3.1326 \(\int \frac{(1-2 x)^2 (2+3 x)^3}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=52 \[ \frac{36 x^3}{125}-\frac{216 x^2}{625}-\frac{153 x}{3125}-\frac{209}{3125 (5 x+3)}-\frac{121}{31250 (5 x+3)^2}+\frac{23}{125} \log (5 x+3) \]

[Out]

(-153*x)/3125 - (216*x^2)/625 + (36*x^3)/125 - 121/(31250*(3 + 5*x)^2) - 209/(3125*(3 + 5*x)) + (23*Log[3 + 5*
x])/125

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Rubi [A]  time = 0.0238402, antiderivative size = 52, 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} \[ \frac{36 x^3}{125}-\frac{216 x^2}{625}-\frac{153 x}{3125}-\frac{209}{3125 (5 x+3)}-\frac{121}{31250 (5 x+3)^2}+\frac{23}{125} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-153*x)/3125 - (216*x^2)/625 + (36*x^3)/125 - 121/(31250*(3 + 5*x)^2) - 209/(3125*(3 + 5*x)) + (23*Log[3 + 5*
x])/125

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 (2+3 x)^3}{(3+5 x)^3} \, dx &=\int \left (-\frac{153}{3125}-\frac{432 x}{625}+\frac{108 x^2}{125}+\frac{121}{3125 (3+5 x)^3}+\frac{209}{625 (3+5 x)^2}+\frac{23}{25 (3+5 x)}\right ) \, dx\\ &=-\frac{153 x}{3125}-\frac{216 x^2}{625}+\frac{36 x^3}{125}-\frac{121}{31250 (3+5 x)^2}-\frac{209}{3125 (3+5 x)}+\frac{23}{125} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0301417, size = 48, normalized size = 0.92 \[ \frac{45000 x^5-56250 x^3-4050 x^2+24640 x+1150 (5 x+3)^2 \log (6 (5 x+3))+7567}{6250 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7567 + 24640*x - 4050*x^2 - 56250*x^3 + 45000*x^5 + 1150*(3 + 5*x)^2*Log[6*(3 + 5*x)])/(6250*(3 + 5*x)^2)

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Maple [A]  time = 0.005, size = 41, normalized size = 0.8 \begin{align*} -{\frac{153\,x}{3125}}-{\frac{216\,{x}^{2}}{625}}+{\frac{36\,{x}^{3}}{125}}-{\frac{121}{31250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{209}{9375+15625\,x}}+{\frac{23\,\ln \left ( 3+5\,x \right ) }{125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-153/3125*x-216/625*x^2+36/125*x^3-121/31250/(3+5*x)^2-209/3125/(3+5*x)+23/125*ln(3+5*x)

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Maxima [A]  time = 1.12938, size = 55, normalized size = 1.06 \begin{align*} \frac{36}{125} \, x^{3} - \frac{216}{625} \, x^{2} - \frac{153}{3125} \, x - \frac{11 \,{\left (950 \, x + 581\right )}}{31250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{23}{125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

36/125*x^3 - 216/625*x^2 - 153/3125*x - 11/31250*(950*x + 581)/(25*x^2 + 30*x + 9) + 23/125*log(5*x + 3)

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Fricas [A]  time = 1.43366, size = 169, normalized size = 3.25 \begin{align*} \frac{225000 \, x^{5} - 281250 \, x^{3} - 143100 \, x^{2} + 5750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 24220 \, x - 6391}{31250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/31250*(225000*x^5 - 281250*x^3 - 143100*x^2 + 5750*(25*x^2 + 30*x + 9)*log(5*x + 3) - 24220*x - 6391)/(25*x^
2 + 30*x + 9)

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Sympy [A]  time = 0.118578, size = 42, normalized size = 0.81 \begin{align*} \frac{36 x^{3}}{125} - \frac{216 x^{2}}{625} - \frac{153 x}{3125} - \frac{10450 x + 6391}{781250 x^{2} + 937500 x + 281250} + \frac{23 \log{\left (5 x + 3 \right )}}{125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

36*x**3/125 - 216*x**2/625 - 153*x/3125 - (10450*x + 6391)/(781250*x**2 + 937500*x + 281250) + 23*log(5*x + 3)
/125

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Giac [A]  time = 2.31745, size = 50, normalized size = 0.96 \begin{align*} \frac{36}{125} \, x^{3} - \frac{216}{625} \, x^{2} - \frac{153}{3125} \, x - \frac{11 \,{\left (950 \, x + 581\right )}}{31250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{23}{125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

36/125*x^3 - 216/625*x^2 - 153/3125*x - 11/31250*(950*x + 581)/(5*x + 3)^2 + 23/125*log(abs(5*x + 3))