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

Optimal. Leaf size=34 \[ \frac{c \log (x)}{a}-\frac{(b c-a d) \log \left (a+b x^2\right )}{2 a b} \]

[Out]

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

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Rubi [A]  time = 0.0895671, antiderivative size = 34, 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 \[ \frac{c \log (x)}{a}-\frac{(b c-a d) \log \left (a+b x^2\right )}{2 a b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.987, size = 29, normalized size = 0.85 \[ \frac{c \log{\left (x^{2} \right )}}{2 a} + \frac{\left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.019751, size = 34, normalized size = 1. \[ \frac{(a d-b c) \log \left (a+b x^2\right )}{2 a b}+\frac{c \log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 37, normalized size = 1.1 \[{\frac{c\ln \left ( x \right ) }{a}}+{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\,b}}-{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/x/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.34795, size = 47, normalized size = 1.38 \[ \frac{c \log \left (x^{2}\right )}{2 \, a} - \frac{{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237207, size = 45, normalized size = 1.32 \[ \frac{2 \, b c \log \left (x\right ) -{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.23361, size = 26, normalized size = 0.76 \[ \frac{c \log{\left (x \right )}}{a} + \frac{\left (a d - b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.245467, size = 49, normalized size = 1.44 \[ \frac{c{\rm ln}\left (x^{2}\right )}{2 \, a} - \frac{{\left (b c - a d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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