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

Optimal result

Integrand size = 7, antiderivative size = 15 \[ \int c (a+b x) \, dx=\frac {c (a+b x)^2}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {9} \[ \int c (a+b x) \, dx=\frac {c (a+b x)^2}{2 b} \]

[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*} \text {integral}& = \frac {c (a+b x)^2}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int c (a+b x) \, dx=c \left (a x+\frac {b x^2}{2}\right ) \]

[In]

Integrate[c*(a + b*x),x]

[Out]

c*(a*x + (b*x^2)/2)

Maple [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int c (a+b x) \, dx=\frac {1}{2} x^{2} c b + x c a \]

[In]

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

[Out]

1/2*x^2*c*b + x*c*a

Sympy [A] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int c (a+b x) \, dx=a c x + \frac {b c x^{2}}{2} \]

[In]

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

[Out]

a*c*x + b*c*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int c (a+b x) \, dx=\frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c \]

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*c

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int c (a+b x) \, dx=\frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c \]

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*c

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int c (a+b x) \, dx=\frac {c\,x\,\left (2\,a+b\,x\right )}{2} \]

[In]

int(c*(a + b*x),x)

[Out]

(c*x*(2*a + b*x))/2