\(\int \frac {a+3 b x^2}{a x+b x^3} \, dx\) [435]

Optimal result

Integrand size = 20, antiderivative size = 10 \[ \int \frac {a+3 b x^2}{a x+b x^3} \, dx=\log \left (a x+b x^3\right ) \]

[Out]

ln(b*x^3+a*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1601} \[ \int \frac {a+3 b x^2}{a x+b x^3} \, dx=\log \left (a x+b x^3\right ) \]

[In]

Int[(a + 3*b*x^2)/(a*x + b*x^3),x]

[Out]

Log[a*x + b*x^3]

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps \begin{align*} \text {integral}& = \log \left (a x+b x^3\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {a+3 b x^2}{a x+b x^3} \, dx=\log (x)+\log \left (a+b x^2\right ) \]

[In]

Integrate[(a + 3*b*x^2)/(a*x + b*x^3),x]

[Out]

Log[x] + Log[a + b*x^2]

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\ln \left (b \,x^{3}+a x \right )\)	\(11\)
default	\(\ln \left (x \left (b \,x^{2}+a \right )\right )\)	\(11\)
risch	\(\ln \left (b \,x^{3}+a x \right )\)	\(11\)
norman	\(\ln \left (x \right )+\ln \left (b \,x^{2}+a \right )\)	\(12\)
parallelrisch	\(\ln \left (x \right )+\ln \left (b \,x^{2}+a \right )\)	\(12\)

[In]

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

[Out]

ln(b*x^3+a*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {a+3 b x^2}{a x+b x^3} \, dx=\log \left (b x^{3} + a x\right ) \]

[In]

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

[Out]

log(b*x^3 + a*x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {a+3 b x^2}{a x+b x^3} \, dx=\log {\left (a x + b x^{3} \right )} \]

[In]

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

[Out]

log(a*x + b*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {a+3 b x^2}{a x+b x^3} \, dx=\log \left (b x^{3} + a x\right ) \]

[In]

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

[Out]

log(b*x^3 + a*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {a+3 b x^2}{a x+b x^3} \, dx=\log \left ({\left | b x^{3} + a x \right |}\right ) \]

[In]

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

[Out]

log(abs(b*x^3 + a*x))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {a+3 b x^2}{a x+b x^3} \, dx=\ln \left (b\,x^3+a\,x\right ) \]

[In]

int((a + 3*b*x^2)/(a*x + b*x^3),x)

[Out]

log(a*x + b*x^3)