3.13.3 \(\int (1-2 x)^3 (2+3 x)^4 (3+5 x) \, dx\)

Optimal. Leaf size=56 \[ -\frac {40 (3 x+2)^9}{2187}+\frac {107}{486} (3 x+2)^8-\frac {74}{81} (3 x+2)^7+\frac {2009 (3 x+2)^6}{1458}-\frac {343 (3 x+2)^5}{1215} \]

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\frac {40 (3 x+2)^9}{2187}+\frac {107}{486} (3 x+2)^8-\frac {74}{81} (3 x+2)^7+\frac {2009 (3 x+2)^6}{1458}-\frac {343 (3 x+2)^5}{1215} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

(-343*(2 + 3*x)^5)/1215 + (2009*(2 + 3*x)^6)/1458 - (74*(2 + 3*x)^7)/81 + (107*(2 + 3*x)^8)/486 - (40*(2 + 3*x
)^9)/2187

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)^3 (2+3 x)^4 (3+5 x) \, dx &=\int \left (-\frac {343}{81} (2+3 x)^4+\frac {2009}{81} (2+3 x)^5-\frac {518}{27} (2+3 x)^6+\frac {428}{81} (2+3 x)^7-\frac {40}{81} (2+3 x)^8\right ) \, dx\\ &=-\frac {343 (2+3 x)^5}{1215}+\frac {2009 (2+3 x)^6}{1458}-\frac {74}{81} (2+3 x)^7+\frac {107}{486} (2+3 x)^8-\frac {40 (2+3 x)^9}{2187}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 50, normalized size = 0.89 \begin {gather*} -360 x^9-\frac {1431 x^8}{2}-54 x^7+\frac {1393 x^6}{2}+\frac {1547 x^5}{5}-252 x^4-168 x^3+40 x^2+48 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

48*x + 40*x^2 - 168*x^3 - 252*x^4 + (1547*x^5)/5 + (1393*x^6)/2 - 54*x^7 - (1431*x^8)/2 - 360*x^9

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (1-2 x)^3 (2+3 x)^4 (3+5 x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x), x]

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fricas [A]  time = 1.15, size = 44, normalized size = 0.79 \begin {gather*} -360 x^{9} - \frac {1431}{2} x^{8} - 54 x^{7} + \frac {1393}{2} x^{6} + \frac {1547}{5} x^{5} - 252 x^{4} - 168 x^{3} + 40 x^{2} + 48 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^4*(3+5*x),x, algorithm="fricas")

[Out]

-360*x^9 - 1431/2*x^8 - 54*x^7 + 1393/2*x^6 + 1547/5*x^5 - 252*x^4 - 168*x^3 + 40*x^2 + 48*x

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giac [A]  time = 0.89, size = 44, normalized size = 0.79 \begin {gather*} -360 \, x^{9} - \frac {1431}{2} \, x^{8} - 54 \, x^{7} + \frac {1393}{2} \, x^{6} + \frac {1547}{5} \, x^{5} - 252 \, x^{4} - 168 \, x^{3} + 40 \, x^{2} + 48 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^4*(3+5*x),x, algorithm="giac")

[Out]

-360*x^9 - 1431/2*x^8 - 54*x^7 + 1393/2*x^6 + 1547/5*x^5 - 252*x^4 - 168*x^3 + 40*x^2 + 48*x

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maple [A]  time = 0.00, size = 45, normalized size = 0.80 \begin {gather*} -360 x^{9}-\frac {1431}{2} x^{8}-54 x^{7}+\frac {1393}{2} x^{6}+\frac {1547}{5} x^{5}-252 x^{4}-168 x^{3}+40 x^{2}+48 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(3*x+2)^4*(5*x+3),x)

[Out]

-360*x^9-1431/2*x^8-54*x^7+1393/2*x^6+1547/5*x^5-252*x^4-168*x^3+40*x^2+48*x

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maxima [A]  time = 0.55, size = 44, normalized size = 0.79 \begin {gather*} -360 \, x^{9} - \frac {1431}{2} \, x^{8} - 54 \, x^{7} + \frac {1393}{2} \, x^{6} + \frac {1547}{5} \, x^{5} - 252 \, x^{4} - 168 \, x^{3} + 40 \, x^{2} + 48 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^4*(3+5*x),x, algorithm="maxima")

[Out]

-360*x^9 - 1431/2*x^8 - 54*x^7 + 1393/2*x^6 + 1547/5*x^5 - 252*x^4 - 168*x^3 + 40*x^2 + 48*x

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mupad [B]  time = 0.03, size = 44, normalized size = 0.79 \begin {gather*} -360\,x^9-\frac {1431\,x^8}{2}-54\,x^7+\frac {1393\,x^6}{2}+\frac {1547\,x^5}{5}-252\,x^4-168\,x^3+40\,x^2+48\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)^3*(3*x + 2)^4*(5*x + 3),x)

[Out]

48*x + 40*x^2 - 168*x^3 - 252*x^4 + (1547*x^5)/5 + (1393*x^6)/2 - 54*x^7 - (1431*x^8)/2 - 360*x^9

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sympy [A]  time = 0.07, size = 48, normalized size = 0.86 \begin {gather*} - 360 x^{9} - \frac {1431 x^{8}}{2} - 54 x^{7} + \frac {1393 x^{6}}{2} + \frac {1547 x^{5}}{5} - 252 x^{4} - 168 x^{3} + 40 x^{2} + 48 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**4*(3+5*x),x)

[Out]

-360*x**9 - 1431*x**8/2 - 54*x**7 + 1393*x**6/2 + 1547*x**5/5 - 252*x**4 - 168*x**3 + 40*x**2 + 48*x

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