3.1366 \(\int \frac{(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=52 \[ -\frac{200 x^3}{81}+\frac{230 x^2}{27}-\frac{1546 x}{81}+\frac{3724}{729 (3 x+2)}-\frac{343}{1458 (3 x+2)^2}+\frac{11599}{729} \log (3 x+2) \]

[Out]

(-1546*x)/81 + (230*x^2)/27 - (200*x^3)/81 - 343/(1458*(2 + 3*x)^2) + 3724/(729*(2 + 3*x)) + (11599*Log[2 + 3*
x])/729

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Rubi [A]  time = 0.0228997, antiderivative size = 52, 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} \[ -\frac{200 x^3}{81}+\frac{230 x^2}{27}-\frac{1546 x}{81}+\frac{3724}{729 (3 x+2)}-\frac{343}{1458 (3 x+2)^2}+\frac{11599}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1546*x)/81 + (230*x^2)/27 - (200*x^3)/81 - 343/(1458*(2 + 3*x)^2) + 3724/(729*(2 + 3*x)) + (11599*Log[2 + 3*
x])/729

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)^3 (3+5 x)^2}{(2+3 x)^3} \, dx &=\int \left (-\frac{1546}{81}+\frac{460 x}{27}-\frac{200 x^2}{27}+\frac{343}{243 (2+3 x)^3}-\frac{3724}{243 (2+3 x)^2}+\frac{11599}{243 (2+3 x)}\right ) \, dx\\ &=-\frac{1546 x}{81}+\frac{230 x^2}{27}-\frac{200 x^3}{81}-\frac{343}{1458 (2+3 x)^2}+\frac{3724}{729 (2+3 x)}+\frac{11599}{729} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.032216, size = 51, normalized size = 0.98 \[ -\frac{97200 x^5-205740 x^4+347436 x^3+1531512 x^2+1171896 x-69594 (3 x+2)^2 \log (30 x+20)+258005}{4374 (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(258005 + 1171896*x + 1531512*x^2 + 347436*x^3 - 205740*x^4 + 97200*x^5 - 69594*(2 + 3*x)^2*Log[20 + 30*x])/(
4374*(2 + 3*x)^2)

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Maple [A]  time = 0.005, size = 41, normalized size = 0.8 \begin{align*} -{\frac{1546\,x}{81}}+{\frac{230\,{x}^{2}}{27}}-{\frac{200\,{x}^{3}}{81}}-{\frac{343}{1458\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{3724}{1458+2187\,x}}+{\frac{11599\,\ln \left ( 2+3\,x \right ) }{729}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1546/81*x+230/27*x^2-200/81*x^3-343/1458/(2+3*x)^2+3724/729/(2+3*x)+11599/729*ln(2+3*x)

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Maxima [A]  time = 2.44276, size = 55, normalized size = 1.06 \begin{align*} -\frac{200}{81} \, x^{3} + \frac{230}{27} \, x^{2} - \frac{1546}{81} \, x + \frac{49 \,{\left (152 \, x + 99\right )}}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{11599}{729} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200/81*x^3 + 230/27*x^2 - 1546/81*x + 49/486*(152*x + 99)/(9*x^2 + 12*x + 4) + 11599/729*log(3*x + 2)

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Fricas [A]  time = 1.23852, size = 184, normalized size = 3.54 \begin{align*} -\frac{32400 \, x^{5} - 68580 \, x^{4} + 115812 \, x^{3} + 284256 \, x^{2} - 23198 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 88968 \, x - 14553}{1458 \,{\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*(3+5*x)^2/(2+3*x)^3,x, algorithm="fricas")

[Out]

-1/1458*(32400*x^5 - 68580*x^4 + 115812*x^3 + 284256*x^2 - 23198*(9*x^2 + 12*x + 4)*log(3*x + 2) + 88968*x - 1
4553)/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 0.123793, size = 42, normalized size = 0.81 \begin{align*} - \frac{200 x^{3}}{81} + \frac{230 x^{2}}{27} - \frac{1546 x}{81} + \frac{7448 x + 4851}{4374 x^{2} + 5832 x + 1944} + \frac{11599 \log{\left (3 x + 2 \right )}}{729} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200*x**3/81 + 230*x**2/27 - 1546*x/81 + (7448*x + 4851)/(4374*x**2 + 5832*x + 1944) + 11599*log(3*x + 2)/729

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Giac [A]  time = 3.13657, size = 50, normalized size = 0.96 \begin{align*} -\frac{200}{81} \, x^{3} + \frac{230}{27} \, x^{2} - \frac{1546}{81} \, x + \frac{49 \,{\left (152 \, x + 99\right )}}{486 \,{\left (3 \, x + 2\right )}^{2}} + \frac{11599}{729} \, \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*(3+5*x)^2/(2+3*x)^3,x, algorithm="giac")

[Out]

-200/81*x^3 + 230/27*x^2 - 1546/81*x + 49/486*(152*x + 99)/(3*x + 2)^2 + 11599/729*log(abs(3*x + 2))