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

Optimal. Leaf size=64 \[ \frac{b \tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{\log \left (a-b x^2+c x^4\right )}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.134761, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{b \tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{\log \left (a-b x^2+c x^4\right )}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a - b*x^2 + c*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.6457, size = 54, normalized size = 0.84 \[ - \frac{b \operatorname{atanh}{\left (\frac{- b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (a - b x^{2} + c x^{4} \right )}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0419203, size = 65, normalized size = 1.02 \[ \frac{\frac{2 b \tan ^{-1}\left (\frac{2 c x^2-b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\log \left (a-b x^2+c x^4\right )}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 63, normalized size = 1. \[{\frac{\ln \left ( c{x}^{4}-b{x}^{2}+a \right ) }{4\,c}}+{\frac{b}{2\,c}\arctan \left ({(2\,c{x}^{2}-b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^4-b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265208, size = 1, normalized size = 0.02 \[ \left [\frac{b \log \left (\frac{b^{3} - 4 \, a b c - 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} - 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} - b x^{2} + a}\right ) + \sqrt{b^{2} - 4 \, a c} \log \left (c x^{4} - b x^{2} + a\right )}{4 \, \sqrt{b^{2} - 4 \, a c} c}, \frac{2 \, b \arctan \left (-\frac{{\left (2 \, c x^{2} - b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c} \log \left (c x^{4} - b x^{2} + a\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.51163, size = 223, normalized size = 3.48 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right ) \log{\left (x^{2} + \frac{8 a c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right ) - 2 a - 2 b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right )}{b} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right ) \log{\left (x^{2} + \frac{8 a c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right ) - 2 a - 2 b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right )}{b} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-b*sqrt(-4*a*c + b**2)/(4*c*(4*a*c - b**2)) + 1/(4*c))*log(x**2 + (8*a*c*(-b*sq
rt(-4*a*c + b**2)/(4*c*(4*a*c - b**2)) + 1/(4*c)) - 2*a - 2*b**2*(-b*sqrt(-4*a*c
+ b**2)/(4*c*(4*a*c - b**2)) + 1/(4*c)))/b) + (b*sqrt(-4*a*c + b**2)/(4*c*(4*a*
c - b**2)) + 1/(4*c))*log(x**2 + (8*a*c*(b*sqrt(-4*a*c + b**2)/(4*c*(4*a*c - b**
2)) + 1/(4*c)) - 2*a - 2*b**2*(b*sqrt(-4*a*c + b**2)/(4*c*(4*a*c - b**2)) + 1/(4
*c)))/b)

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GIAC/XCAS [A]  time = 0.295392, size = 84, normalized size = 1.31 \[ \frac{b \arctan \left (\frac{2 \, c x^{2} - b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c} + \frac{{\rm ln}\left (c x^{4} - b x^{2} + a\right )}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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