3.1170 \(\int \frac{(1-2 x) (3+5 x)^2}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=44 \[ -\frac{65}{27 (3 x+2)}+\frac{4}{9 (3 x+2)^2}-\frac{7}{243 (3 x+2)^3}-\frac{50}{81} \log (3 x+2) \]

[Out]

-7/(243*(2 + 3*x)^3) + 4/(9*(2 + 3*x)^2) - 65/(27*(2 + 3*x)) - (50*Log[2 + 3*x])/81

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Rubi [A]  time = 0.0174397, 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} \[ -\frac{65}{27 (3 x+2)}+\frac{4}{9 (3 x+2)^2}-\frac{7}{243 (3 x+2)^3}-\frac{50}{81} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(243*(2 + 3*x)^3) + 4/(9*(2 + 3*x)^2) - 65/(27*(2 + 3*x)) - (50*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) (3+5 x)^2}{(2+3 x)^4} \, dx &=\int \left (\frac{7}{27 (2+3 x)^4}-\frac{8}{3 (2+3 x)^3}+\frac{65}{9 (2+3 x)^2}-\frac{50}{27 (2+3 x)}\right ) \, dx\\ &=-\frac{7}{243 (2+3 x)^3}+\frac{4}{9 (2+3 x)^2}-\frac{65}{27 (2+3 x)}-\frac{50}{81} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0129345, size = 36, normalized size = 0.82 \[ -\frac{5265 x^2+6696 x+150 (3 x+2)^3 \log (3 x+2)+2131}{243 (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2131 + 6696*x + 5265*x^2 + 150*(2 + 3*x)^3*Log[2 + 3*x])/(243*(2 + 3*x)^3)

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Maple [A]  time = 0.006, size = 37, normalized size = 0.8 \begin{align*} -{\frac{7}{243\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{4}{9\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{65}{54+81\,x}}-{\frac{50\,\ln \left ( 2+3\,x \right ) }{81}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3+5*x)^2/(2+3*x)^4,x)

[Out]

-7/243/(2+3*x)^3+4/9/(2+3*x)^2-65/27/(2+3*x)-50/81*ln(2+3*x)

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Maxima [A]  time = 1.14627, size = 51, normalized size = 1.16 \begin{align*} -\frac{5265 \, x^{2} + 6696 \, x + 2131}{243 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac{50}{81} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/243*(5265*x^2 + 6696*x + 2131)/(27*x^3 + 54*x^2 + 36*x + 8) - 50/81*log(3*x + 2)

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Fricas [A]  time = 1.69936, size = 151, normalized size = 3.43 \begin{align*} -\frac{5265 \, x^{2} + 150 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 6696 \, x + 2131}{243 \,{\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)*(3+5*x)^2/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/243*(5265*x^2 + 150*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) + 6696*x + 2131)/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]  time = 0.127129, size = 36, normalized size = 0.82 \begin{align*} - \frac{5265 x^{2} + 6696 x + 2131}{6561 x^{3} + 13122 x^{2} + 8748 x + 1944} - \frac{50 \log{\left (3 x + 2 \right )}}{81} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)**2/(2+3*x)**4,x)

[Out]

-(5265*x**2 + 6696*x + 2131)/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 50*log(3*x + 2)/81

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Giac [A]  time = 1.88928, size = 39, normalized size = 0.89 \begin{align*} -\frac{5265 \, x^{2} + 6696 \, x + 2131}{243 \,{\left (3 \, x + 2\right )}^{3}} - \frac{50}{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)*(3+5*x)^2/(2+3*x)^4,x, algorithm="giac")

[Out]

-1/243*(5265*x^2 + 6696*x + 2131)/(3*x + 2)^3 - 50/81*log(abs(3*x + 2))