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

Optimal. Leaf size=3 \[ c x \]

[Out]

c*x

________________________________________________________________________________________

Rubi [A]  time = 0.0009539, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 8} \[ c x \]

Antiderivative was successfully verified.

[In]

Int[(a*c + b*c*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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0001705, size = 3, normalized size = 1. \[ c x \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*x

________________________________________________________________________________________

Maple [A]  time = 0., size = 4, normalized size = 1.3 \begin{align*} cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x

________________________________________________________________________________________

Maxima [A]  time = 0.988227, size = 4, normalized size = 1.33 \begin{align*} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x

________________________________________________________________________________________

Fricas [A]  time = 1.24767, size = 7, normalized size = 2.33 \begin{align*} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x

________________________________________________________________________________________

Sympy [A]  time = 0.061941, size = 2, normalized size = 0.67 \begin{align*} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x

________________________________________________________________________________________

Giac [A]  time = 1.12315, size = 4, normalized size = 1.33 \begin{align*} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x