3.1264 \(\int \frac{(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=45 \[ \frac{50 x^2}{27}-\frac{20 x}{3}+\frac{518}{243 (3 x+2)}-\frac{49}{486 (3 x+2)^2}+\frac{503}{81} \log (3 x+2) \]

[Out]

(-20*x)/3 + (50*x^2)/27 - 49/(486*(2 + 3*x)^2) + 518/(243*(2 + 3*x)) + (503*Log[2 + 3*x])/81

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Rubi [A]  time = 0.0212613, antiderivative size = 45, normalized size of antiderivative = 1., 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} \[ \frac{50 x^2}{27}-\frac{20 x}{3}+\frac{518}{243 (3 x+2)}-\frac{49}{486 (3 x+2)^2}+\frac{503}{81} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-20*x)/3 + (50*x^2)/27 - 49/(486*(2 + 3*x)^2) + 518/(243*(2 + 3*x)) + (503*Log[2 + 3*x])/81

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)^2}{(2+3 x)^3} \, dx &=\int \left (-\frac{20}{3}+\frac{100 x}{27}+\frac{49}{81 (2+3 x)^3}-\frac{518}{81 (2+3 x)^2}+\frac{503}{27 (2+3 x)}\right ) \, dx\\ &=-\frac{20 x}{3}+\frac{50 x^2}{27}-\frac{49}{486 (2+3 x)^2}+\frac{518}{243 (2+3 x)}+\frac{503}{81} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0167365, size = 42, normalized size = 0.93 \[ \frac{503}{81} \log (3 x+2)-\frac{-900 x^4+2040 x^3+6480 x^2+4508 x+913}{54 (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(913 + 4508*x + 6480*x^2 + 2040*x^3 - 900*x^4)/(54*(2 + 3*x)^2) + (503*Log[2 + 3*x])/81

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{20\,x}{3}}+{\frac{50\,{x}^{2}}{27}}-{\frac{49}{486\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{518}{486+729\,x}}+{\frac{503\,\ln \left ( 2+3\,x \right ) }{81}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20/3*x+50/27*x^2-49/486/(2+3*x)^2+518/243/(2+3*x)+503/81*ln(2+3*x)

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Maxima [A]  time = 2.45356, size = 49, normalized size = 1.09 \begin{align*} \frac{50}{27} \, x^{2} - \frac{20}{3} \, x + \frac{7 \,{\left (444 \, x + 289\right )}}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{503}{81} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/27*x^2 - 20/3*x + 7/486*(444*x + 289)/(9*x^2 + 12*x + 4) + 503/81*log(3*x + 2)

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Fricas [A]  time = 1.47322, size = 157, normalized size = 3.49 \begin{align*} \frac{8100 \, x^{4} - 18360 \, x^{3} - 35280 \, x^{2} + 3018 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 9852 \, x + 2023}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/486*(8100*x^4 - 18360*x^3 - 35280*x^2 + 3018*(9*x^2 + 12*x + 4)*log(3*x + 2) - 9852*x + 2023)/(9*x^2 + 12*x
+ 4)

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Sympy [A]  time = 0.115924, size = 36, normalized size = 0.8 \begin{align*} \frac{50 x^{2}}{27} - \frac{20 x}{3} + \frac{3108 x + 2023}{4374 x^{2} + 5832 x + 1944} + \frac{503 \log{\left (3 x + 2 \right )}}{81} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50*x**2/27 - 20*x/3 + (3108*x + 2023)/(4374*x**2 + 5832*x + 1944) + 503*log(3*x + 2)/81

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Giac [A]  time = 2.83239, size = 43, normalized size = 0.96 \begin{align*} \frac{50}{27} \, x^{2} - \frac{20}{3} \, x + \frac{7 \,{\left (444 \, x + 289\right )}}{486 \,{\left (3 \, x + 2\right )}^{2}} + \frac{503}{81} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/27*x^2 - 20/3*x + 7/486*(444*x + 289)/(3*x + 2)^2 + 503/81*log(abs(3*x + 2))