3.13.11 \(\int \frac {(1-2 x)^3 (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=49 \[ -\frac {40 x}{81}+\frac {518}{81 (3 x+2)}-\frac {2009}{486 (3 x+2)^2}+\frac {343}{729 (3 x+2)^3}+\frac {428}{243} \log (3 x+2) \]

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Rubi [A]  time = 0.02, antiderivative size = 49, 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}{81}+\frac {518}{81 (3 x+2)}-\frac {2009}{486 (3 x+2)^2}+\frac {343}{729 (3 x+2)^3}+\frac {428}{243} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-40*x)/81 + 343/(729*(2 + 3*x)^3) - 2009/(486*(2 + 3*x)^2) + 518/(81*(2 + 3*x)) + (428*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)^4} \, dx &=\int \left (-\frac {40}{81}-\frac {343}{81 (2+3 x)^4}+\frac {2009}{81 (2+3 x)^3}-\frac {518}{27 (2+3 x)^2}+\frac {428}{81 (2+3 x)}\right ) \, dx\\ &=-\frac {40 x}{81}+\frac {343}{729 (2+3 x)^3}-\frac {2009}{486 (2+3 x)^2}+\frac {518}{81 (2+3 x)}+\frac {428}{243} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 46, normalized size = 0.94 \begin {gather*} \frac {-19440 x^4-51840 x^3+32076 x^2+70767 x+2568 (3 x+2)^3 \log (3 x+2)+22088}{1458 (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(22088 + 70767*x + 32076*x^2 - 51840*x^3 - 19440*x^4 + 2568*(2 + 3*x)^3*Log[2 + 3*x])/(1458*(2 + 3*x)^3)

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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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.93, size = 62, normalized size = 1.27 \begin {gather*} -\frac {19440 \, x^{4} + 38880 \, x^{3} - 57996 \, x^{2} - 2568 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 88047 \, x - 25928}{1458 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

-1/1458*(19440*x^4 + 38880*x^3 - 57996*x^2 - 2568*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) - 88047*x - 25928)
/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 0.80, size = 32, normalized size = 0.65 \begin {gather*} -\frac {40}{81} \, x + \frac {7 \, {\left (11988 \, x^{2} + 13401 \, x + 3704\right )}}{1458 \, {\left (3 \, x + 2\right )}^{3}} + \frac {428}{243} \, \log \left ({\left | 3 \, x + 2 \right |}\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)^4,x, algorithm="giac")

[Out]

-40/81*x + 7/1458*(11988*x^2 + 13401*x + 3704)/(3*x + 2)^3 + 428/243*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 40, normalized size = 0.82 \begin {gather*} -\frac {40 x}{81}+\frac {428 \ln \left (3 x +2\right )}{243}+\frac {343}{729 \left (3 x +2\right )^{3}}-\frac {2009}{486 \left (3 x +2\right )^{2}}+\frac {518}{81 \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)^4,x)

[Out]

-40/81*x+343/729/(3*x+2)^3-2009/486/(3*x+2)^2+518/81/(3*x+2)+428/243*ln(3*x+2)

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maxima [A]  time = 0.54, size = 41, normalized size = 0.84 \begin {gather*} -\frac {40}{81} \, x + \frac {7 \, {\left (11988 \, x^{2} + 13401 \, x + 3704\right )}}{1458 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {428}{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)^4,x, algorithm="maxima")

[Out]

-40/81*x + 7/1458*(11988*x^2 + 13401*x + 3704)/(27*x^3 + 54*x^2 + 36*x + 8) + 428/243*log(3*x + 2)

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mupad [B]  time = 0.04, size = 36, normalized size = 0.73 \begin {gather*} \frac {428\,\ln \left (x+\frac {2}{3}\right )}{243}-\frac {40\,x}{81}+\frac {\frac {518\,x^2}{243}+\frac {10423\,x}{4374}+\frac {12964}{19683}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{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)^4,x)

[Out]

(428*log(x + 2/3))/243 - (40*x)/81 + ((10423*x)/4374 + (518*x^2)/243 + 12964/19683)/((4*x)/3 + 2*x^2 + x^3 + 8
/27)

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sympy [A]  time = 0.14, size = 41, normalized size = 0.84 \begin {gather*} - \frac {40 x}{81} - \frac {- 83916 x^{2} - 93807 x - 25928}{39366 x^{3} + 78732 x^{2} + 52488 x + 11664} + \frac {428 \log {\left (3 x + 2 \right )}}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40*x/81 - (-83916*x**2 - 93807*x - 25928)/(39366*x**3 + 78732*x**2 + 52488*x + 11664) + 428*log(3*x + 2)/243

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