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

Optimal. Leaf size=10 \[ \log \left (a x+b x^3\right ) \]

[Out]

Log[a*x + b*x^3]

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Rubi [A]  time = 0.0090229, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1587} \[ \log \left (a x+b x^3\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + 3*b*x^2)/(a*x + b*x^3),x]

[Out]

Log[a*x + b*x^3]

Rule 1587

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]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

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

Mathematica [A]  time = 0.0059126, size = 11, normalized size = 1.1 \[ \log \left (a+b x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[a + b*x^2]

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Maple [A]  time = 0.002, size = 11, normalized size = 1.1 \begin{align*} \ln \left ( x \left ( b{x}^{2}+a \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x*(b*x^2+a))

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Maxima [A]  time = 0.965325, size = 14, normalized size = 1.4 \begin{align*} \log \left (b x^{3} + a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x^3 + a*x)

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Fricas [A]  time = 0.978201, size = 24, normalized size = 2.4 \begin{align*} \log \left (b x^{3} + a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x^3 + a*x)

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Sympy [A]  time = 0.27362, size = 8, normalized size = 0.8 \begin{align*} \log{\left (a x + b x^{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b*x**3)

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Giac [A]  time = 1.09781, size = 22, normalized size = 2.2 \begin{align*} \frac{1}{2} \, \log \left (x^{2}\right ) + \log \left ({\left | b x^{2} + a \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2) + log(abs(b*x^2 + a))