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

Optimal. Leaf size=6 \[ -\frac {c}{x} \]

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Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {c}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 6, normalized size = 1.00 \begin {gather*} -\frac {c}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/x)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 6, normalized size = 1.00 \begin {gather*} -\frac {c}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/x

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giac [A]  time = 0.32, size = 6, normalized size = 1.00 \begin {gather*} -\frac {c}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/x

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maple [A]  time = 0.00, size = 7, normalized size = 1.17 \begin {gather*} -\frac {c}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/x

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maxima [A]  time = 1.09, size = 6, normalized size = 1.00 \begin {gather*} -\frac {c}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/x

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mupad [B]  time = 0.01, size = 6, normalized size = 1.00 \begin {gather*} -\frac {c}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/x

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sympy [A]  time = 0.07, size = 3, normalized size = 0.50 \begin {gather*} - \frac {c}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/x

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