3.995 \(\int \frac{x^3}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=58 \[ \frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]

[Out]

Sqrt[b*x^2 + c*x^4]/(2*c) - (b*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(2*c^
(3/2))

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Rubi [A]  time = 0.13859, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]

[Out]

Sqrt[b*x^2 + c*x^4]/(2*c) - (b*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(2*c^
(3/2))

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Rubi in Sympy [A]  time = 12.3511, size = 48, normalized size = 0.83 \[ - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 c^{\frac{3}{2}}} + \frac{\sqrt{b x^{2} + c x^{4}}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(c*x**4+b*x**2)**(1/2),x)

[Out]

-b*atanh(sqrt(c)*x**2/sqrt(b*x**2 + c*x**4))/(2*c**(3/2)) + sqrt(b*x**2 + c*x**4
)/(2*c)

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Mathematica [A]  time = 0.0582484, size = 76, normalized size = 1.31 \[ \frac{x \left (\sqrt{c} x \left (b+c x^2\right )-b \sqrt{b+c x^2} \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{2 c^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[2 + 2*a - 2*(1 + a) + b*x^2 + c*x^4],x]

[Out]

(x*(Sqrt[c]*x*(b + c*x^2) - b*Sqrt[b + c*x^2]*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]
))/(2*c^(3/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.007, size = 64, normalized size = 1.1 \[ -{\frac{x}{2}\sqrt{c{x}^{2}+b} \left ( -x\sqrt{c{x}^{2}+b}{c}^{{\frac{3}{2}}}+b\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/2*x*(c*x^2+b)^(1/2)*(-x*(c*x^2+b)^(1/2)*c^(3/2)+b*ln(x*c^(1/2)+(c*x^2+b)^(1/2
))*c)/(c*x^4+b*x^2)^(1/2)/c^(5/2)

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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(x^3/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284583, size = 1, normalized size = 0.02 \[ \left [\frac{b \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) + 2 \, \sqrt{c x^{4} + b x^{2}} c}{4 \, c^{2}}, \frac{b \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} c}{2 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(b*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(c) + 2*sqrt(c*x^4 + b*x^2)*c) + 2*sqrt(c
*x^4 + b*x^2)*c)/c^2, 1/2*(b*sqrt(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + b*x^2)) +
sqrt(c*x^4 + b*x^2)*c)/c^2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(x**3/sqrt(x**2*(b + c*x**2)), x)

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GIAC/XCAS [A]  time = 0.298499, size = 80, normalized size = 1.38 \[ \frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2}}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{4} + b x^{2}}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b*ln(abs(-2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2))*sqrt(c) - b))/c^(3/2) + 1/2*
sqrt(c*x^4 + b*x^2)/c