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

Optimal. Leaf size=29 \[ -\frac{\log (a+b x)}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0420813, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{\log (a+b x)}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a*x^2 + b*x^3)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.6504, size = 24, normalized size = 0.83 \[ \frac{1}{a \left (a + b x\right )} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (a + b x \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0161748, size = 24, normalized size = 0.83 \[ \frac{\frac{a}{a+b x}-\log (a+b x)+\log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 30, normalized size = 1. \[{\frac{1}{a \left ( bx+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( bx+a \right ) }{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^3+a*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.40641, size = 38, normalized size = 1.31 \[ \frac{1}{a b x + a^{2}} - \frac{\log \left (b x + a\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.212578, size = 53, normalized size = 1.83 \[ -\frac{{\left (b x + a\right )} \log \left (b x + a\right ) -{\left (b x + a\right )} \log \left (x\right ) - a}{a^{2} b x + a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.37572, size = 22, normalized size = 0.76 \[ \frac{1}{a^{2} + a b x} + \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219702, size = 42, normalized size = 1.45 \[ -\frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{1}{{\left (b x + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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