3.104 \(\int \frac{x \left (A+B x^2\right )}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=71 \[ \frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.159537, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x^2))/(a + b*x^2 + c*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 23.2393, size = 63, normalized size = 0.89 \[ \frac{B \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c} - \frac{\left (2 A c - B b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0797695, size = 71, normalized size = 1. \[ \frac{B \log \left (a+b x^2+c x^4\right )-\frac{2 (b B-2 A c) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 98, normalized size = 1.4 \[{\frac{B\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,c}}+{A\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bB}{2\,c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((B*x^2 + A)*x/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271636, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{b^{2} - 4 \, a c} B \log \left (c x^{4} + b x^{2} + a\right ) -{\left (B b - 2 \, A c\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{4 \, \sqrt{b^{2} - 4 \, a c} c}, \frac{\sqrt{-b^{2} + 4 \, a c} B \log \left (c x^{4} + b x^{2} + a\right ) - 2 \,{\left (B b - 2 \, A c\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.68165, size = 287, normalized size = 4.04 \[ \left (\frac{B}{4 c} - \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- A b + 2 B a - 8 a c \left (\frac{B}{4 c} - \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} \left (\frac{B}{4 c} - \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right )}{- 2 A c + B b} \right )} + \left (\frac{B}{4 c} + \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- A b + 2 B a - 8 a c \left (\frac{B}{4 c} + \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} \left (\frac{B}{4 c} + \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right )}{- 2 A c + B b} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B/(4*c) - (-2*A*c + B*b)*sqrt(-4*a*c + b**2)/(4*c*(4*a*c - b**2)))*log(x**2 + (
-A*b + 2*B*a - 8*a*c*(B/(4*c) - (-2*A*c + B*b)*sqrt(-4*a*c + b**2)/(4*c*(4*a*c -
b**2))) + 2*b**2*(B/(4*c) - (-2*A*c + B*b)*sqrt(-4*a*c + b**2)/(4*c*(4*a*c - b*
*2))))/(-2*A*c + B*b)) + (B/(4*c) + (-2*A*c + B*b)*sqrt(-4*a*c + b**2)/(4*c*(4*a
*c - b**2)))*log(x**2 + (-A*b + 2*B*a - 8*a*c*(B/(4*c) + (-2*A*c + B*b)*sqrt(-4*
a*c + b**2)/(4*c*(4*a*c - b**2))) + 2*b**2*(B/(4*c) + (-2*A*c + B*b)*sqrt(-4*a*c
+ b**2)/(4*c*(4*a*c - b**2))))/(-2*A*c + B*b))

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GIAC/XCAS [A]  time = 0.292196, size = 90, normalized size = 1.27 \[ \frac{B{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c} - \frac{{\left (B b - 2 \, A c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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