3.1248 \(\int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=44 \[ \frac{16}{9 (3 x+2)}-\frac{91}{54 (3 x+2)^2}+\frac{49}{243 (3 x+2)^3}+\frac{20}{81} \log (3 x+2) \]

[Out]

49/(243*(2 + 3*x)^3) - 91/(54*(2 + 3*x)^2) + 16/(9*(2 + 3*x)) + (20*Log[2 + 3*x])/81

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Rubi [A]  time = 0.0170495, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{16}{9 (3 x+2)}-\frac{91}{54 (3 x+2)^2}+\frac{49}{243 (3 x+2)^3}+\frac{20}{81} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

49/(243*(2 + 3*x)^3) - 91/(54*(2 + 3*x)^2) + 16/(9*(2 + 3*x)) + (20*Log[2 + 3*x])/81

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^4} \, dx &=\int \left (-\frac{49}{27 (2+3 x)^4}+\frac{91}{9 (2+3 x)^3}-\frac{16}{3 (2+3 x)^2}+\frac{20}{27 (2+3 x)}\right ) \, dx\\ &=\frac{49}{243 (2+3 x)^3}-\frac{91}{54 (2+3 x)^2}+\frac{16}{9 (2+3 x)}+\frac{20}{81} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0132563, size = 36, normalized size = 0.82 \[ \frac{7776 x^2+7911 x+120 (3 x+2)^3 \log (3 x+2)+1916}{486 (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

(1916 + 7911*x + 7776*x^2 + 120*(2 + 3*x)^3*Log[2 + 3*x])/(486*(2 + 3*x)^3)

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Maple [A]  time = 0.005, size = 37, normalized size = 0.8 \begin{align*}{\frac{49}{243\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{91}{54\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{16}{18+27\,x}}+{\frac{20\,\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+3*x)^4,x)

[Out]

49/243/(2+3*x)^3-91/54/(2+3*x)^2+16/9/(2+3*x)+20/81*ln(2+3*x)

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Maxima [A]  time = 2.18492, size = 51, normalized size = 1.16 \begin{align*} \frac{7776 \, x^{2} + 7911 \, x + 1916}{486 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{20}{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+3*x)^4,x, algorithm="maxima")

[Out]

1/486*(7776*x^2 + 7911*x + 1916)/(27*x^3 + 54*x^2 + 36*x + 8) + 20/81*log(3*x + 2)

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Fricas [A]  time = 1.54492, size = 150, normalized size = 3.41 \begin{align*} \frac{7776 \, x^{2} + 120 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 7911 \, x + 1916}{486 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/486*(7776*x^2 + 120*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) + 7911*x + 1916)/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]  time = 0.123864, size = 34, normalized size = 0.77 \begin{align*} \frac{7776 x^{2} + 7911 x + 1916}{13122 x^{3} + 26244 x^{2} + 17496 x + 3888} + \frac{20 \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+3*x)**4,x)

[Out]

(7776*x**2 + 7911*x + 1916)/(13122*x**3 + 26244*x**2 + 17496*x + 3888) + 20*log(3*x + 2)/81

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Giac [A]  time = 2.50807, size = 39, normalized size = 0.89 \begin{align*} \frac{7776 \, x^{2} + 7911 \, x + 1916}{486 \,{\left (3 \, x + 2\right )}^{3}} + \frac{20}{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+3*x)^4,x, algorithm="giac")

[Out]

1/486*(7776*x^2 + 7911*x + 1916)/(3*x + 2)^3 + 20/81*log(abs(3*x + 2))