\(\int (a c+(b c+d) x) \, dx\) [429]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int (a c+(b c+d) x) \, dx=a c x+\frac {1}{2} (b c+d) x^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a c+(b c+d) x) \, dx=a c x+\frac {1}{2} x^2 (b c+d) \]

[In]

Int[a*c + (b*c + d)*x,x]

[Out]

a*c*x + ((b*c + d)*x^2)/2

Rubi steps \begin{align*} \text {integral}& = a c x+\frac {1}{2} (b c+d) x^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int (a c+(b c+d) x) \, dx=a c x+\frac {1}{2} b c x^2+\frac {d x^2}{2} \]

[In]

Integrate[a*c + (b*c + d)*x,x]

[Out]

a*c*x + (b*c*x^2)/2 + (d*x^2)/2

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
gosper	\(\frac {x \left (b c x +2 a c +d x \right )}{2}\)	\(16\)
parallelrisch	\(a c x +\frac {\left (b c +d \right ) x^{2}}{2}\)	\(16\)
norman	\(\left (\frac {b c}{2}+\frac {d}{2}\right ) x^{2}+a c x\)	\(18\)
default	\(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\)	\(19\)
risch	\(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\)	\(19\)
parts	\(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\)	\(19\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int (a c+(b c+d) x) \, dx=\frac {1}{2} x^{2} c b + \frac {1}{2} x^{2} d + x c a \]

[In]

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

[Out]

1/2*x^2*c*b + 1/2*x^2*d + x*c*a

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int (a c+(b c+d) x) \, dx=a c x + x^{2} \left (\frac {b c}{2} + \frac {d}{2}\right ) \]

[In]

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

[Out]

a*c*x + x**2*(b*c/2 + d/2)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int (a c+(b c+d) x) \, dx=a c x + \frac {1}{2} \, {\left (b c + d\right )} x^{2} \]

[In]

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

[Out]

a*c*x + 1/2*(b*c + d)*x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int (a c+(b c+d) x) \, dx=a c x + \frac {1}{2} \, {\left (b c + d\right )} x^{2} \]

[In]

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

[Out]

a*c*x + 1/2*(b*c + d)*x^2

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int (a c+(b c+d) x) \, dx=\left (\frac {d}{2}+\frac {b\,c}{2}\right )\,x^2+a\,c\,x \]

[In]

int(a*c + x*(d + b*c),x)

[Out]

x^2*(d/2 + (b*c)/2) + a*c*x