3.1310 \(\int \frac{(1-2 x)^2 (2+3 x)^2}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=41 \[ \frac{12 x^3}{25}-\frac{78 x^2}{125}+\frac{37 x}{625}-\frac{121}{3125 (5 x+3)}+\frac{682 \log (5 x+3)}{3125} \]

[Out]

(37*x)/625 - (78*x^2)/125 + (12*x^3)/25 - 121/(3125*(3 + 5*x)) + (682*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0186623, antiderivative size = 41, 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} \[ \frac{12 x^3}{25}-\frac{78 x^2}{125}+\frac{37 x}{625}-\frac{121}{3125 (5 x+3)}+\frac{682 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(37*x)/625 - (78*x^2)/125 + (12*x^3)/25 - 121/(3125*(3 + 5*x)) + (682*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)^2}{(3+5 x)^2} \, dx &=\int \left (\frac{37}{625}-\frac{156 x}{125}+\frac{36 x^2}{25}+\frac{121}{625 (3+5 x)^2}+\frac{682}{625 (3+5 x)}\right ) \, dx\\ &=\frac{37 x}{625}-\frac{78 x^2}{125}+\frac{12 x^3}{25}-\frac{121}{3125 (3+5 x)}+\frac{682 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0160155, size = 42, normalized size = 1.02 \[ \frac{\frac{5 \left (1500 x^4-1050 x^3-985 x^2+1248 x+658\right )}{5 x+3}+682 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((5*(658 + 1248*x - 985*x^2 - 1050*x^3 + 1500*x^4))/(3 + 5*x) + 682*Log[3 + 5*x])/3125

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \begin{align*}{\frac{37\,x}{625}}-{\frac{78\,{x}^{2}}{125}}+{\frac{12\,{x}^{3}}{25}}-{\frac{121}{9375+15625\,x}}+{\frac{682\,\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)^2/(3+5*x)^2,x)

[Out]

37/625*x-78/125*x^2+12/25*x^3-121/3125/(3+5*x)+682/3125*ln(3+5*x)

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Maxima [A]  time = 1.13228, size = 42, normalized size = 1.02 \begin{align*} \frac{12}{25} \, x^{3} - \frac{78}{125} \, x^{2} + \frac{37}{625} \, x - \frac{121}{3125 \,{\left (5 \, x + 3\right )}} + \frac{682}{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)^2/(3+5*x)^2,x, algorithm="maxima")

[Out]

12/25*x^3 - 78/125*x^2 + 37/625*x - 121/3125/(5*x + 3) + 682/3125*log(5*x + 3)

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Fricas [A]  time = 1.48463, size = 127, normalized size = 3.1 \begin{align*} \frac{7500 \, x^{4} - 5250 \, x^{3} - 4925 \, x^{2} + 682 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 555 \, x - 121}{3125 \,{\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)^2/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/3125*(7500*x^4 - 5250*x^3 - 4925*x^2 + 682*(5*x + 3)*log(5*x + 3) + 555*x - 121)/(5*x + 3)

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Sympy [A]  time = 0.098975, size = 34, normalized size = 0.83 \begin{align*} \frac{12 x^{3}}{25} - \frac{78 x^{2}}{125} + \frac{37 x}{625} + \frac{682 \log{\left (5 x + 3 \right )}}{3125} - \frac{121}{15625 x + 9375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*x**3/25 - 78*x**2/125 + 37*x/625 + 682*log(5*x + 3)/3125 - 121/(15625*x + 9375)

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Giac [A]  time = 2.3531, size = 77, normalized size = 1.88 \begin{align*} -\frac{1}{3125} \,{\left (5 \, x + 3\right )}^{3}{\left (\frac{186}{5 \, x + 3} - \frac{829}{{\left (5 \, x + 3\right )}^{2}} - 12\right )} - \frac{121}{3125 \,{\left (5 \, x + 3\right )}} - \frac{682}{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)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/3125*(5*x + 3)^3*(186/(5*x + 3) - 829/(5*x + 3)^2 - 12) - 121/3125/(5*x + 3) - 682/3125*log(1/5*abs(5*x + 3
)/(5*x + 3)^2)