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

Optimal. Leaf size=58 \[ \frac{b^2}{a^3 (a+b x)}+\frac{3 b^2 \log (x)}{a^4}-\frac{3 b^2 \log (a+b x)}{a^4}+\frac{2 b}{a^3 x}-\frac{1}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0315931, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1584, 44} \[ \frac{b^2}{a^3 (a+b x)}+\frac{3 b^2 \log (x)}{a^4}-\frac{3 b^2 \log (a+b x)}{a^4}+\frac{2 b}{a^3 x}-\frac{1}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a*x^2 + b*x^3)^2,x]

[Out]

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

Mathematica [A]  time = 0.0500844, size = 53, normalized size = 0.91 \[ \frac{a \left (\frac{2 b^2}{a+b x}-\frac{a}{x^2}+\frac{4 b}{x}\right )-6 b^2 \log (a+b x)+6 b^2 \log (x)}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a*x^2 + b*x^3)^2,x]

[Out]

(a*(-(a/x^2) + (4*b)/x + (2*b^2)/(a + b*x)) + 6*b^2*Log[x] - 6*b^2*Log[a + b*x])/(2*a^4)

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Maple [A]  time = 0.01, size = 57, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{b}{x{a}^{3}}}+{\frac{{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^3+a*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.01718, size = 86, normalized size = 1.48 \begin{align*} \frac{6 \, b^{2} x^{2} + 3 \, a b x - a^{2}}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac{3 \, b^{2} \log \left (b x + a\right )}{a^{4}} + \frac{3 \, b^{2} \log \left (x\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.791073, size = 177, normalized size = 3.05 \begin{align*} \frac{6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} - 6 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*a*b^2*x^2 + 3*a^2*b*x - a^3 - 6*(b^3*x^3 + a*b^2*x^2)*log(b*x + a) + 6*(b^3*x^3 + a*b^2*x^2)*log(x))/(a
^4*b*x^3 + a^5*x^2)

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Sympy [A]  time = 0.759451, size = 54, normalized size = 0.93 \begin{align*} \frac{- a^{2} + 3 a b x + 6 b^{2} x^{2}}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} + \frac{3 b^{2} \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(x/(b*x**3+a*x**2)**2,x)

[Out]

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

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Giac [A]  time = 1.18427, size = 86, normalized size = 1.48 \begin{align*} -\frac{3 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{4}} + \frac{3 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3}}{2 \,{\left (b x + a\right )} a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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