3.225 \(\int \frac{x^5}{\left (a x^2+b x^3\right )^2} \, dx\)

Optimal. Leaf size=23 \[ \frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2} \]

[Out]

a/(b^2*(a + b*x)) + Log[a + b*x]/b^2

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Rubi [A]  time = 0.0397896, antiderivative size = 23, 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 \[ \frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a*x^2 + b*x^3)^2,x]

[Out]

a/(b^2*(a + b*x)) + Log[a + b*x]/b^2

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Rubi in Sympy [A]  time = 6.83493, size = 19, normalized size = 0.83 \[ \frac{a}{b^{2} \left (a + b x\right )} + \frac{\log{\left (a + b x \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**3+a*x**2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0102577, size = 20, normalized size = 0.87 \[ \frac{\frac{a}{a+b x}+\log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a*x^2 + b*x^3)^2,x]

[Out]

(a/(a + b*x) + Log[a + b*x])/b^2

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Maple [A]  time = 0.01, size = 24, normalized size = 1. \[{\frac{a}{{b}^{2} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.45213, size = 35, normalized size = 1.52 \[ \frac{a}{b^{3} x + a b^{2}} + \frac{\log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a*x^2)^2,x, algorithm="maxima")

[Out]

a/(b^3*x + a*b^2) + log(b*x + a)/b^2

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Fricas [A]  time = 0.205626, size = 38, normalized size = 1.65 \[ \frac{{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b^{3} x + a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a*x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.18563, size = 20, normalized size = 0.87 \[ \frac{a}{a b^{2} + b^{3} x} + \frac{\log{\left (a + b x \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**3+a*x**2)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217479, size = 32, normalized size = 1.39 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} + \frac{a}{{\left (b x + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a*x^2)^2,x, algorithm="giac")

[Out]

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