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

Optimal. Leaf size=24 \[ \frac{c \log (x)}{a}-\frac{c \log \left (a+b x^2\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0368758, antiderivative size = 24, 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 \[ \frac{c \log (x)}{a}-\frac{c \log \left (a+b x^2\right )}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.09932, size = 22, normalized size = 0.92 \[ \frac{c \log{\left (x^{2} \right )}}{2 a} - \frac{c \log{\left (a + b x^{2} \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00873298, size = 24, normalized size = 1. \[ c \left (\frac{\log (x)}{a}-\frac{\log \left (a+b x^2\right )}{2 a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

c*(Log[x]/a - Log[a + b*x^2]/(2*a))

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Maple [A]  time = 0.006, size = 23, normalized size = 1. \[{\frac{c\ln \left ( x \right ) }{a}}-{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.333, size = 34, normalized size = 1.42 \[ -\frac{c \log \left (b x^{2} + a\right )}{2 \, a} + \frac{c \log \left (x^{2}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231777, size = 28, normalized size = 1.17 \[ -\frac{c \log \left (b x^{2} + a\right ) - 2 \, c \log \left (x\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.536742, size = 17, normalized size = 0.71 \[ c \left (\frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a}\right ) \]

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)

[Out]

c*(log(x)/a - log(a/b + x**2)/(2*a))

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GIAC/XCAS [A]  time = 0.223636, size = 35, normalized size = 1.46 \[ \frac{c{\rm ln}\left (x^{2}\right )}{2 \, a} - \frac{c{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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