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

Optimal. Leaf size=34 \[ -\frac {5}{81} (3 x+2)^6+\frac {37}{135} (3 x+2)^5-\frac {7}{108} (3 x+2)^4 \]

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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \begin {gather*} -\frac {5}{81} (3 x+2)^6+\frac {37}{135} (3 x+2)^5-\frac {7}{108} (3 x+2)^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 + 3*x)^4)/108 + (37*(2 + 3*x)^5)/135 - (5*(2 + 3*x)^6)/81

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx &=\int \left (-\frac {7}{9} (2+3 x)^3+\frac {37}{9} (2+3 x)^4-\frac {10}{9} (2+3 x)^5\right ) \, dx\\ &=-\frac {7}{108} (2+3 x)^4+\frac {37}{135} (2+3 x)^5-\frac {5}{81} (2+3 x)^6\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 35, normalized size = 1.03 \begin {gather*} -45 x^6-\frac {567 x^5}{5}-\frac {333 x^4}{4}+\frac {46 x^3}{3}+50 x^2+24 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

24*x + 50*x^2 + (46*x^3)/3 - (333*x^4)/4 - (567*x^5)/5 - 45*x^6

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.20, size = 29, normalized size = 0.85 \begin {gather*} -45 x^{6} - \frac {567}{5} x^{5} - \frac {333}{4} x^{4} + \frac {46}{3} x^{3} + 50 x^{2} + 24 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45*x^6 - 567/5*x^5 - 333/4*x^4 + 46/3*x^3 + 50*x^2 + 24*x

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giac [A]  time = 1.19, size = 29, normalized size = 0.85 \begin {gather*} -45 \, x^{6} - \frac {567}{5} \, x^{5} - \frac {333}{4} \, x^{4} + \frac {46}{3} \, x^{3} + 50 \, x^{2} + 24 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45*x^6 - 567/5*x^5 - 333/4*x^4 + 46/3*x^3 + 50*x^2 + 24*x

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maple [A]  time = 0.00, size = 30, normalized size = 0.88 \begin {gather*} -45 x^{6}-\frac {567}{5} x^{5}-\frac {333}{4} x^{4}+\frac {46}{3} x^{3}+50 x^{2}+24 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45*x^6-567/5*x^5-333/4*x^4+46/3*x^3+50*x^2+24*x

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maxima [A]  time = 0.59, size = 29, normalized size = 0.85 \begin {gather*} -45 \, x^{6} - \frac {567}{5} \, x^{5} - \frac {333}{4} \, x^{4} + \frac {46}{3} \, x^{3} + 50 \, x^{2} + 24 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45*x^6 - 567/5*x^5 - 333/4*x^4 + 46/3*x^3 + 50*x^2 + 24*x

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mupad [B]  time = 0.02, size = 29, normalized size = 0.85 \begin {gather*} -45\,x^6-\frac {567\,x^5}{5}-\frac {333\,x^4}{4}+\frac {46\,x^3}{3}+50\,x^2+24\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

24*x + 50*x^2 + (46*x^3)/3 - (333*x^4)/4 - (567*x^5)/5 - 45*x^6

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sympy [A]  time = 0.07, size = 32, normalized size = 0.94 \begin {gather*} - 45 x^{6} - \frac {567 x^{5}}{5} - \frac {333 x^{4}}{4} + \frac {46 x^{3}}{3} + 50 x^{2} + 24 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45*x**6 - 567*x**5/5 - 333*x**4/4 + 46*x**3/3 + 50*x**2 + 24*x

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