3.100.5 \(\int \frac {1}{4} (-1+8 x) \, dx\)

Optimal. Leaf size=19 \[ 6+\frac {1}{4} \left (-x+4 x^2\right )-\log (5) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.58, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {9} \begin {gather*} \frac {1}{64} (1-8 x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 8*x)/4,x]

[Out]

(1 - 8*x)^2/64

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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{64} (1-8 x)^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.47 \begin {gather*} -\frac {x}{4}+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 8*x)/4,x]

[Out]

-1/4*x + x^2

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fricas [A]  time = 0.56, size = 7, normalized size = 0.37 \begin {gather*} x^{2} - \frac {1}{4} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x-1/4,x, algorithm="fricas")

[Out]

x^2 - 1/4*x

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giac [A]  time = 0.13, size = 7, normalized size = 0.37 \begin {gather*} x^{2} - \frac {1}{4} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x-1/4,x, algorithm="giac")

[Out]

x^2 - 1/4*x

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maple [A]  time = 0.01, size = 8, normalized size = 0.42

method	result	size
default	\(x^{2}-\frac {1}{4} x\)	\(8\)
norman	\(x^{2}-\frac {1}{4} x\)	\(8\)
risch	\(x^{2}-\frac {1}{4} x\)	\(8\)
gosper	\(\frac {x \left (4 x -1\right )}{4}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x-1/4,x,method=_RETURNVERBOSE)

[Out]

x^2-1/4*x

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maxima [A]  time = 0.34, size = 7, normalized size = 0.37 \begin {gather*} x^{2} - \frac {1}{4} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x-1/4,x, algorithm="maxima")

[Out]

x^2 - 1/4*x

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mupad [B]  time = 0.04, size = 8, normalized size = 0.42 \begin {gather*} \frac {x\,\left (4\,x-1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - 1/4,x)

[Out]

(x*(4*x - 1))/4

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sympy [A]  time = 0.04, size = 5, normalized size = 0.26 \begin {gather*} x^{2} - \frac {x}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x-1/4,x)

[Out]

x**2 - x/4

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