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

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

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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} \begin {gather*} c x+\frac {d x^2}{2} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} c x+\frac {d x^2}{2} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.03, size = 14, normalized size = 1.17 \begin {gather*} \frac {(c+d x)^2}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^2/(2*d)

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fricas [A]  time = 0.39, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, d x^{2} + c x \end {gather*}

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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giac [A]  time = 0.15, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, d x^{2} + c x \end {gather*}

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

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maple [A]  time = 0.04, size = 11, normalized size = 0.92 \begin {gather*} \frac {1}{2} d \,x^{2}+c x \end {gather*}

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]

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

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maxima [A]  time = 1.04, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, d x^{2} + c x \end {gather*}

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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mupad [B]  time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} \frac {d\,x^2}{2}+c\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x + (d*x^2)/2

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sympy [A]  time = 0.10, size = 8, normalized size = 0.67 \begin {gather*} c x + \frac {d x^{2}}{2} \end {gather*}

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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