\(\int (a+b x) \, dx\) [46]

Optimal result

Integrand size = 5, antiderivative size = 12 \[ \int (a+b x) \, dx=a x+\frac {b x^2}{2} \]

[Out]

a*x+1/2*b*x^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, 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 = {} \[ \int (a+b x) \, dx=a x+\frac {b x^2}{2} \]

[In]

Int[a + b*x,x]

[Out]

a*x + (b*x^2)/2

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int (a+b x) \, dx=a x+\frac {b x^2}{2} \]

[In]

Integrate[a + b*x,x]

[Out]

a*x + (b*x^2)/2

Maple [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
gosper	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
default	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
norman	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
risch	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
parallelrisch	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
parts	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)

[In]

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

[Out]

a*x+1/2*b*x^2

Fricas [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int (a+b x) \, dx=\frac {1}{2} x^{2} b + x a \]

[In]

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

[Out]

1/2*x^2*b + x*a

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int (a+b x) \, dx=a x + \frac {b x^{2}}{2} \]

[In]

integrate(b*x+a,x)

[Out]

a*x + b*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int (a+b x) \, dx=\frac {1}{2} \, b x^{2} + a x \]

[In]

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

[Out]

1/2*b*x^2 + a*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int (a+b x) \, dx=\frac {1}{2} \, b x^{2} + a x \]

[In]

integrate(b*x+a,x, algorithm="giac")

[Out]

1/2*b*x^2 + a*x

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int (a+b x) \, dx=\frac {b\,x^2}{2}+a\,x \]

[In]

int(a + b*x,x)

[Out]

a*x + (b*x^2)/2