3.124 \(\int \frac{a c+b c x^2}{x (a+b x^2)} \, dx\)

Optimal. Leaf size=4 \[ c \log (x) \]

[Out]

c*Log[x]

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Rubi [A]  time = 0.0010773, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {21, 29} \[ c \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a*c + b*c*x^2)/(x*(a + b*x^2)),x]

[Out]

c*Log[x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0002694, size = 4, normalized size = 1. \[ c \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*Log[x]

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Maple [A]  time = 0.002, size = 5, normalized size = 1.3 \begin{align*} c\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*ln(x)

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Maxima [A]  time = 0.998049, size = 9, normalized size = 2.25 \begin{align*} \frac{1}{2} \, c \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*log(x^2)

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Fricas [A]  time = 1.22363, size = 14, normalized size = 3.5 \begin{align*} c \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*log(x)

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Sympy [A]  time = 0.067502, size = 3, normalized size = 0.75 \begin{align*} c \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*log(x)

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Giac [A]  time = 1.14353, size = 7, normalized size = 1.75 \begin{align*} c \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*log(abs(x))