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

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

[Out]

c*x

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Rubi [A]  time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 8

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

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*}

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Mathematica [A]  time = 0.00, size = 3, normalized size = 1.00 \[ c x \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*x

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fricas [A]  time = 0.45, size = 3, normalized size = 1.00 \[ c x \]

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

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giac [A]  time = 0.29, size = 3, normalized size = 1.00 \[ c x \]

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

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maple [A]  time = 0.00, size = 4, normalized size = 1.33 \[ c x \]

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

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maxima [A]  time = 1.05, size = 3, normalized size = 1.00 \[ c x \]

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

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mupad [B]  time = 0.01, size = 3, normalized size = 1.00 \[ c\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c + b*c*x^2)/(a + b*x^2),x)

[Out]

c*x

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sympy [A]  time = 0.07, size = 2, normalized size = 0.67 \[ c x \]

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

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