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

Optimal. Leaf size=51 \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]

[Out]

-7*d^2*x - (4*d*e*x^3)/3 - (e^2*x^5)/5 + (8*d^(5/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d]]
)/Sqrt[e]

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Rubi [A]  time = 0.0724048, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]

Antiderivative was successfully verified.

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

[Out]

-7*d^2*x - (4*d*e*x^3)/3 - (e^2*x^5)/5 + (8*d^(5/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d]]
)/Sqrt[e]

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Rubi in Sympy [A]  time = 16.6056, size = 49, normalized size = 0.96 \[ \frac{8 d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{\sqrt{e}} - 7 d^{2} x - \frac{4 d e x^{3}}{3} - \frac{e^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*d**(5/2)*atanh(sqrt(e)*x/sqrt(d))/sqrt(e) - 7*d**2*x - 4*d*e*x**3/3 - e**2*x**
5/5

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Mathematica [A]  time = 0.037631, size = 51, normalized size = 1. \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]

Antiderivative was successfully verified.

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

[Out]

-7*d^2*x - (4*d*e*x^3)/3 - (e^2*x^5)/5 + (8*d^(5/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d]]
)/Sqrt[e]

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Maple [A]  time = 0.005, size = 42, normalized size = 0.8 \[ -{\frac{{e}^{2}{x}^{5}}{5}}-{\frac{4\,de{x}^{3}}{3}}-7\,{d}^{2}x+8\,{\frac{{d}^{3}}{\sqrt{de}}{\it Artanh} \left ({\frac{ex}{\sqrt{de}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297931, size = 1, normalized size = 0.02 \[ \left [-\frac{1}{5} \, e^{2} x^{5} - \frac{4}{3} \, d e x^{3} + 4 \, d^{2} \sqrt{\frac{d}{e}} \log \left (\frac{e x^{2} + 2 \, e x \sqrt{\frac{d}{e}} + d}{e x^{2} - d}\right ) - 7 \, d^{2} x, -\frac{1}{5} \, e^{2} x^{5} - \frac{4}{3} \, d e x^{3} + 8 \, d^{2} \sqrt{-\frac{d}{e}} \arctan \left (\frac{x}{\sqrt{-\frac{d}{e}}}\right ) - 7 \, d^{2} x\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/5*e^2*x^5 - 4/3*d*e*x^3 + 4*d^2*sqrt(d/e)*log((e*x^2 + 2*e*x*sqrt(d/e) + d)/
(e*x^2 - d)) - 7*d^2*x, -1/5*e^2*x^5 - 4/3*d*e*x^3 + 8*d^2*sqrt(-d/e)*arctan(x/s
qrt(-d/e)) - 7*d^2*x]

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Sympy [A]  time = 1.73923, size = 75, normalized size = 1.47 \[ - 7 d^{2} x - \frac{4 d e x^{3}}{3} - \frac{e^{2} x^{5}}{5} - 4 \sqrt{\frac{d^{5}}{e}} \log{\left (x - \frac{\sqrt{\frac{d^{5}}{e}}}{d^{2}} \right )} + 4 \sqrt{\frac{d^{5}}{e}} \log{\left (x + \frac{\sqrt{\frac{d^{5}}{e}}}{d^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*d**2*x - 4*d*e*x**3/3 - e**2*x**5/5 - 4*sqrt(d**5/e)*log(x - sqrt(d**5/e)/d**
2) + 4*sqrt(d**5/e)*log(x + sqrt(d**5/e)/d**2)

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GIAC/XCAS [A]  time = 0.281494, size = 194, normalized size = 3.8 \[ 4 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{1}{4}} d{\left | d \right |} e^{\frac{11}{2}}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )} + 2 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{15}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} d e^{\frac{15}{2}}\right )} e^{\left (-8\right )}{\rm ln}\left ({\left |{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right ) - 2 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{1}{4}} d{\left | d \right |} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | -{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right ) - \frac{1}{15} \,{\left (3 \, x^{5} e^{12} + 20 \, d x^{3} e^{11} + 105 \, d^{2} x e^{10}\right )} e^{\left (-10\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*((d^2)^(1/4)*d^2*e^(11/2) - (d^2)^(1/4)*d*abs(d)*e^(11/2))*arctan(x*e^(1/2)/(d
^2)^(1/4))*e^(-6) + 2*((d^2)^(1/4)*d^2*e^(15/2) + (d^2)^(3/4)*d*e^(15/2))*e^(-8)
*ln(abs((d^2)^(1/4)*e^(-1/2) + x)) - 2*((d^2)^(1/4)*d^2*e^(11/2) + (d^2)^(1/4)*d
*abs(d)*e^(11/2))*e^(-6)*ln(abs(-(d^2)^(1/4)*e^(-1/2) + x)) - 1/15*(3*x^5*e^12 +
20*d*x^3*e^11 + 105*d^2*x*e^10)*e^(-10)