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

Optimal. Leaf size=12 \[ a x+\frac {b x^2}{2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} a x+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*x,x]

[Out]

a*x + (b*x^2)/2

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} a x+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*x,x]

[Out]

a*x + (b*x^2)/2

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Maple [A]
time = 0.01, size = 11, normalized size = 0.92

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x

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Fricas [A]
time = 0.80, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} x^{2} b + x a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*b + x*a

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Sympy [A]
time = 0.00, size = 8, normalized size = 0.67 \begin {gather*} a x + \frac {b x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*x+a,x)

[Out]

a*x + b*x**2/2

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Giac [A]
time = 1.46, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*x+a,x, algorithm="giac")

[Out]

1/2*b*x^2 + a*x

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Mupad [B]
time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} \frac {b\,x^2}{2}+a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*x,x)

[Out]

a*x + (b*x^2)/2

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