3.129 \(\int \frac{x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{c \log \left (a+b x^2\right )}{2 b} \]

[Out]

(c*Log[a + b*x^2])/(2*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.012239, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{c \log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

(c*Log[a + b*x^2])/(2*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.77827, size = 12, normalized size = 0.75 \[ \frac{c \log{\left (a + b x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(a + b*x**2)/(2*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00323439, size = 16, normalized size = 1. \[ \frac{c \log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

(c*Log[a + b*x^2])/(2*b)

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \[{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*ln(b*x^2+a)/b

_______________________________________________________________________________________

Maxima [A]  time = 1.3302, size = 19, normalized size = 1.19 \[ \frac{c \log \left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*log(b*x^2 + a)/b

_______________________________________________________________________________________

Fricas [A]  time = 0.227455, size = 19, normalized size = 1.19 \[ \frac{c \log \left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*log(b*x^2 + a)/b

_______________________________________________________________________________________

Sympy [A]  time = 0.271459, size = 12, normalized size = 0.75 \[ \frac{c \log{\left (a + b x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(a + b*x**2)/(2*b)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222329, size = 85, normalized size = 5.31 \[ -\frac{1}{2} \, c{\left (\frac{{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{2} + a\right )} b}\right )} - \frac{a c}{2 \,{\left (b x^{2} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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