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

Optimal. Leaf size=51 \[ -\frac{A \log \left (a+b x^2\right )}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{2 a b \left (a+b x^2\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.113601, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{A \log \left (a+b x^2\right )}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.503, size = 44, normalized size = 0.86 \[ \frac{A \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{A \log{\left (a + b x^{2} \right )}}{2 a^{2}} + \frac{A b - B a}{2 a b \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x**2)/(2*a**2) - A*log(a + b*x**2)/(2*a**2) + (A*b - B*a)/(2*a*b*(a + b*x*
*2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0500623, size = 46, normalized size = 0.9 \[ \frac{\frac{a (A b-a B)}{b \left (a+b x^2\right )}-A \log \left (a+b x^2\right )+2 A \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 53, normalized size = 1. \[{\frac{A\ln \left ( x \right ) }{{a}^{2}}}-{\frac{A\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}}+{\frac{A}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{B}{2\,b \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x/(b*x^2+a)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34773, size = 69, normalized size = 1.35 \[ -\frac{B a - A b}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )}} - \frac{A \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{A \log \left (x^{2}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(B*a - A*b)/(a*b^2*x^2 + a^2*b) - 1/2*A*log(b*x^2 + a)/a^2 + 1/2*A*log(x^2)
/a^2

_______________________________________________________________________________________

Fricas [A]  time = 0.233235, size = 95, normalized size = 1.86 \[ -\frac{B a^{2} - A a b +{\left (A b^{2} x^{2} + A a b\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (A b^{2} x^{2} + A a b\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{2} x^{2} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(B*a^2 - A*a*b + (A*b^2*x^2 + A*a*b)*log(b*x^2 + a) - 2*(A*b^2*x^2 + A*a*b)
*log(x))/(a^2*b^2*x^2 + a^3*b)

_______________________________________________________________________________________

Sympy [A]  time = 2.19838, size = 46, normalized size = 0.9 \[ \frac{A \log{\left (x \right )}}{a^{2}} - \frac{A \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} - \frac{- A b + B a}{2 a^{2} b + 2 a b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x)/a**2 - A*log(a/b + x**2)/(2*a**2) - (-A*b + B*a)/(2*a**2*b + 2*a*b**2*x
**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22716, size = 85, normalized size = 1.67 \[ \frac{A{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{A{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{A b^{2} x^{2} - B a^{2} + 2 \, A a b}{2 \,{\left (b x^{2} + a\right )} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*A*ln(x^2)/a^2 - 1/2*A*ln(abs(b*x^2 + a))/a^2 + 1/2*(A*b^2*x^2 - B*a^2 + 2*A*
a*b)/((b*x^2 + a)*a^2*b)