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

Optimal. Leaf size=78 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{A \log \left (a+b x^2+c x^4\right )}{4 a}+\frac{A \log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.277219, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{A \log \left (a+b x^2+c x^4\right )}{4 a}+\frac{A \log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.5782, size = 73, normalized size = 0.94 \[ \frac{A \log{\left (x^{2} \right )}}{2 a} - \frac{A \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a} + \frac{\left (A b - 2 B a\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.180405, size = 128, normalized size = 1.64 \[ \frac{-\left (A \left (\sqrt{b^2-4 a c}+b\right )-2 a B\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )+\left (A \left (b-\sqrt{b^2-4 a c}\right )-2 a B\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )+4 A \log (x) \sqrt{b^2-4 a c}}{4 a \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 105, normalized size = 1.4 \[ -{\frac{A\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,a}}-{\frac{Ab}{2\,a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{B\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{A\ln \left ( x \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.318328, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, B a - A b\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) + \sqrt{b^{2} - 4 \, a c}{\left (A \log \left (c x^{4} + b x^{2} + a\right ) - 4 \, A \log \left (x\right )\right )}}{4 \, \sqrt{b^{2} - 4 \, a c} a}, \frac{2 \,{\left (2 \, B a - A b\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left (A \log \left (c x^{4} + b x^{2} + a\right ) - 4 \, A \log \left (x\right )\right )}}{4 \, \sqrt{-b^{2} + 4 \, a c} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 133.358, size = 330, normalized size = 4.23 \[ \frac{A \log{\left (x \right )}}{a} + \left (- \frac{A}{4 a} - \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{2 A a c - A b^{2} + B a b + 8 a^{2} c \left (- \frac{A}{4 a} - \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right ) - 2 a b^{2} \left (- \frac{A}{4 a} - \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right )}{- A b c + 2 B a c} \right )} + \left (- \frac{A}{4 a} + \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{2 A a c - A b^{2} + B a b + 8 a^{2} c \left (- \frac{A}{4 a} + \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right ) - 2 a b^{2} \left (- \frac{A}{4 a} + \frac{\left (- A b + 2 B a\right ) \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )}\right )}{- A b c + 2 B a c} \right )} \]

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)

[Out]

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

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GIAC/XCAS [A]  time = 0.289236, size = 105, normalized size = 1.35 \[ -\frac{A{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a} + \frac{A{\rm ln}\left (x^{2}\right )}{2 \, a} + \frac{{\left (2 \, B a - A b\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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