3.202 \(\int \frac{1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx\)

Optimal. Leaf size=80 \[ \frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 \sqrt{2} d^3 \sqrt{e}}+\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}} \]

[Out]

x/(6*d^2*(d + e*x^2)^(3/2)) + (7*x)/(12*d^3*Sqrt[d + e*x^2]) + ArcTanh[(Sqrt[2]*
Sqrt[e]*x)/Sqrt[d + e*x^2]]/(4*Sqrt[2]*d^3*Sqrt[e])

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Rubi [A]  time = 0.207191, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 \sqrt{2} d^3 \sqrt{e}}+\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x^2)^(3/2)*(d^2 - e^2*x^4)),x]

[Out]

x/(6*d^2*(d + e*x^2)^(3/2)) + (7*x)/(12*d^3*Sqrt[d + e*x^2]) + ArcTanh[(Sqrt[2]*
Sqrt[e]*x)/Sqrt[d + e*x^2]]/(4*Sqrt[2]*d^3*Sqrt[e])

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Rubi in Sympy [A]  time = 42.7073, size = 73, normalized size = 0.91 \[ \frac{x}{6 d^{2} \left (d + e x^{2}\right )^{\frac{3}{2}}} + \frac{7 x}{12 d^{3} \sqrt{d + e x^{2}}} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{8 d^{3} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x**2+d)**(3/2)/(-e**2*x**4+d**2),x)

[Out]

x/(6*d**2*(d + e*x**2)**(3/2)) + 7*x/(12*d**3*sqrt(d + e*x**2)) + sqrt(2)*atanh(
sqrt(2)*sqrt(e)*x/sqrt(d + e*x**2))/(8*d**3*sqrt(e))

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Mathematica [A]  time = 0.154576, size = 68, normalized size = 0.85 \[ \frac{\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}}+\frac{2 \left (9 d x+7 e x^3\right )}{\left (d+e x^2\right )^{3/2}}}{24 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x^2)^(3/2)*(d^2 - e^2*x^4)),x]

[Out]

((2*(9*d*x + 7*e*x^3))/(d + e*x^2)^(3/2) + (3*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[e]*x
)/Sqrt[d + e*x^2]])/Sqrt[e])/(24*d^3)

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Maple [B]  time = 0.033, size = 911, normalized size = 11.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x^2+d)^(3/2)/(-e^2*x^4+d^2),x)

[Out]

1/4*e/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(d*e)^(1/2)/d/((x-(
d*e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*d)^(1/2)-1/4*e/((d*e)^(1/2)+
(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/d^2/((x-(d*e)^(1/2)/e)^2*e+2*(d*e)^(1/
2)*(x-(d*e)^(1/2)/e)+2*d)^(1/2)*x-1/8*e/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)
+(-d*e)^(1/2))/(d*e)^(1/2)/d^(3/2)*2^(1/2)*ln((4*d+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/
e)+2*2^(1/2)*d^(1/2)*((x-(d*e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*d)
^(1/2))/(x-(d*e)^(1/2)/e))-1/4*e/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)
^(1/2))/(d*e)^(1/2)/d/((x+(d*e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*d
)^(1/2)-1/4*e/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/d^2/((x+(d*
e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*d)^(1/2)*x+1/8*e/((d*e)^(1/2)+
(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(d*e)^(1/2)/d^(3/2)*2^(1/2)*ln((4*d-2*
(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*2^(1/2)*d^(1/2)*((x+(d*e)^(1/2)/e)^2*e-2*(d*e)^(
1/2)*(x+(d*e)^(1/2)/e)+2*d)^(1/2))/(x+(d*e)^(1/2)/e))-1/6/d/((d*e)^(1/2)+(-d*e)^
(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(x-1/e*(-d*e)^(1/2))/((x-1/e*(-d*e)^(1/2))^2*
e+2*(-d*e)^(1/2)*(x-1/e*(-d*e)^(1/2)))^(1/2)-1/3*e/d^2/((d*e)^(1/2)+(-d*e)^(1/2)
)/(-(d*e)^(1/2)+(-d*e)^(1/2))/((x-1/e*(-d*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-
d*e)^(1/2)))^(1/2)*x-1/6/d/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2)
)/(x+1/e*(-d*e)^(1/2))/((x+1/e*(-d*e)^(1/2))^2*e-2*(-d*e)^(1/2)*(x+1/e*(-d*e)^(1
/2)))^(1/2)-1/3*e/d^2/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/((x
+1/e*(-d*e)^(1/2))^2*e-2*(-d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(1/2)*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (e^{2} x^{4} - d^{2}\right )}{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((e^2*x^4 - d^2)*(e*x^2 + d)^(3/2)),x, algorithm="maxima")

