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

Optimal. Leaf size=76 \[ \frac{(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac{c}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.177952, antiderivative size = 76, 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 \[ \frac{(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac{c}{2 a^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.2363, size = 68, normalized size = 0.89 \[ - \frac{c}{2 a^{2} x^{2}} + \frac{a d - b c}{2 a^{2} \left (a + b x^{2}\right )} + \frac{\left (a d - 2 b c\right ) \log{\left (x^{2} \right )}}{2 a^{3}} - \frac{\left (a d - 2 b c\right ) \log{\left (a + b x^{2} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0833559, size = 64, normalized size = 0.84 \[ \frac{\frac{a (a d-b c)}{a+b x^2}+(2 b c-a d) \log \left (a+b x^2\right )+2 \log (x) (a d-2 b c)-\frac{a c}{x^2}}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 86, normalized size = 1.1 \[ -{\frac{c}{2\,{a}^{2}{x}^{2}}}+{\frac{\ln \left ( x \right ) d}{{a}^{2}}}-2\,{\frac{bc\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\,{a}^{2}}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{{a}^{3}}}+{\frac{d}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bc}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34356, size = 105, normalized size = 1.38 \[ -\frac{{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac{{\left (2 \, b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} - \frac{{\left (2 \, b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231165, size = 165, normalized size = 2.17 \[ -\frac{a^{2} c +{\left (2 \, a b c - a^{2} d\right )} x^{2} -{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} +{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} +{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.86831, size = 70, normalized size = 0.92 \[ \frac{- a c + x^{2} \left (a d - 2 b c\right )}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} + \frac{\left (a d - 2 b c\right ) \log{\left (x \right )}}{a^{3}} - \frac{\left (a d - 2 b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23943, size = 113, normalized size = 1.49 \[ -\frac{{\left (2 \, b c - a d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} - \frac{2 \, b c x^{2} - a d x^{2} + a c}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2}} + \frac{{\left (2 \, b^{2} c - a b d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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