\(\int \frac {x^2}{a x^2+b x^3} \, dx\) [217]

Optimal result

Integrand size = 17, antiderivative size = 10 \[ \int \frac {x^2}{a x^2+b x^3} \, dx=\frac {\log (a+b x)}{b} \]

[Out]

ln(b*x+a)/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 31} \[ \int \frac {x^2}{a x^2+b x^3} \, dx=\frac {\log (a+b x)}{b} \]

[In]

Int[x^2/(a*x^2 + b*x^3),x]

[Out]

Log[a + b*x]/b

Rule 31

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

Rule 1598

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{a+b x} \, dx \\ & = \frac {\log (a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{a x^2+b x^3} \, dx=\frac {\log (a+b x)}{b} \]

[In]

Integrate[x^2/(a*x^2 + b*x^3),x]

[Out]

Log[a + b*x]/b

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
default	\(\frac {\ln \left (b x +a \right )}{b}\)	\(11\)
norman	\(\frac {\ln \left (b x +a \right )}{b}\)	\(11\)
risch	\(\frac {\ln \left (b x +a \right )}{b}\)	\(11\)
parallelrisch	\(\frac {\ln \left (b x +a \right )}{b}\)	\(11\)

[In]

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

[Out]

ln(b*x+a)/b

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{a x^2+b x^3} \, dx=\frac {\log \left (b x + a\right )}{b} \]

[In]

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

[Out]

log(b*x + a)/b

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{a x^2+b x^3} \, dx=\frac {\log {\left (a + b x \right )}}{b} \]

[In]

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

[Out]

log(a + b*x)/b

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{a x^2+b x^3} \, dx=\frac {\log \left (b x + a\right )}{b} \]

[In]

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

[Out]

log(b*x + a)/b

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {x^2}{a x^2+b x^3} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{b} \]

[In]

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

[Out]

log(abs(b*x + a))/b

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{a x^2+b x^3} \, dx=\frac {\ln \left (a+b\,x\right )}{b} \]

[In]

int(x^2/(a*x^2 + b*x^3),x)

[Out]

log(a + b*x)/b