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

Optimal. Leaf size=51 \[ -\frac{c \log \left (a+b x^2\right )}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{b c-a d}{2 a b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.113667, antiderivative size = 51, 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 \[ -\frac{c \log \left (a+b x^2\right )}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{b c-a d}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.6897, size = 44, normalized size = 0.86 \[ - \frac{a d - b c}{2 a b \left (a + b x^{2}\right )} + \frac{c \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{c \log{\left (a + b x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0488924, size = 46, normalized size = 0.9 \[ \frac{\frac{a (b c-a d)}{b \left (a+b x^2\right )}-c \log \left (a+b x^2\right )+2 c \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 53, normalized size = 1. \[{\frac{c\ln \left ( x \right ) }{{a}^{2}}}-{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}}-{\frac{d}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{c}{2\,a \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/x/(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.34692, size = 69, normalized size = 1.35 \[ \frac{b c - a d}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )}} - \frac{c \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{c \log \left (x^{2}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231858, size = 96, normalized size = 1.88 \[ \frac{a b c - a^{2} d -{\left (b^{2} c x^{2} + a b c\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (b^{2} c x^{2} + a b c\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{2} x^{2} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.2005, size = 46, normalized size = 0.9 \[ - \frac{a d - b c}{2 a^{2} b + 2 a b^{2} x^{2}} + \frac{c \log{\left (x \right )}}{a^{2}} - \frac{c \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.251333, size = 85, normalized size = 1.67 \[ \frac{c{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{c{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{b^{2} c x^{2} + 2 \, a b c - a^{2} d}{2 \,{\left (b x^{2} + a\right )} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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