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

Optimal. Leaf size=70 \[ -\frac{b \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{4 c^{3/2}}-\frac{\sqrt{a+b x^2-c x^4}}{2 c} \]

[Out]

-Sqrt[a + b*x^2 - c*x^4]/(2*c) - (b*ArcTan[(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 = 0.114199, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{b \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{4 c^{3/2}}-\frac{\sqrt{a+b x^2-c x^4}}{2 c} \]

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*ArcTan[(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 = 12.5072, size = 60, normalized size = 0.86 \[ - \frac{b \operatorname{atan}{\left (\frac{b - 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} - c x^{4}}} \right )}}{4 c^{\frac{3}{2}}} - \frac{\sqrt{a + b x^{2} - c x^{4}}}{2 c} \]

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*atan((b - 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 - c*x**4)))/(4*c**(3/2)) - sqr
t(a + b*x**2 - c*x**4)/(2*c)

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Mathematica [C]  time = 0.0709748, size = 77, normalized size = 1.1 \[ -\frac{\sqrt{a+b x^2-c x^4}}{2 c}+\frac{i b \log \left (2 \sqrt{a+b x^2-c x^4}-\frac{i \left (2 c x^2-b\right )}{\sqrt{c}}\right )}{4 c^{3/2}} \]

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

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Maple [A]  time = 0.017, size = 58, normalized size = 0.8 \[ -{\frac{1}{2\,c}\sqrt{-c{x}^{4}+b{x}^{2}+a}}+{\frac{b}{4}\arctan \left ({1\sqrt{c} \left ({x}^{2}-{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \right ){c}^{-{\frac{3}{2}}}} \]

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)*arctan(c^(1/2)*(x^2-1/2*b/c)/(-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 = 0.292197, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (4 \, \sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} - b c\right )} +{\left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, a c\right )} \sqrt{-c}\right ) - 4 \, \sqrt{-c x^{4} + b x^{2} + a} \sqrt{-c}}{8 \, \sqrt{-c} c}, \frac{b \arctan \left (\frac{2 \, c x^{2} - b}{2 \, \sqrt{-c x^{4} + b x^{2} + a} \sqrt{c}}\right ) - 2 \, \sqrt{-c x^{4} + b x^{2} + a} \sqrt{c}}{4 \, c^{\frac{3}{2}}}\right ] \]

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))/(sqrt(-c)*c),
1/4*(b*arctan(1/2*(2*c*x^2 - b)/(sqrt(-c*x^4 + b*x^2 + a)*sqrt(c))) - 2*sqrt(-c*
x^4 + b*x^2 + a)*sqrt(c))/c^(3/2)]

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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 = 0.293335, size = 95, normalized size = 1.36 \[ -\frac{b{\rm ln}\left ({\left | 2 \,{\left (\sqrt{-c} x^{2} - \sqrt{-c x^{4} + b x^{2} + a}\right )} \sqrt{-c} + b \right |}\right )}{4 \, \sqrt{-c} c} - \frac{\sqrt{-c x^{4} + b x^{2} + a}}{2 \, c} \]

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