3.1186 \(\int \frac{(1-2 x) (3+5 x)^3}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=45 \[ -\frac{125 x^2}{27}+\frac{175 x}{27}-\frac{107}{243 (3 x+2)}+\frac{7}{486 (3 x+2)^2}-\frac{185}{81} \log (3 x+2) \]

[Out]

(175*x)/27 - (125*x^2)/27 + 7/(486*(2 + 3*x)^2) - 107/(243*(2 + 3*x)) - (185*Log[2 + 3*x])/81

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Rubi [A]  time = 0.0198191, antiderivative size = 45, 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} \[ -\frac{125 x^2}{27}+\frac{175 x}{27}-\frac{107}{243 (3 x+2)}+\frac{7}{486 (3 x+2)^2}-\frac{185}{81} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(175*x)/27 - (125*x^2)/27 + 7/(486*(2 + 3*x)^2) - 107/(243*(2 + 3*x)) - (185*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)^3}{(2+3 x)^3} \, dx &=\int \left (\frac{175}{27}-\frac{250 x}{27}-\frac{7}{81 (2+3 x)^3}+\frac{107}{81 (2+3 x)^2}-\frac{185}{27 (2+3 x)}\right ) \, dx\\ &=\frac{175 x}{27}-\frac{125 x^2}{27}+\frac{7}{486 (2+3 x)^2}-\frac{107}{243 (2+3 x)}-\frac{185}{81} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0109489, size = 46, normalized size = 1.02 \[ \frac{-6750 x^4+450 x^3+18900 x^2+16386 x-370 (3 x+2)^2 \log (3 x+2)+3993}{162 (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3993 + 16386*x + 18900*x^2 + 450*x^3 - 6750*x^4 - 370*(2 + 3*x)^2*Log[2 + 3*x])/(162*(2 + 3*x)^2)

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\frac{175\,x}{27}}-{\frac{125\,{x}^{2}}{27}}+{\frac{7}{486\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{107}{486+729\,x}}-{\frac{185\,\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)^3/(2+3*x)^3,x)

[Out]

175/27*x-125/27*x^2+7/486/(2+3*x)^2-107/243/(2+3*x)-185/81*ln(2+3*x)

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Maxima [A]  time = 1.15465, size = 49, normalized size = 1.09 \begin{align*} -\frac{125}{27} \, x^{2} + \frac{175}{27} \, x - \frac{642 \, x + 421}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{185}{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)^3/(2+3*x)^3,x, algorithm="maxima")

[Out]

-125/27*x^2 + 175/27*x - 1/486*(642*x + 421)/(9*x^2 + 12*x + 4) - 185/81*log(3*x + 2)

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Fricas [A]  time = 1.71187, size = 158, normalized size = 3.51 \begin{align*} -\frac{20250 \, x^{4} - 1350 \, x^{3} - 28800 \, x^{2} + 1110 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 11958 \, x + 421}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/486*(20250*x^4 - 1350*x^3 - 28800*x^2 + 1110*(9*x^2 + 12*x + 4)*log(3*x + 2) - 11958*x + 421)/(9*x^2 + 12*x
+ 4)

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Sympy [A]  time = 0.120531, size = 36, normalized size = 0.8 \begin{align*} - \frac{125 x^{2}}{27} + \frac{175 x}{27} - \frac{642 x + 421}{4374 x^{2} + 5832 x + 1944} - \frac{185 \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)**3/(2+3*x)**3,x)

[Out]

-125*x**2/27 + 175*x/27 - (642*x + 421)/(4374*x**2 + 5832*x + 1944) - 185*log(3*x + 2)/81

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Giac [A]  time = 2.3589, size = 43, normalized size = 0.96 \begin{align*} -\frac{125}{27} \, x^{2} + \frac{175}{27} \, x - \frac{642 \, x + 421}{486 \,{\left (3 \, x + 2\right )}^{2}} - \frac{185}{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)^3/(2+3*x)^3,x, algorithm="giac")

[Out]

-125/27*x^2 + 175/27*x - 1/486*(642*x + 421)/(3*x + 2)^2 - 185/81*log(abs(3*x + 2))