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

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

[Out]

(c*x^2)/2

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Rubi [A]  time = 0.0015419, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {21, 30} \[ \frac{c x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0002229, size = 8, normalized size = 1. \[ \frac{c x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^2)/2

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Maple [A]  time = 0., size = 7, normalized size = 0.9 \begin{align*}{\frac{c{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*x^2

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Maxima [A]  time = 0.993909, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, c x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*x^2

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Fricas [A]  time = 1.18695, size = 15, normalized size = 1.88 \begin{align*} \frac{1}{2} \, c x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*x^2

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Sympy [A]  time = 0.064669, size = 5, normalized size = 0.62 \begin{align*} \frac{c x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x**2/2

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Giac [A]  time = 1.14358, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, c x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*x^2