3.289 \(\int \frac{x^3}{\sqrt{-3 x^2+4 x^4}} \, dx\)

Optimal. Leaf size=45 \[ \frac{1}{8} \sqrt{4 x^4-3 x^2}+\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{4 x^4-3 x^2}}\right ) \]

[Out]

Sqrt[-3*x^2 + 4*x^4]/8 + (3*ArcTanh[(2*x^2)/Sqrt[-3*x^2 + 4*x^4]])/16

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Rubi [A]  time = 0.101116, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{1}{8} \sqrt{4 x^4-3 x^2}+\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{4 x^4-3 x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[-3*x^2 + 4*x^4],x]

[Out]

Sqrt[-3*x^2 + 4*x^4]/8 + (3*ArcTanh[(2*x^2)/Sqrt[-3*x^2 + 4*x^4]])/16

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Rubi in Sympy [A]  time = 10.7373, size = 37, normalized size = 0.82 \[ \frac{\sqrt{4 x^{4} - 3 x^{2}}}{8} + \frac{3 \operatorname{atanh}{\left (\frac{2 x^{2}}{\sqrt{4 x^{4} - 3 x^{2}}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(4*x**4 - 3*x**2)/8 + 3*atanh(2*x**2/sqrt(4*x**4 - 3*x**2))/16

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Mathematica [A]  time = 0.0214401, size = 58, normalized size = 1.29 \[ \frac{x \left (8 x^3+3 \sqrt{4 x^2-3} \log \left (\sqrt{4 x^2-3}+2 x\right )-6 x\right )}{16 \sqrt{x^2 \left (4 x^2-3\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[-3*x^2 + 4*x^4],x]

[Out]

(x*(-6*x + 8*x^3 + 3*Sqrt[-3 + 4*x^2]*Log[2*x + Sqrt[-3 + 4*x^2]]))/(16*Sqrt[x^2
*(-3 + 4*x^2)])

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Maple [A]  time = 0.01, size = 60, normalized size = 1.3 \[{\frac{x}{32}\sqrt{4\,{x}^{2}-3} \left ( 3\,\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-3} \right ) \sqrt{4}+4\,x\sqrt{4\,{x}^{2}-3} \right ){\frac{1}{\sqrt{4\,{x}^{4}-3\,{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(4*x^4-3*x^2)^(1/2),x)

[Out]

1/32*x*(4*x^2-3)^(1/2)*(3*ln(x*4^(1/2)+(4*x^2-3)^(1/2))*4^(1/2)+4*x*(4*x^2-3)^(1
/2))/(4*x^4-3*x^2)^(1/2)

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Maxima [A]  time = 0.764865, size = 55, normalized size = 1.22 \[ \frac{1}{8} \, \sqrt{4 \, x^{4} - 3 \, x^{2}} + \frac{3}{32} \, \log \left (8 \, x^{2} + 4 \, \sqrt{4 \, x^{4} - 3 \, x^{2}} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(4*x^4 - 3*x^2),x, algorithm="maxima")

[Out]

1/8*sqrt(4*x^4 - 3*x^2) + 3/32*log(8*x^2 + 4*sqrt(4*x^4 - 3*x^2) - 3)

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Fricas [A]  time = 0.261799, size = 61, normalized size = 1.36 \[ \frac{1}{8} \, \sqrt{4 \, x^{4} - 3 \, x^{2}} - \frac{3}{16} \, \log \left (-\frac{2 \, x^{2} - \sqrt{4 \, x^{4} - 3 \, x^{2}}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(4*x^4 - 3*x^2) - 3/16*log(-(2*x^2 - sqrt(4*x^4 - 3*x^2))/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{x^{2} \left (4 x^{2} - 3\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3/sqrt(x**2*(4*x**2 - 3)), x)

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GIAC/XCAS [A]  time = 0.282338, size = 57, normalized size = 1.27 \[ \frac{1}{8} \, \sqrt{4 \, x^{4} - 3 \, x^{2}} - \frac{3}{32} \,{\rm ln}\left ({\left | -8 \, x^{2} + 4 \, \sqrt{4 \, x^{4} - 3 \, x^{2}} + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(4*x^4 - 3*x^2) - 3/32*ln(abs(-8*x^2 + 4*sqrt(4*x^4 - 3*x^2) + 3))