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

Optimal. Leaf size=50 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.120203, antiderivative size = 50, 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 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.0644, size = 44, normalized size = 0.88 \[ - \frac{c}{2 a x^{2}} + \frac{\left (a d - b c\right ) \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{\left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0423776, size = 49, normalized size = 0.98 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}+\frac{\log (x) (a d-b c)}{a^2}-\frac{c}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 56, normalized size = 1.1 \[ -{\frac{c}{2\,a{x}^{2}}}+{\frac{\ln \left ( x \right ) d}{a}}-{\frac{bc\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\,a}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/x^3/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.35141, size = 65, normalized size = 1.3 \[ \frac{{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac{{\left (b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{c}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.242194, size = 65, normalized size = 1.3 \[ \frac{{\left (b c - a d\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (b c - a d\right )} x^{2} \log \left (x\right ) - a c}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.01857, size = 41, normalized size = 0.82 \[ - \frac{c}{2 a x^{2}} + \frac{\left (a d - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224113, size = 97, normalized size = 1.94 \[ -\frac{{\left (b c - a d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} c - a b d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac{b c x^{2} - a d x^{2} - a c}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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