3.2.55 \(\int c (a+b x) \, dx\) [155]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {c (a+b x)^2}{2 b} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 0.93 \begin {gather*} c \left (a x+\frac {b x^2}{2}\right ) \end {gather*}

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 = 0.01, size = 13, normalized size = 0.87

method	result	size
gosper	\(\frac {x \left (b x +2 a \right ) c}{2}\)	\(12\)
default	\(\left (\frac {1}{2} x^{2} b +a x \right ) c\)	\(13\)
norman	\(a c x +\frac {1}{2} b c \,x^{2}\)	\(13\)
risch	\(a c x +\frac {1}{2} b c \,x^{2}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c \end {gather*}

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 = 0.89, size = 12, normalized size = 0.80 \begin {gather*} \frac {1}{2} x^{2} c b + x c a \end {gather*}

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 = 0.01, size = 12, normalized size = 0.80 \begin {gather*} a c x + \frac {b c x^{2}}{2} \end {gather*}

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 = 1.30, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c \end {gather*}

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

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Mupad [B]
time = 0.02, size = 11, normalized size = 0.73 \begin {gather*} \frac {c\,x\,\left (2\,a+b\,x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*(a + b*x),x)

[Out]

(c*x*(2*a + b*x))/2

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