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

Optimal. Leaf size=8 \[ -\frac{c}{2 x^2} \]

[Out]

-c/(2*x^2)

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Rubi [A]  time = 0.0014944, antiderivative size = 8, 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, 30} \[ -\frac{c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*x^2)

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0002812, size = 8, normalized size = 1. \[ -\frac{c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*x^2)

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Maple [A]  time = 0.001, size = 7, normalized size = 0.9 \begin{align*} -{\frac{c}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c/x^2

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Maxima [A]  time = 0.996967, size = 8, normalized size = 1. \begin{align*} -\frac{c}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c/x^2

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Fricas [A]  time = 1.19666, size = 16, normalized size = 2. \begin{align*} -\frac{c}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c/x^2

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Sympy [A]  time = 0.07011, size = 7, normalized size = 0.88 \begin{align*} - \frac{c}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/(2*x**2)

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Giac [A]  time = 1.16327, size = 8, normalized size = 1. \begin{align*} -\frac{c}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c/x^2