[Out]

-integrate(1/((e^2*x^4 - d^2)*(e*x^2 + d)^(3/2)), x)

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Fricas [A]  time = 0.327713, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2}{\left (4 \, \sqrt{2}{\left (7 \, e x^{3} + 9 \, d x\right )} \sqrt{e x^{2} + d} \sqrt{e} + 3 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \log \left (\frac{\sqrt{2}{\left (17 \, e^{2} x^{4} + 14 \, d e x^{2} + d^{2}\right )} \sqrt{e} + 8 \,{\left (3 \, e^{2} x^{3} + d e x\right )} \sqrt{e x^{2} + d}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )\right )}}{96 \,{\left (d^{3} e^{2} x^{4} + 2 \, d^{4} e x^{2} + d^{5}\right )} \sqrt{e}}, \frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (7 \, e x^{3} + 9 \, d x\right )} \sqrt{e x^{2} + d} \sqrt{-e} + 3 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (3 \, e x^{2} + d\right )} \sqrt{-e}}{4 \, \sqrt{e x^{2} + d} e x}\right )\right )}}{48 \,{\left (d^{3} e^{2} x^{4} + 2 \, d^{4} e x^{2} + d^{5}\right )} \sqrt{-e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((e^2*x^4 - d^2)*(e*x^2 + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/96*sqrt(2)*(4*sqrt(2)*(7*e*x^3 + 9*d*x)*sqrt(e*x^2 + d)*sqrt(e) + 3*(e^2*x^4
+ 2*d*e*x^2 + d^2)*log((sqrt(2)*(17*e^2*x^4 + 14*d*e*x^2 + d^2)*sqrt(e) + 8*(3*e
^2*x^3 + d*e*x)*sqrt(e*x^2 + d))/(e^2*x^4 - 2*d*e*x^2 + d^2)))/((d^3*e^2*x^4 + 2
*d^4*e*x^2 + d^5)*sqrt(e)), 1/48*sqrt(2)*(2*sqrt(2)*(7*e*x^3 + 9*d*x)*sqrt(e*x^2
+ d)*sqrt(-e) + 3*(e^2*x^4 + 2*d*e*x^2 + d^2)*arctan(1/4*sqrt(2)*(3*e*x^2 + d)*
sqrt(-e)/(sqrt(e*x^2 + d)*e*x)))/((d^3*e^2*x^4 + 2*d^4*e*x^2 + d^5)*sqrt(-e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{- d^{3} \sqrt{d + e x^{2}} - d^{2} e x^{2} \sqrt{d + e x^{2}} + d e^{2} x^{4} \sqrt{d + e x^{2}} + e^{3} x^{6} \sqrt{d + e x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x**2+d)**(3/2)/(-e**2*x**4+d**2),x)

[Out]

-Integral(1/(-d**3*sqrt(d + e*x**2) - d**2*e*x**2*sqrt(d + e*x**2) + d*e**2*x**4
*sqrt(d + e*x**2) + e**3*x**6*sqrt(d + e*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, -1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((e^2*x^4 - d^2)*(e*x^2 + d)^(3/2)),x, algorithm="giac")

[Out]

[undef, undef, undef, -1]