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

Optimal. Leaf size=56 \[ -\frac {4}{243} (3 x+2)^{10}+\frac {428 (3 x+2)^9}{2187}-\frac {259}{324} (3 x+2)^8+\frac {287}{243} (3 x+2)^7-\frac {343 (3 x+2)^6}{1458} \]

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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 {4}{243} (3 x+2)^{10}+\frac {428 (3 x+2)^9}{2187}-\frac {259}{324} (3 x+2)^8+\frac {287}{243} (3 x+2)^7-\frac {343 (3 x+2)^6}{1458} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-343*(2 + 3*x)^6)/1458 + (287*(2 + 3*x)^7)/243 - (259*(2 + 3*x)^8)/324 + (428*(2 + 3*x)^9)/2187 - (4*(2 + 3*x
)^10)/243

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)^5 (3+5 x) \, dx &=\int \left (-\frac {343}{81} (2+3 x)^5+\frac {2009}{81} (2+3 x)^6-\frac {518}{27} (2+3 x)^7+\frac {428}{81} (2+3 x)^8-\frac {40}{81} (2+3 x)^9\right ) \, dx\\ &=-\frac {343 (2+3 x)^6}{1458}+\frac {287}{243} (2+3 x)^7-\frac {259}{324} (2+3 x)^8+\frac {428 (2+3 x)^9}{2187}-\frac {4}{243} (2+3 x)^{10}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 53, normalized size = 0.95 \begin {gather*} -972 x^{10}-2628 x^9-\frac {6291 x^8}{4}+1683 x^7+\frac {4333 x^6}{2}+14 x^5-882 x^4-256 x^3+152 x^2+96 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

96*x + 152*x^2 - 256*x^3 - 882*x^4 + 14*x^5 + (4333*x^6)/2 + 1683*x^7 - (6291*x^8)/4 - 2628*x^9 - 972*x^10

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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)^5 (3+5 x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.90, size = 49, normalized size = 0.88 \begin {gather*} -972 x^{10} - 2628 x^{9} - \frac {6291}{4} x^{8} + 1683 x^{7} + \frac {4333}{2} x^{6} + 14 x^{5} - 882 x^{4} - 256 x^{3} + 152 x^{2} + 96 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-972*x^10 - 2628*x^9 - 6291/4*x^8 + 1683*x^7 + 4333/2*x^6 + 14*x^5 - 882*x^4 - 256*x^3 + 152*x^2 + 96*x

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giac [A]  time = 0.93, size = 49, normalized size = 0.88 \begin {gather*} -972 \, x^{10} - 2628 \, x^{9} - \frac {6291}{4} \, x^{8} + 1683 \, x^{7} + \frac {4333}{2} \, x^{6} + 14 \, x^{5} - 882 \, x^{4} - 256 \, x^{3} + 152 \, x^{2} + 96 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-972*x^10 - 2628*x^9 - 6291/4*x^8 + 1683*x^7 + 4333/2*x^6 + 14*x^5 - 882*x^4 - 256*x^3 + 152*x^2 + 96*x

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maple [A]  time = 0.00, size = 50, normalized size = 0.89 \begin {gather*} -972 x^{10}-2628 x^{9}-\frac {6291}{4} x^{8}+1683 x^{7}+\frac {4333}{2} x^{6}+14 x^{5}-882 x^{4}-256 x^{3}+152 x^{2}+96 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-972*x^10-2628*x^9-6291/4*x^8+1683*x^7+4333/2*x^6+14*x^5-882*x^4-256*x^3+152*x^2+96*x

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maxima [A]  time = 0.45, size = 49, normalized size = 0.88 \begin {gather*} -972 \, x^{10} - 2628 \, x^{9} - \frac {6291}{4} \, x^{8} + 1683 \, x^{7} + \frac {4333}{2} \, x^{6} + 14 \, x^{5} - 882 \, x^{4} - 256 \, x^{3} + 152 \, x^{2} + 96 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-972*x^10 - 2628*x^9 - 6291/4*x^8 + 1683*x^7 + 4333/2*x^6 + 14*x^5 - 882*x^4 - 256*x^3 + 152*x^2 + 96*x

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mupad [B]  time = 0.04, size = 49, normalized size = 0.88 \begin {gather*} -972\,x^{10}-2628\,x^9-\frac {6291\,x^8}{4}+1683\,x^7+\frac {4333\,x^6}{2}+14\,x^5-882\,x^4-256\,x^3+152\,x^2+96\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

96*x + 152*x^2 - 256*x^3 - 882*x^4 + 14*x^5 + (4333*x^6)/2 + 1683*x^7 - (6291*x^8)/4 - 2628*x^9 - 972*x^10

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sympy [A]  time = 0.07, size = 51, normalized size = 0.91 \begin {gather*} - 972 x^{10} - 2628 x^{9} - \frac {6291 x^{8}}{4} + 1683 x^{7} + \frac {4333 x^{6}}{2} + 14 x^{5} - 882 x^{4} - 256 x^{3} + 152 x^{2} + 96 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-972*x**10 - 2628*x**9 - 6291*x**8/4 + 1683*x**7 + 4333*x**6/2 + 14*x**5 - 882*x**4 - 256*x**3 + 152*x**2 + 96
*x

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