3.1309 \(\int \frac{(1-2 x)^2 (2+3 x)^3}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{27 x^4}{25}-\frac{36 x^3}{125}-\frac{1449 x^2}{1250}+\frac{2416 x}{3125}-\frac{121}{15625 (5 x+3)}+\frac{209 \log (5 x+3)}{3125} \]

[Out]

(2416*x)/3125 - (1449*x^2)/1250 - (36*x^3)/125 + (27*x^4)/25 - 121/(15625*(3 + 5*x)) + (209*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0220635, 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} \[ \frac{27 x^4}{25}-\frac{36 x^3}{125}-\frac{1449 x^2}{1250}+\frac{2416 x}{3125}-\frac{121}{15625 (5 x+3)}+\frac{209 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2416*x)/3125 - (1449*x^2)/1250 - (36*x^3)/125 + (27*x^4)/25 - 121/(15625*(3 + 5*x)) + (209*Log[3 + 5*x])/3125

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)^2} \, dx &=\int \left (\frac{2416}{3125}-\frac{1449 x}{625}-\frac{108 x^2}{125}+\frac{108 x^3}{25}+\frac{121}{3125 (3+5 x)^2}+\frac{209}{625 (3+5 x)}\right ) \, dx\\ &=\frac{2416 x}{3125}-\frac{1449 x^2}{1250}-\frac{36 x^3}{125}+\frac{27 x^4}{25}-\frac{121}{15625 (3+5 x)}+\frac{209 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0286559, size = 51, normalized size = 1.06 \[ \frac{33750 x^5+11250 x^4-41625 x^3+2425 x^2+35715 x+418 (5 x+3) \log (6 (5 x+3))+12683}{6250 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12683 + 35715*x + 2425*x^2 - 41625*x^3 + 11250*x^4 + 33750*x^5 + 418*(3 + 5*x)*Log[6*(3 + 5*x)])/(6250*(3 + 5
*x))

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Maple [A]  time = 0.006, size = 37, normalized size = 0.8 \begin{align*}{\frac{2416\,x}{3125}}-{\frac{1449\,{x}^{2}}{1250}}-{\frac{36\,{x}^{3}}{125}}+{\frac{27\,{x}^{4}}{25}}-{\frac{121}{46875+78125\,x}}+{\frac{209\,\ln \left ( 3+5\,x \right ) }{3125}} \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)^2,x)

[Out]

2416/3125*x-1449/1250*x^2-36/125*x^3+27/25*x^4-121/15625/(3+5*x)+209/3125*ln(3+5*x)

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Maxima [A]  time = 1.13786, size = 49, normalized size = 1.02 \begin{align*} \frac{27}{25} \, x^{4} - \frac{36}{125} \, x^{3} - \frac{1449}{1250} \, x^{2} + \frac{2416}{3125} \, x - \frac{121}{15625 \,{\left (5 \, x + 3\right )}} + \frac{209}{3125} \, \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)^2,x, algorithm="maxima")

[Out]

27/25*x^4 - 36/125*x^3 - 1449/1250*x^2 + 2416/3125*x - 121/15625/(5*x + 3) + 209/3125*log(5*x + 3)

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Fricas [A]  time = 1.56279, size = 155, normalized size = 3.23 \begin{align*} \frac{168750 \, x^{5} + 56250 \, x^{4} - 208125 \, x^{3} + 12125 \, x^{2} + 2090 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 72480 \, x - 242}{31250 \,{\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)^2,x, algorithm="fricas")

[Out]

1/31250*(168750*x^5 + 56250*x^4 - 208125*x^3 + 12125*x^2 + 2090*(5*x + 3)*log(5*x + 3) + 72480*x - 242)/(5*x +
3)

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Sympy [A]  time = 0.100625, size = 41, normalized size = 0.85 \begin{align*} \frac{27 x^{4}}{25} - \frac{36 x^{3}}{125} - \frac{1449 x^{2}}{1250} + \frac{2416 x}{3125} + \frac{209 \log{\left (5 x + 3 \right )}}{3125} - \frac{121}{78125 x + 46875} \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)**2,x)

[Out]

27*x**4/25 - 36*x**3/125 - 1449*x**2/1250 + 2416*x/3125 + 209*log(5*x + 3)/3125 - 121/(78125*x + 46875)

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Giac [A]  time = 2.86704, size = 89, normalized size = 1.85 \begin{align*} -\frac{1}{31250} \,{\left (5 \, x + 3\right )}^{4}{\left (\frac{720}{5 \, x + 3} - \frac{2115}{{\left (5 \, x + 3\right )}^{2}} - \frac{5750}{{\left (5 \, x + 3\right )}^{3}} - 54\right )} - \frac{121}{15625 \,{\left (5 \, x + 3\right )}} - \frac{209}{3125} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\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)^2,x, algorithm="giac")

[Out]

-1/31250*(5*x + 3)^4*(720/(5*x + 3) - 2115/(5*x + 3)^2 - 5750/(5*x + 3)^3 - 54) - 121/15625/(5*x + 3) - 209/31
25*log(1/5*abs(5*x + 3)/(5*x + 3)^2)