3.1282 \(\int \frac{(1-2 x)^2 (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{125 x^4}{9}-\frac{800 x^3}{81}-\frac{305 x^2}{54}+\frac{1271 x}{243}+\frac{49}{729 (3 x+2)}+\frac{763}{729} \log (3 x+2) \]

[Out]

(1271*x)/243 - (305*x^2)/54 - (800*x^3)/81 + (125*x^4)/9 + 49/(729*(2 + 3*x)) + (763*Log[2 + 3*x])/729

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Rubi [A]  time = 0.0230949, antiderivative size = 48, 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} \[ \frac{125 x^4}{9}-\frac{800 x^3}{81}-\frac{305 x^2}{54}+\frac{1271 x}{243}+\frac{49}{729 (3 x+2)}+\frac{763}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1271*x)/243 - (305*x^2)/54 - (800*x^3)/81 + (125*x^4)/9 + 49/(729*(2 + 3*x)) + (763*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)^2} \, dx &=\int \left (\frac{1271}{243}-\frac{305 x}{27}-\frac{800 x^2}{27}+\frac{500 x^3}{9}-\frac{49}{243 (2+3 x)^2}+\frac{763}{243 (2+3 x)}\right ) \, dx\\ &=\frac{1271 x}{243}-\frac{305 x^2}{54}-\frac{800 x^3}{81}+\frac{125 x^4}{9}+\frac{49}{729 (2+3 x)}+\frac{763}{729} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0285264, size = 49, normalized size = 1.02 \[ \frac{182250 x^5-8100 x^4-160515 x^3+19224 x^2+50052 x+4578 (3 x+2) \log (30 x+20)+3158}{4374 (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3158 + 50052*x + 19224*x^2 - 160515*x^3 - 8100*x^4 + 182250*x^5 + 4578*(2 + 3*x)*Log[20 + 30*x])/(4374*(2 + 3
*x))

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Maple [A]  time = 0.005, size = 37, normalized size = 0.8 \begin{align*}{\frac{1271\,x}{243}}-{\frac{305\,{x}^{2}}{54}}-{\frac{800\,{x}^{3}}{81}}+{\frac{125\,{x}^{4}}{9}}+{\frac{49}{1458+2187\,x}}+{\frac{763\,\ln \left ( 2+3\,x \right ) }{729}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1271/243*x-305/54*x^2-800/81*x^3+125/9*x^4+49/729/(2+3*x)+763/729*ln(2+3*x)

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Maxima [A]  time = 0.995059, size = 49, normalized size = 1.02 \begin{align*} \frac{125}{9} \, x^{4} - \frac{800}{81} \, x^{3} - \frac{305}{54} \, x^{2} + \frac{1271}{243} \, x + \frac{49}{729 \,{\left (3 \, x + 2\right )}} + \frac{763}{729} \, \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)^3/(2+3*x)^2,x, algorithm="maxima")

[Out]

125/9*x^4 - 800/81*x^3 - 305/54*x^2 + 1271/243*x + 49/729/(3*x + 2) + 763/729*log(3*x + 2)

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Fricas [A]  time = 1.52547, size = 147, normalized size = 3.06 \begin{align*} \frac{60750 \, x^{5} - 2700 \, x^{4} - 53505 \, x^{3} + 6408 \, x^{2} + 1526 \,{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 15252 \, x + 98}{1458 \,{\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)^3/(2+3*x)^2,x, algorithm="fricas")

[Out]

1/1458*(60750*x^5 - 2700*x^4 - 53505*x^3 + 6408*x^2 + 1526*(3*x + 2)*log(3*x + 2) + 15252*x + 98)/(3*x + 2)

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Sympy [A]  time = 0.103381, size = 41, normalized size = 0.85 \begin{align*} \frac{125 x^{4}}{9} - \frac{800 x^{3}}{81} - \frac{305 x^{2}}{54} + \frac{1271 x}{243} + \frac{763 \log{\left (3 x + 2 \right )}}{729} + \frac{49}{2187 x + 1458} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*x**4/9 - 800*x**3/81 - 305*x**2/54 + 1271*x/243 + 763*log(3*x + 2)/729 + 49/(2187*x + 1458)

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Giac [A]  time = 2.60323, size = 89, normalized size = 1.85 \begin{align*} -\frac{1}{4374} \,{\left (3 \, x + 2\right )}^{4}{\left (\frac{7600}{3 \, x + 2} - \frac{24855}{{\left (3 \, x + 2\right )}^{2}} + \frac{24594}{{\left (3 \, x + 2\right )}^{3}} - 750\right )} + \frac{49}{729 \,{\left (3 \, x + 2\right )}} - \frac{763}{729} \, \log \left (\frac{{\left | 3 \, x + 2 \right |}}{3 \,{\left (3 \, x + 2\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4374*(3*x + 2)^4*(7600/(3*x + 2) - 24855/(3*x + 2)^2 + 24594/(3*x + 2)^3 - 750) + 49/729/(3*x + 2) - 763/72
9*log(1/3*abs(3*x + 2)/(3*x + 2)^2)