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

Optimal. Leaf size=34 \[ \frac {2}{375} (5 x+3)^6-\frac {44}{625} (5 x+3)^5+\frac {121}{500} (5 x+3)^4 \]

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} \frac {2}{375} (5 x+3)^6-\frac {44}{625} (5 x+3)^5+\frac {121}{500} (5 x+3)^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2*(3 + 5*x)^3,x]

[Out]

(121*(3 + 5*x)^4)/500 - (44*(3 + 5*x)^5)/625 + (2*(3 + 5*x)^6)/375

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)^3 \, dx &=\int \left (\frac {121}{25} (3+5 x)^3-\frac {44}{25} (3+5 x)^4+\frac {4}{25} (3+5 x)^5\right ) \, dx\\ &=\frac {121}{500} (3+5 x)^4-\frac {44}{625} (3+5 x)^5+\frac {2}{375} (3+5 x)^6\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 35, normalized size = 1.03 \begin {gather*} \frac {250 x^6}{3}+80 x^5-\frac {235 x^4}{4}-69 x^3+\frac {27 x^2}{2}+27 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2*(3 + 5*x)^3,x]

[Out]

27*x + (27*x^2)/2 - 69*x^3 - (235*x^4)/4 + 80*x^5 + (250*x^6)/3

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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)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.17, size = 29, normalized size = 0.85 \begin {gather*} \frac {250}{3} x^{6} + 80 x^{5} - \frac {235}{4} x^{4} - 69 x^{3} + \frac {27}{2} x^{2} + 27 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250/3*x^6 + 80*x^5 - 235/4*x^4 - 69*x^3 + 27/2*x^2 + 27*x

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giac [A]  time = 0.96, size = 29, normalized size = 0.85 \begin {gather*} \frac {250}{3} \, x^{6} + 80 \, x^{5} - \frac {235}{4} \, x^{4} - 69 \, x^{3} + \frac {27}{2} \, x^{2} + 27 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250/3*x^6 + 80*x^5 - 235/4*x^4 - 69*x^3 + 27/2*x^2 + 27*x

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maple [A]  time = 0.00, size = 30, normalized size = 0.88 \begin {gather*} \frac {250}{3} x^{6}+80 x^{5}-\frac {235}{4} x^{4}-69 x^{3}+\frac {27}{2} x^{2}+27 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250/3*x^6+80*x^5-235/4*x^4-69*x^3+27/2*x^2+27*x

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maxima [A]  time = 0.52, size = 29, normalized size = 0.85 \begin {gather*} \frac {250}{3} \, x^{6} + 80 \, x^{5} - \frac {235}{4} \, x^{4} - 69 \, x^{3} + \frac {27}{2} \, x^{2} + 27 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250/3*x^6 + 80*x^5 - 235/4*x^4 - 69*x^3 + 27/2*x^2 + 27*x

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mupad [B]  time = 0.02, size = 29, normalized size = 0.85 \begin {gather*} \frac {250\,x^6}{3}+80\,x^5-\frac {235\,x^4}{4}-69\,x^3+\frac {27\,x^2}{2}+27\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)^2*(5*x + 3)^3,x)

[Out]

27*x + (27*x^2)/2 - 69*x^3 - (235*x^4)/4 + 80*x^5 + (250*x^6)/3

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sympy [A]  time = 0.07, size = 32, normalized size = 0.94 \begin {gather*} \frac {250 x^{6}}{3} + 80 x^{5} - \frac {235 x^{4}}{4} - 69 x^{3} + \frac {27 x^{2}}{2} + 27 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250*x**6/3 + 80*x**5 - 235*x**4/4 - 69*x**3 + 27*x**2/2 + 27*x

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