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

Optimal. Leaf size=56 \[ -\frac {100 x^2}{81}+\frac {1780 x}{243}-\frac {11599}{729 (3 x+2)}+\frac {1862}{729 (3 x+2)^2}-\frac {343}{2187 (3 x+2)^3}-\frac {8198}{729} \log (3 x+2) \]

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \begin {gather*} -\frac {100 x^2}{81}+\frac {1780 x}{243}-\frac {11599}{729 (3 x+2)}+\frac {1862}{729 (3 x+2)^2}-\frac {343}{2187 (3 x+2)^3}-\frac {8198}{729} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1780*x)/243 - (100*x^2)/81 - 343/(2187*(2 + 3*x)^3) + 1862/(729*(2 + 3*x)^2) - 11599/(729*(2 + 3*x)) - (8198*
Log[2 + 3*x])/729

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx &=\int \left (\frac {1780}{243}-\frac {200 x}{81}+\frac {343}{243 (2+3 x)^4}-\frac {3724}{243 (2+3 x)^3}+\frac {11599}{243 (2+3 x)^2}-\frac {8198}{243 (2+3 x)}\right ) \, dx\\ &=\frac {1780 x}{243}-\frac {100 x^2}{81}-\frac {343}{2187 (2+3 x)^3}+\frac {1862}{729 (2+3 x)^2}-\frac {11599}{729 (2+3 x)}-\frac {8198}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 51, normalized size = 0.91 \begin {gather*} \frac {-72900 x^5+286740 x^4+1088640 x^3+883467 x^2+155034 x-24594 (3 x+2)^3 \log (30 x+20)-33319}{2187 (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-33319 + 155034*x + 883467*x^2 + 1088640*x^3 + 286740*x^4 - 72900*x^5 - 24594*(2 + 3*x)^3*Log[20 + 30*x])/(21
87*(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}{(2+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.27, size = 67, normalized size = 1.20 \begin {gather*} -\frac {72900 \, x^{5} - 286740 \, x^{4} - 767880 \, x^{3} - 241947 \, x^{2} + 24594 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 272646 \, x + 128359}{2187 \, {\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/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/2187*(72900*x^5 - 286740*x^4 - 767880*x^3 - 241947*x^2 + 24594*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) +
272646*x + 128359)/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 0.92, size = 37, normalized size = 0.66 \begin {gather*} -\frac {100}{81} \, x^{2} + \frac {1780}{243} \, x - \frac {7 \, {\left (44739 \, x^{2} + 57258 \, x + 18337\right )}}{2187 \, {\left (3 \, x + 2\right )}^{3}} - \frac {8198}{729} \, \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/(2+3*x)^4,x, algorithm="giac")

[Out]

-100/81*x^2 + 1780/243*x - 7/2187*(44739*x^2 + 57258*x + 18337)/(3*x + 2)^3 - 8198/729*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 45, normalized size = 0.80 \begin {gather*} -\frac {100 x^{2}}{81}+\frac {1780 x}{243}-\frac {8198 \ln \left (3 x +2\right )}{729}-\frac {343}{2187 \left (3 x +2\right )^{3}}+\frac {1862}{729 \left (3 x +2\right )^{2}}-\frac {11599}{729 \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)^2/(3*x+2)^4,x)

[Out]

1780/243*x-100/81*x^2-343/2187/(3*x+2)^3+1862/729/(3*x+2)^2-11599/729/(3*x+2)-8198/729*ln(3*x+2)

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maxima [A]  time = 0.62, size = 46, normalized size = 0.82 \begin {gather*} -\frac {100}{81} \, x^{2} + \frac {1780}{243} \, x - \frac {7 \, {\left (44739 \, x^{2} + 57258 \, x + 18337\right )}}{2187 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {8198}{729} \, \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/(2+3*x)^4,x, algorithm="maxima")

[Out]

-100/81*x^2 + 1780/243*x - 7/2187*(44739*x^2 + 57258*x + 18337)/(27*x^3 + 54*x^2 + 36*x + 8) - 8198/729*log(3*
x + 2)

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mupad [B]  time = 0.04, size = 42, normalized size = 0.75 \begin {gather*} \frac {1780\,x}{243}-\frac {8198\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {\frac {11599\,x^2}{2187}+\frac {44534\,x}{6561}+\frac {128359}{59049}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}}-\frac {100\,x^2}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1780*x)/243 - (8198*log(x + 2/3))/729 - ((44534*x)/6561 + (11599*x^2)/2187 + 128359/59049)/((4*x)/3 + 2*x^2 +
x^3 + 8/27) - (100*x^2)/81

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sympy [A]  time = 0.15, size = 46, normalized size = 0.82 \begin {gather*} - \frac {100 x^{2}}{81} + \frac {1780 x}{243} - \frac {313173 x^{2} + 400806 x + 128359}{59049 x^{3} + 118098 x^{2} + 78732 x + 17496} - \frac {8198 \log {\left (3 x + 2 \right )}}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-100*x**2/81 + 1780*x/243 - (313173*x**2 + 400806*x + 128359)/(59049*x**3 + 118098*x**2 + 78732*x + 17496) - 8
198*log(3*x + 2)/729

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