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

Optimal. Leaf size=68 \[ \frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.106927, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.2634, size = 58, normalized size = 0.85 \[ - \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 c^{\frac{3}{2}}} + \frac{\sqrt{a + b x^{2} + c x^{4}}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.036847, size = 66, normalized size = 0.97 \[ \frac{\sqrt{a+b x^2+c x^4}}{2 c}-\frac{b \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{4 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 56, normalized size = 0.8 \[{\frac{1}{2\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{b}{4}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(c*x^4+b*x^2+a)^(1/2)/c-1/4*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+
a)^(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(x^3/sqrt(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29296, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + 4 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{c}}{8 \, c^{\frac{3}{2}}}, -\frac{b \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right ) - 2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c}}{4 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(b*log(4*sqrt(c*x^4 + b*x^2 + a)*(2*c^2*x^2 + b*c) - (8*c^2*x^4 + 8*b*c*x^2
+ b^2 + 4*a*c)*sqrt(c)) + 4*sqrt(c*x^4 + b*x^2 + a)*sqrt(c))/c^(3/2), -1/4*(b*a
rctan(1/2*(2*c*x^2 + b)*sqrt(-c)/(sqrt(c*x^4 + b*x^2 + a)*c)) - 2*sqrt(c*x^4 + b
*x^2 + a)*sqrt(-c))/(sqrt(-c)*c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3/sqrt(a + b*x**2 + c*x**4), x)

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GIAC/XCAS [A]  time = 0.294992, size = 82, normalized size = 1.21 \[ \frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{4} + b x^{2} + a}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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