size = 533, normalized size = 3.15 \[ -\frac{24 \, e^{10} x^{10} + 15 \, d e^{9} x^{9} - 300 \, d^{2} e^{8} x^{8} - 185 \, d^{3} e^{7} x^{7} + 520 \, d^{4} e^{6} x^{6} + 290 \, d^{5} e^{5} x^{5} - 100 \, d^{6} e^{4} x^{4} + 40 \, d^{7} e^{3} x^{3} - 80 \, d^{8} e^{2} x^{2} - 160 \, d^{9} e x - 64 \, d^{10} - 15 \,{\left (5 \, d e^{9} x^{9} - 20 \, d^{3} e^{7} x^{7} + 16 \, d^{5} e^{5} x^{5} -{\left (e^{9} x^{9} - 12 \, d^{2} e^{7} x^{7} + 16 \, d^{4} e^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (120 \, d e^{8} x^{8} + 75 \, d^{2} e^{7} x^{7} - 420 \, d^{3} e^{6} x^{6} - 250 \, d^{4} e^{5} x^{5} + 164 \, d^{5} e^{4} x^{4} + 40 \, d^{6} e^{3} x^{3} + 112 \, d^{7} e^{2} x^{2} + 160 \, d^{8} e x + 64 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{20 \,{\left (5 \, d^{5} e^{4} x^{9} - 20 \, d^{7} e^{2} x^{7} + 16 \, d^{9} x^{5} -{\left (d^{4} e^{4} x^{9} - 12 \, d^{6} e^{2} x^{7} + 16 \, d^{8} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(24*e^10*x^10 + 15*d*e^9*x^9 - 300*d^2*e^8*x^8 - 185*d^3*e^7*x^7 + 520*d^4
*e^6*x^6 + 290*d^5*e^5*x^5 - 100*d^6*e^4*x^4 + 40*d^7*e^3*x^3 - 80*d^8*e^2*x^2 -
160*d^9*e*x - 64*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) + (120*d*e^8*x^8 + 75*d^2*e^7*x^7 - 420*d^3*e^6*x^6 - 250*d^4*
e^5*x^5 + 164*d^5*e^4*x^4 + 40*d^6*e^3*x^3 + 112*d^7*e^2*x^2 + 160*d^8*e*x + 64*
d^9)*sqrt(-e^2*x^2 + d^2))/(5*d^5*e^4*x^9 - 20*d^7*e^2*x^7 + 16*d^9*x^5 - (d^4*e
^4*x^9 - 12*d^6*e^2*x^7 + 16*d^8*x^5)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 22.0968, size = 510, normalized size = 3.02 \[ d^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{5 d^{2} x^{4}} - \frac{4 e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{15 d^{4} x^{2}} - \frac{8 e^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{15 d^{6}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{5 d^{2} x^{4}} - \frac{4 i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{15 d^{4} x^{2}} - \frac{8 i e^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{15 d^{6}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{1}{4 e x^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{e}{8 d^{2} x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{3 e^{3}}{8 d^{4} x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{3 e^{4} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{8 d^{5}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i}{4 e x^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{i e}{8 d^{2} x^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{3 i e^{3}}{8 d^{4} x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{3 i e^{4} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{8 d^{5}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac{2 e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac{2 i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/x**6/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(5*d**2*x**4) - 4*e**3*sqrt(d**2/(
e**2*x**2) - 1)/(15*d**4*x**2) - 8*e**5*sqrt(d**2/(e**2*x**2) - 1)/(15*d**6), Ab
s(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(5*d**2*x**4) - 4*I*
e**3*sqrt(-d**2/(e**2*x**2) + 1)/(15*d**4*x**2) - 8*I*e**5*sqrt(-d**2/(e**2*x**2
) + 1)/(15*d**6), True)) + 2*d*e*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*acosh(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)) + e**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))

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GIAC/XCAS [A]  time = 0.298571, size = 493, normalized size = 2.92 \[ \frac{x^{5}{\left (\frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{10}}{x} + \frac{15 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{8}}{x^{2}} + \frac{40 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{6}}{x^{3}} + \frac{110 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{4}}{x^{4}} + e^{12}\right )} e^{3}}{160 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{4}} - \frac{3 \, e^{5}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{4 \, d^{4}} - \frac{{\left (\frac{110 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{16} e^{38}}{x} + \frac{40 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{16} e^{36}}{x^{2}} + \frac{15 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{16} e^{34}}{x^{3}} + \frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{16} e^{32}}{x^{4}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{16} e^{30}}{x^{5}}\right )} e^{\left (-35\right )}}{160 \, d^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/160*x^5*(5*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^10/x + 15*(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 + 110*(d*e + sqrt
(-x^2*e^2 + d^2)*e)^4*e^4/x^4 + e^12)*e^3/((d*e + sqrt(-x^2*e^2 + d^2)*e)^5*d^4)
- 3/4*e^5*ln(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^4 - 1/
160*(110*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^16*e^38/x + 40*(d*e + sqrt(-x^2*e^2 +
d^2)*e)^2*d^16*e^36/x^2 + 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^16*e^34/x^3 + 5*
(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^16*e^32/x^4 + (d*e + sqrt(-x^2*e^2 + d^2)*e)^
5*d^16*e^30/x^5)*e^(-35)/d^20