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

Optimal. Leaf size=62 \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.120745, antiderivative size = 62, 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 \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.5816, size = 56, normalized size = 0.9 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{\sqrt{e}} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{\sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(e)*x/sqrt(d + e*x**2))/sqrt(e) + sqrt(2)*atanh(sqrt(2)*sqrt(e)*x/sqr
t(d + e*x**2))/sqrt(e)

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Mathematica [A]  time = 0.0421149, size = 61, normalized size = 0.98 \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )-\log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{\sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]] - Log[e*x + Sqrt[e]*Sqrt[d
+ e*x^2]])/Sqrt[e]

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Maple [B]  time = 0.066, size = 1442, normalized size = 23.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*e/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(d*e)^(1/2)*((x-(d*
e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*d)^(3/2)+1/4*e/((d*e)^(1/2)+(-
d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(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+5/4*e^(1/2)/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)
+(-d*e)^(1/2))*d*ln(((x-(d*e)^(1/2)/e)*e+(d*e)^(1/2))/e^(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))+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)-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*e/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(d*e
)^(1/2)*((x+(d*e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*d)^(3/2)+1/4*e/
((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(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+5/4*e^(1/2)/((d*e)^(1/2)+(-d*e)^(1/2)
)/(-(d*e)^(1/2)+(-d*e)^(1/2))*d*ln(((x+(d*e)^(1/2)/e)*e-(d*e)^(1/2))/e^(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))-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)+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*e/(-d*e)^(1/2)/((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)))^(3/2)-1/4*e/((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-1/4*e^(1/2)/((d
*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))*d*ln(((x-1/e*(-d*e)^(1/2))*e
+(-d*e)^(1/2))/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))+1/6*e/(-d*e)^(1/2)/((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)))^(3/2)-1/4*
e/((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-1/4*e^(1/2)/((d*e)^(1/2)+(-d*e)^
(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))*d*ln(((x+1/e*(-d*e)^(1/2))*e-(-d*e)^(1/2))/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))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307087, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{2} \log \left (\frac{17 \, e^{2} x^{4} + 14 \, d e x^{2} + d^{2} + \frac{4 \, \sqrt{2}{\left (3 \, e^{2} x^{3} + d e x\right )} \sqrt{e x^{2} + d}}{\sqrt{e}}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 2 \, \log \left (2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{4 \, \sqrt{e}}, \frac{\sqrt{2} \sqrt{-e} \sqrt{-\frac{1}{e}} \arctan \left (\frac{\sqrt{2}{\left (3 \, e x^{2} + d\right )}}{4 \, \sqrt{e x^{2} + d} e x \sqrt{-\frac{1}{e}}}\right ) - 2 \, \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{2 \, \sqrt{-e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(sqrt(2)*log((17*e^2*x^4 + 14*d*e*x^2 + d^2 + 4*sqrt(2)*(3*e^2*x^3 + d*e*x)
*sqrt(e*x^2 + d)/sqrt(e))/(e^2*x^4 - 2*d*e*x^2 + d^2)) + 2*log(2*sqrt(e*x^2 + d)
*e*x - (2*e*x^2 + d)*sqrt(e)))/sqrt(e), 1/2*(sqrt(2)*sqrt(-e)*sqrt(-1/e)*arctan(
1/4*sqrt(2)*(3*e*x^2 + d)/(sqrt(e*x^2 + d)*e*x*sqrt(-1/e))) - 2*arctan(sqrt(-e)*
x/sqrt(e*x^2 + d)))/sqrt(-e)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(sqrt(d + e*x**2)/(-d + e*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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