3.133 \(\int \frac{a c+b c x^2}{x^3 \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{b c \log \left (a+b x^2\right )}{2 a^2}-\frac{b c \log (x)}{a^2}-\frac{c}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0626028, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{b c \log \left (a+b x^2\right )}{2 a^2}-\frac{b c \log (x)}{a^2}-\frac{c}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + b*c*x^2)/(x^3*(a + b*x^2)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.4026, size = 37, normalized size = 0.97 \[ - \frac{c}{2 a x^{2}} - \frac{b c \log{\left (x^{2} \right )}}{2 a^{2}} + \frac{b c \log{\left (a + b x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*c*x**2+a*c)/x**3/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0112903, size = 37, normalized size = 0.97 \[ c \left (\frac{b \log \left (a+b x^2\right )}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + b*c*x^2)/(x^3*(a + b*x^2)^2),x]

[Out]

c*(-1/(2*a*x^2) - (b*Log[x])/a^2 + (b*Log[a + b*x^2])/(2*a^2))

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \[ -{\frac{c}{2\,a{x}^{2}}}-{\frac{bc\ln \left ( x \right ) }{{a}^{2}}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*c*x^2+a*c)/x^3/(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.32238, size = 49, normalized size = 1.29 \[ \frac{b c \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac{b c \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{c}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x^2 + a*c)/((b*x^2 + a)^2*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.230648, size = 49, normalized size = 1.29 \[ \frac{b c x^{2} \log \left (b x^{2} + a\right ) - 2 \, b c x^{2} \log \left (x\right ) - a c}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x^2 + a*c)/((b*x^2 + a)^2*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.63474, size = 32, normalized size = 0.84 \[ c \left (- \frac{1}{2 a x^{2}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x**2+a*c)/x**3/(b*x**2+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224729, size = 63, normalized size = 1.66 \[ -\frac{b c{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} + \frac{b c{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{b c x^{2} - a c}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x^2 + a*c)/((b*x^2 + a)^2*x^3),x, algorithm="giac")

[Out]

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