3.12.5 \(\int (1-2 x)^2 (3+5 x) \, dx\)

Optimal. Leaf size=23 \[ 5 x^4-\frac {8 x^3}{3}-\frac {7 x^2}{2}+3 x \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} 5 x^4-\frac {8 x^3}{3}-\frac {7 x^2}{2}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2*(3 + 5*x),x]

[Out]

3*x - (7*x^2)/2 - (8*x^3)/3 + 5*x^4

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (1-2 x)^2 (3+5 x) \, dx &=\int \left (3-7 x-8 x^2+20 x^3\right ) \, dx\\ &=3 x-\frac {7 x^2}{2}-\frac {8 x^3}{3}+5 x^4\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} 5 x^4-\frac {8 x^3}{3}-\frac {7 x^2}{2}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2*(3 + 5*x),x]

[Out]

3*x - (7*x^2)/2 - (8*x^3)/3 + 5*x^4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (1-2 x)^2 (3+5 x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^2*(3 + 5*x),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^2*(3 + 5*x), x]

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fricas [A]  time = 0.96, size = 19, normalized size = 0.83 \begin {gather*} 5 x^{4} - \frac {8}{3} x^{3} - \frac {7}{2} x^{2} + 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x),x, algorithm="fricas")

[Out]

5*x^4 - 8/3*x^3 - 7/2*x^2 + 3*x

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giac [A]  time = 1.21, size = 19, normalized size = 0.83 \begin {gather*} 5 \, x^{4} - \frac {8}{3} \, x^{3} - \frac {7}{2} \, x^{2} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x),x, algorithm="giac")

[Out]

5*x^4 - 8/3*x^3 - 7/2*x^2 + 3*x

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maple [A]  time = 0.00, size = 20, normalized size = 0.87 \begin {gather*} 5 x^{4}-\frac {8}{3} x^{3}-\frac {7}{2} x^{2}+3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(5*x+3),x)

[Out]

3*x-7/2*x^2-8/3*x^3+5*x^4

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maxima [A]  time = 0.71, size = 19, normalized size = 0.83 \begin {gather*} 5 \, x^{4} - \frac {8}{3} \, x^{3} - \frac {7}{2} \, x^{2} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x),x, algorithm="maxima")

[Out]

5*x^4 - 8/3*x^3 - 7/2*x^2 + 3*x

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mupad [B]  time = 0.03, size = 19, normalized size = 0.83 \begin {gather*} 5\,x^4-\frac {8\,x^3}{3}-\frac {7\,x^2}{2}+3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)^2*(5*x + 3),x)

[Out]

3*x - (7*x^2)/2 - (8*x^3)/3 + 5*x^4

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sympy [A]  time = 0.06, size = 20, normalized size = 0.87 \begin {gather*} 5 x^{4} - \frac {8 x^{3}}{3} - \frac {7 x^{2}}{2} + 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(3+5*x),x)

[Out]

5*x**4 - 8*x**3/3 - 7*x**2/2 + 3*x

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