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

Optimal. Leaf size=10 \[ \frac{\log (a+b x)}{b} \]

[Out]

Log[a + b*x]/b

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Rubi [A]  time = 0.0072511, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 31} \[ \frac{\log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a*x^2 + b*x^3),x]

[Out]

Log[a + b*x]/b

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0009937, size = 10, normalized size = 1. \[ \frac{\log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a*x^2 + b*x^3),x]

[Out]

Log[a + b*x]/b

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Maple [A]  time = 0.001, size = 11, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(b*x+a)/b

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Maxima [A]  time = 1.09744, size = 14, normalized size = 1.4 \begin{align*} \frac{\log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + a)/b

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Fricas [A]  time = 0.761894, size = 22, normalized size = 2.2 \begin{align*} \frac{\log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + a)/b

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Sympy [A]  time = 0.229684, size = 7, normalized size = 0.7 \begin{align*} \frac{\log{\left (a + b x \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x**3+a*x**2),x)

[Out]

log(a + b*x)/b

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Giac [A]  time = 1.15313, size = 15, normalized size = 1.5 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*x + a))/b