3.218 \(\int \frac{x}{a x^2+b x^3} \, dx\)

Optimal. Leaf size=18 \[ \frac{\log (x)}{a}-\frac{\log (a+b x)}{a} \]

[Out]

Log[x]/a - Log[a + b*x]/a

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Rubi [A]  time = 0.0219345, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\log (x)}{a}-\frac{\log (a+b x)}{a} \]

Antiderivative was successfully verified.

[In]  Int[x/(a*x^2 + b*x^3),x]

[Out]

Log[x]/a - Log[a + b*x]/a

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Rubi in Sympy [A]  time = 4.5208, size = 12, normalized size = 0.67 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (a + b x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/a - log(a + b*x)/a

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Mathematica [A]  time = 0.0056813, size = 18, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log (a+b x)}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a*x^2 + b*x^3),x]

[Out]

Log[x]/a - Log[a + b*x]/a

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Maple [A]  time = 0.008, size = 19, normalized size = 1.1 \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( bx+a \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^3+a*x^2),x)

[Out]

ln(x)/a-ln(b*x+a)/a

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Maxima [A]  time = 1.38077, size = 24, normalized size = 1.33 \[ -\frac{\log \left (b x + a\right )}{a} + \frac{\log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(b*x + a)/a + log(x)/a

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Fricas [A]  time = 0.213362, size = 22, normalized size = 1.22 \[ -\frac{\log \left (b x + a\right ) - \log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(log(b*x + a) - log(x))/a

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Sympy [A]  time = 0.335426, size = 10, normalized size = 0.56 \[ \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**3+a*x**2),x)

[Out]

(log(x) - log(a/b + x))/a

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GIAC/XCAS [A]  time = 0.218869, size = 27, normalized size = 1.5 \[ -\frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(b*x + a))/a + ln(abs(x))/a