3.155 \(\int c (a+b x) \, dx\)

Optimal. Leaf size=15 \[ \frac{c (a+b x)^2}{2 b} \]

[Out]

(c*(a + b*x)^2)/(2*b)

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Rubi [A]  time = 0.0016425, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {9} \[ \frac{c (a+b x)^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[c*(a + b*x),x]

[Out]

(c*(a + b*x)^2)/(2*b)

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]

Rubi steps

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

Mathematica [A]  time = 0.001267, size = 14, normalized size = 0.93 \[ c \left (a x+\frac{b x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[c*(a + b*x),x]

[Out]

c*(a*x + (b*x^2)/2)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \begin{align*} c \left ( ax+{\frac{b{x}^{2}}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*(b*x+a),x)

[Out]

c*(a*x+1/2*b*x^2)

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Maxima [A]  time = 0.996894, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*c

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Fricas [A]  time = 1.26331, size = 28, normalized size = 1.87 \begin{align*} \frac{1}{2} x^{2} c b + x c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*c*b + x*c*a

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Sympy [A]  time = 0.074225, size = 12, normalized size = 0.8 \begin{align*} a c x + \frac{b c x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x + b*c*x**2/2

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Giac [A]  time = 1.17623, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*c