### 3.1763 $$\int \frac{a c+(b c+a d) x+b d x^2}{a+b x} \, dx$$

Optimal. Leaf size=12 $c x+\frac{d x^2}{2}$

[Out]

c*x + (d*x^2)/2

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Rubi [A]  time = 0.0065882, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.037, Rules used = {24} $c x+\frac{d x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

c*x + (d*x^2)/2

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0006399, size = 12, normalized size = 1. $c x+\frac{d x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

c*x + (d*x^2)/2

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Maple [A]  time = 0.038, size = 11, normalized size = 0.9 \begin{align*} cx+{\frac{d{x}^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x+1/2*d*x^2

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Maxima [A]  time = 1.1512, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*x^2 + c*x

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Fricas [A]  time = 1.77442, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*x^2 + c*x

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Sympy [A]  time = 0.110716, size = 8, normalized size = 0.67 \begin{align*} c x + \frac{d x^{2}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x + d*x**2/2

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Giac [A]  time = 1.21701, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*x^2 + c*x