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

Optimal. Leaf size=60 \[ \frac {500 x}{243}-\frac {8285}{729 (3 x+2)}+\frac {4099}{1458 (3 x+2)^2}-\frac {763}{2187 (3 x+2)^3}+\frac {49}{2916 (3 x+2)^4}-\frac {3800}{729} \log (3 x+2) \]

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Rubi [A]  time = 0.03, antiderivative size = 60, 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 {500 x}{243}-\frac {8285}{729 (3 x+2)}+\frac {4099}{1458 (3 x+2)^2}-\frac {763}{2187 (3 x+2)^3}+\frac {49}{2916 (3 x+2)^4}-\frac {3800}{729} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(500*x)/243 + 49/(2916*(2 + 3*x)^4) - 763/(2187*(2 + 3*x)^3) + 4099/(1458*(2 + 3*x)^2) - 8285/(729*(2 + 3*x))
- (3800*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)^2 (3+5 x)^3}{(2+3 x)^5} \, dx &=\int \left (\frac {500}{243}-\frac {49}{243 (2+3 x)^5}+\frac {763}{243 (2+3 x)^4}-\frac {4099}{243 (2+3 x)^3}+\frac {8285}{243 (2+3 x)^2}-\frac {3800}{243 (2+3 x)}\right ) \, dx\\ &=\frac {500 x}{243}+\frac {49}{2916 (2+3 x)^4}-\frac {763}{2187 (2+3 x)^3}+\frac {4099}{1458 (2+3 x)^2}-\frac {8285}{729 (2+3 x)}-\frac {3800}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 51, normalized size = 0.85 \begin {gather*} \frac {1458000 x^5+4860000 x^4+3795660 x^3-827334 x^2-1853148 x-45600 (3 x+2)^4 \log (30 x+20)-510941}{8748 (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-510941 - 1853148*x - 827334*x^2 + 3795660*x^3 + 4860000*x^4 + 1458000*x^5 - 45600*(2 + 3*x)^4*Log[20 + 30*x]
)/(8748*(2 + 3*x)^4)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.30, size = 77, normalized size = 1.28 \begin {gather*} \frac {1458000 \, x^{5} + 3888000 \, x^{4} + 1203660 \, x^{3} - 3419334 \, x^{2} - 45600 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) - 3005148 \, x - 702941}{8748 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8748*(1458000*x^5 + 3888000*x^4 + 1203660*x^3 - 3419334*x^2 - 45600*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
*log(3*x + 2) - 3005148*x - 702941)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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giac [A]  time = 1.13, size = 59, normalized size = 0.98 \begin {gather*} \frac {500}{243} \, x - \frac {8285}{729 \, {\left (3 \, x + 2\right )}} + \frac {4099}{1458 \, {\left (3 \, x + 2\right )}^{2}} - \frac {763}{2187 \, {\left (3 \, x + 2\right )}^{3}} + \frac {49}{2916 \, {\left (3 \, x + 2\right )}^{4}} + \frac {3800}{729} \, \log \left (\frac {{\left | 3 \, x + 2 \right |}}{3 \, {\left (3 \, x + 2\right )}^{2}}\right ) + \frac {1000}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/243*x - 8285/729/(3*x + 2) + 4099/1458/(3*x + 2)^2 - 763/2187/(3*x + 2)^3 + 49/2916/(3*x + 2)^4 + 3800/729
*log(1/3*abs(3*x + 2)/(3*x + 2)^2) + 1000/729

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maple [A]  time = 0.01, size = 49, normalized size = 0.82 \begin {gather*} \frac {500 x}{243}-\frac {3800 \ln \left (3 x +2\right )}{729}+\frac {49}{2916 \left (3 x +2\right )^{4}}-\frac {763}{2187 \left (3 x +2\right )^{3}}+\frac {4099}{1458 \left (3 x +2\right )^{2}}-\frac {8285}{729 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/243*x+49/2916/(3*x+2)^4-763/2187/(3*x+2)^3+4099/1458/(3*x+2)^2-8285/729/(3*x+2)-3800/729*ln(3*x+2)

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maxima [A]  time = 0.56, size = 51, normalized size = 0.85 \begin {gather*} \frac {500}{243} \, x - \frac {2684340 \, x^{3} + 5147334 \, x^{2} + 3293148 \, x + 702941}{8748 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {3800}{729} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/243*x - 1/8748*(2684340*x^3 + 5147334*x^2 + 3293148*x + 702941)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) -
3800/729*log(3*x + 2)

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mupad [B]  time = 0.04, size = 47, normalized size = 0.78 \begin {gather*} \frac {500\,x}{243}-\frac {3800\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {\frac {8285\,x^3}{2187}+\frac {95321\,x^2}{13122}+\frac {274429\,x}{59049}+\frac {702941}{708588}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(500*x)/243 - (3800*log(x + 2/3))/729 - ((274429*x)/59049 + (95321*x^2)/13122 + (8285*x^3)/2187 + 702941/70858
8)/((32*x)/27 + (8*x^2)/3 + (8*x^3)/3 + x^4 + 16/81)

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sympy [A]  time = 0.16, size = 51, normalized size = 0.85 \begin {gather*} \frac {500 x}{243} + \frac {- 2684340 x^{3} - 5147334 x^{2} - 3293148 x - 702941}{708588 x^{4} + 1889568 x^{3} + 1889568 x^{2} + 839808 x + 139968} - \frac {3800 \log {\left (3 x + 2 \right )}}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500*x/243 + (-2684340*x**3 - 5147334*x**2 - 3293148*x - 702941)/(708588*x**4 + 1889568*x**3 + 1889568*x**2 + 8
39808*x + 139968) - 3800*log(3*x + 2)/729

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