3.221 \(\int \frac{1}{x^2 \left (a x^2+b x^3\right )} \, dx\)

Optimal. Leaf size=56 \[ -\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}-\frac{b^2}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]

[Out]

-1/(3*a*x^3) + b/(2*a^2*x^2) - b^2/(a^3*x) - (b^3*Log[x])/a^4 + (b^3*Log[a + b*x
])/a^4

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Rubi [A]  time = 0.0584743, antiderivative size = 56, 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 \[ -\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}-\frac{b^2}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a*x^2 + b*x^3)),x]

[Out]

-1/(3*a*x^3) + b/(2*a^2*x^2) - b^2/(a^3*x) - (b^3*Log[x])/a^4 + (b^3*Log[a + b*x
])/a^4

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Rubi in Sympy [A]  time = 10.8652, size = 49, normalized size = 0.88 \[ - \frac{1}{3 a x^{3}} + \frac{b}{2 a^{2} x^{2}} - \frac{b^{2}}{a^{3} x} - \frac{b^{3} \log{\left (x \right )}}{a^{4}} + \frac{b^{3} \log{\left (a + b x \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**3+a*x**2),x)

[Out]

-1/(3*a*x**3) + b/(2*a**2*x**2) - b**2/(a**3*x) - b**3*log(x)/a**4 + b**3*log(a
+ b*x)/a**4

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Mathematica [A]  time = 0.00819892, size = 56, normalized size = 1. \[ -\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}-\frac{b^2}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a*x^2 + b*x^3)),x]

[Out]

-1/(3*a*x^3) + b/(2*a^2*x^2) - b^2/(a^3*x) - (b^3*Log[x])/a^4 + (b^3*Log[a + b*x
])/a^4

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Maple [A]  time = 0.011, size = 53, normalized size = 1. \[ -{\frac{1}{3\,a{x}^{3}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}-{\frac{{b}^{2}}{x{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^3+a*x^2),x)

[Out]

-1/3/a/x^3+1/2*b/a^2/x^2-b^2/a^3/x-b^3*ln(x)/a^4+b^3*ln(b*x+a)/a^4

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Maxima [A]  time = 1.39426, size = 69, normalized size = 1.23 \[ \frac{b^{3} \log \left (b x + a\right )}{a^{4}} - \frac{b^{3} \log \left (x\right )}{a^{4}} - \frac{6 \, b^{2} x^{2} - 3 \, a b x + 2 \, a^{2}}{6 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x^2)*x^2),x, algorithm="maxima")

[Out]

b^3*log(b*x + a)/a^4 - b^3*log(x)/a^4 - 1/6*(6*b^2*x^2 - 3*a*b*x + 2*a^2)/(a^3*x
^3)

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Fricas [A]  time = 0.231339, size = 73, normalized size = 1.3 \[ \frac{6 \, b^{3} x^{3} \log \left (b x + a\right ) - 6 \, b^{3} x^{3} \log \left (x\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x^2)*x^2),x, algorithm="fricas")

[Out]

1/6*(6*b^3*x^3*log(b*x + a) - 6*b^3*x^3*log(x) - 6*a*b^2*x^2 + 3*a^2*b*x - 2*a^3
)/(a^4*x^3)

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Sympy [A]  time = 1.51566, size = 44, normalized size = 0.79 \[ - \frac{2 a^{2} - 3 a b x + 6 b^{2} x^{2}}{6 a^{3} x^{3}} + \frac{b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**3+a*x**2),x)

[Out]

-(2*a**2 - 3*a*b*x + 6*b**2*x**2)/(6*a**3*x**3) + b**3*(-log(x) + log(a/b + x))/
a**4

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GIAC/XCAS [A]  time = 0.217432, size = 76, normalized size = 1.36 \[ \frac{b^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{4}} - \frac{b^{3}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{2} - 3 \, a^{2} b x + 2 \, a^{3}}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x^2)*x^2),x, algorithm="giac")

[Out]

b^3*ln(abs(b*x + a))/a^4 - b^3*ln(abs(x))/a^4 - 1/6*(6*a*b^2*x^2 - 3*a^2*b*x + 2
*a^3)/(a^4*x^3)