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

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

[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 = 0.0267151, 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, Rules used = {1584, 44} \[ -\frac{b^2}{a^3 x}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]

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

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a x^2+b x^3\right )} \, dx &=\int \frac{1}{x^4 (a+b x)} \, dx\\ &=\int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{1}{3 a x^3}+\frac{b}{2 a^2 x^2}-\frac{b^2}{a^3 x}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0048084, size = 56, normalized size = 1. \[ -\frac{b^2}{a^3 x}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]

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 = 0.006, size = 53, normalized size = 1. \begin{align*} -{\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}}} \end{align*}

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 = 0.971695, size = 69, normalized size = 1.23 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^3+a*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 = 0.725751, size = 126, normalized size = 2.25 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^3+a*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 = 1.20975, size = 44, normalized size = 0.79 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.20949, size = 76, normalized size = 1.36 \begin{align*} \frac{b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac{b^{3} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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