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

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 = 0.0188643, 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, Rules used = {1584, 44} \[ -\frac{\log (a+b x)}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a (a+b x)} \]

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

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

Mathematica [A]  time = 0.0099306, 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 = 0.006, size = 30, normalized size = 1. \begin{align*}{\frac{1}{a \left ( bx+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( bx+a \right ) }{{a}^{2}}} \end{align*}

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 = 1.01662, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{a b x + a^{2}} - \frac{\log \left (b x + a\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}} \end{align*}

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 = 0.730418, size = 89, normalized size = 3.07 \begin{align*} -\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}} \end{align*}

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 = 1.03874, size = 22, normalized size = 0.76 \begin{align*} \frac{1}{a^{2} + a b x} + \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a^{2}} \end{align*}

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 [A]  time = 1.14525, size = 42, normalized size = 1.45 \begin{align*} -\frac{\log \left ({\left | b x + a \right |}\right )}{a^{2}} + \frac{\log \left ({\left | x \right |}\right )}{a^{2}} + \frac{1}{{\left (b x + a\right )} a} \end{align*}

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]

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