3.13.9 \(\int \frac {(1-2 x)^3 (3+5 x)}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=41 \[ -\frac {40 x^3}{27}+\frac {134 x^2}{27}-\frac {286 x}{27}+\frac {343}{243 (3 x+2)}+\frac {2009}{243} \log (3 x+2) \]

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Rubi [A]  time = 0.02, antiderivative size = 41, 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 x^3}{27}+\frac {134 x^2}{27}-\frac {286 x}{27}+\frac {343}{243 (3 x+2)}+\frac {2009}{243} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-286*x)/27 + (134*x^2)/27 - (40*x^3)/27 + 343/(243*(2 + 3*x)) + (2009*Log[2 + 3*x])/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 \frac {(1-2 x)^3 (3+5 x)}{(2+3 x)^2} \, dx &=\int \left (-\frac {286}{27}+\frac {268 x}{27}-\frac {40 x^2}{9}-\frac {343}{81 (2+3 x)^2}+\frac {2009}{81 (2+3 x)}\right ) \, dx\\ &=-\frac {286 x}{27}+\frac {134 x^2}{27}-\frac {40 x^3}{27}+\frac {343}{243 (2+3 x)}+\frac {2009}{243} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 1.07 \begin {gather*} \frac {-2160 x^4+5796 x^3-10620 x^2-4113 x+4018 (3 x+2) \log (6 x+4)+4808}{486 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4808 - 4113*x - 10620*x^2 + 5796*x^3 - 2160*x^4 + 4018*(2 + 3*x)*Log[4 + 6*x])/(486*(2 + 3*x))

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.13, size = 42, normalized size = 1.02 \begin {gather*} -\frac {1080 \, x^{4} - 2898 \, x^{3} + 5310 \, x^{2} - 2009 \, {\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 5148 \, x - 343}{243 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/243*(1080*x^4 - 2898*x^3 + 5310*x^2 - 2009*(3*x + 2)*log(3*x + 2) + 5148*x - 343)/(3*x + 2)

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giac [A]  time = 0.95, size = 57, normalized size = 1.39 \begin {gather*} \frac {2}{729} \, {\left (3 \, x + 2\right )}^{3} {\left (\frac {321}{3 \, x + 2} - \frac {2331}{{\left (3 \, x + 2\right )}^{2}} - 20\right )} + \frac {343}{243 \, {\left (3 \, x + 2\right )}} - \frac {2009}{243} \, \log \left (\frac {{\left | 3 \, x + 2 \right |}}{3 \, {\left (3 \, x + 2\right )}^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/729*(3*x + 2)^3*(321/(3*x + 2) - 2331/(3*x + 2)^2 - 20) + 343/243/(3*x + 2) - 2009/243*log(1/3*abs(3*x + 2)/
(3*x + 2)^2)

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maple [A]  time = 0.01, size = 32, normalized size = 0.78 \begin {gather*} -\frac {40 x^{3}}{27}+\frac {134 x^{2}}{27}-\frac {286 x}{27}+\frac {2009 \ln \left (3 x +2\right )}{243}+\frac {343}{243 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-286/27*x+134/27*x^2-40/27*x^3+343/243/(3*x+2)+2009/243*ln(3*x+2)

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maxima [A]  time = 0.55, size = 31, normalized size = 0.76 \begin {gather*} -\frac {40}{27} \, x^{3} + \frac {134}{27} \, x^{2} - \frac {286}{27} \, x + \frac {343}{243 \, {\left (3 \, x + 2\right )}} + \frac {2009}{243} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40/27*x^3 + 134/27*x^2 - 286/27*x + 343/243/(3*x + 2) + 2009/243*log(3*x + 2)

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mupad [B]  time = 0.03, size = 29, normalized size = 0.71 \begin {gather*} \frac {2009\,\ln \left (x+\frac {2}{3}\right )}{243}-\frac {286\,x}{27}+\frac {343}{729\,\left (x+\frac {2}{3}\right )}+\frac {134\,x^2}{27}-\frac {40\,x^3}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2009*log(x + 2/3))/243 - (286*x)/27 + 343/(729*(x + 2/3)) + (134*x^2)/27 - (40*x^3)/27

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sympy [A]  time = 0.11, size = 34, normalized size = 0.83 \begin {gather*} - \frac {40 x^{3}}{27} + \frac {134 x^{2}}{27} - \frac {286 x}{27} + \frac {2009 \log {\left (3 x + 2 \right )}}{243} + \frac {343}{729 x + 486} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40*x**3/27 + 134*x**2/27 - 286*x/27 + 2009*log(3*x + 2)/243 + 343/(729*x + 486)

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