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

Optimal. Leaf size=53 \[ \frac{b c \log \left (a+b x^2\right )}{a^3}-\frac{2 b c \log (x)}{a^3}-\frac{b c}{2 a^2 \left (a+b x^2\right )}-\frac{c}{2 a^2 x^2} \]

[Out]

-c/(2*a^2*x^2) - (b*c)/(2*a^2*(a + b*x^2)) - (2*b*c*Log[x])/a^3 + (b*c*Log[a + b
*x^2])/a^3

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Rubi [A]  time = 0.088698, antiderivative size = 53, 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 \[ \frac{b c \log \left (a+b x^2\right )}{a^3}-\frac{2 b c \log (x)}{a^3}-\frac{b c}{2 a^2 \left (a+b x^2\right )}-\frac{c}{2 a^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-c/(2*a^2*x^2) - (b*c)/(2*a^2*(a + b*x^2)) - (2*b*c*Log[x])/a^3 + (b*c*Log[a + b
*x^2])/a^3

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Rubi in Sympy [A]  time = 15.6278, size = 51, normalized size = 0.96 \[ - \frac{b c}{2 a^{2} \left (a + b x^{2}\right )} - \frac{c}{2 a^{2} x^{2}} - \frac{b c \log{\left (x^{2} \right )}}{a^{3}} + \frac{b c \log{\left (a + b x^{2} \right )}}{a^{3}} \]

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)**3,x)

[Out]

-b*c/(2*a**2*(a + b*x**2)) - c/(2*a**2*x**2) - b*c*log(x**2)/a**3 + b*c*log(a +
b*x**2)/a**3

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Mathematica [A]  time = 0.0604297, size = 42, normalized size = 0.79 \[ -\frac{c \left (a \left (\frac{b}{a+b x^2}+\frac{1}{x^2}\right )-2 b \log \left (a+b x^2\right )+4 b \log (x)\right )}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

-(c*(a*(x^(-2) + b/(a + b*x^2)) + 4*b*Log[x] - 2*b*Log[a + b*x^2]))/(2*a^3)

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Maple [A]  time = 0.013, size = 50, normalized size = 0.9 \[ -{\frac{c}{2\,{a}^{2}{x}^{2}}}-{\frac{bc}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{bc\ln \left ( x \right ) }{{a}^{3}}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/a^2/x^2-1/2*b*c/a^2/(b*x^2+a)-2*b*c*ln(x)/a^3+b*c*ln(b*x^2+a)/a^3

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Maxima [A]  time = 1.35243, size = 77, normalized size = 1.45 \[ -\frac{2 \, b c x^{2} + a c}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac{b c \log \left (b x^{2} + a\right )}{a^{3}} - \frac{b c \log \left (x^{2}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*b*c*x^2 + a*c)/(a^2*b*x^4 + a^3*x^2) + b*c*log(b*x^2 + a)/a^3 - b*c*log(
x^2)/a^3

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Fricas [A]  time = 0.228062, size = 108, normalized size = 2.04 \[ -\frac{2 \, a b c x^{2} + a^{2} c - 2 \,{\left (b^{2} c x^{4} + a b c x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left (b^{2} c x^{4} + a b c x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*a*b*c*x^2 + a^2*c - 2*(b^2*c*x^4 + a*b*c*x^2)*log(b*x^2 + a) + 4*(b^2*c*
x^4 + a*b*c*x^2)*log(x))/(a^3*b*x^4 + a^4*x^2)

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Sympy [A]  time = 2.09321, size = 51, normalized size = 0.96 \[ c \left (- \frac{a + 2 b x^{2}}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(-(a + 2*b*x**2)/(2*a**3*x**2 + 2*a**2*b*x**4) - 2*b*log(x)/a**3 + b*log(a/b +
x**2)/a**3)

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GIAC/XCAS [A]  time = 0.24299, size = 76, normalized size = 1.43 \[ -\frac{b c{\rm ln}\left (x^{2}\right )}{a^{3}} + \frac{b c{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{3}} - \frac{2 \, b c x^{2} + a c}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*c*ln(x^2)/a^3 + b*c*ln(abs(b*x^2 + a))/a^3 - 1/2*(2*b*c*x^2 + a*c)/((b*x^4 +
a*x^2)*a^2)