3.1365 \(\int \frac{(1-2 x)^3 (3+5 x)^2}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac{50 x^4}{9}+\frac{980 x^3}{81}-\frac{313 x^2}{27}+\frac{2323 x}{243}-\frac{343}{729 (3 x+2)}-\frac{3724}{729} \log (3 x+2) \]

[Out]

(2323*x)/243 - (313*x^2)/27 + (980*x^3)/81 - (50*x^4)/9 - 343/(729*(2 + 3*x)) - (3724*Log[2 + 3*x])/729

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Rubi [A]  time = 0.0209315, antiderivative size = 48, 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} \[ -\frac{50 x^4}{9}+\frac{980 x^3}{81}-\frac{313 x^2}{27}+\frac{2323 x}{243}-\frac{343}{729 (3 x+2)}-\frac{3724}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2323*x)/243 - (313*x^2)/27 + (980*x^3)/81 - (50*x^4)/9 - 343/(729*(2 + 3*x)) - (3724*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)^2} \, dx &=\int \left (\frac{2323}{243}-\frac{626 x}{27}+\frac{980 x^2}{27}-\frac{200 x^3}{9}+\frac{343}{243 (2+3 x)^2}-\frac{3724}{243 (2+3 x)}\right ) \, dx\\ &=\frac{2323 x}{243}-\frac{313 x^2}{27}+\frac{980 x^3}{81}-\frac{50 x^4}{9}-\frac{343}{729 (2+3 x)}-\frac{3724}{729} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0305086, size = 49, normalized size = 1.02 \[ \frac{-36450 x^5+55080 x^4-23139 x^3+12015 x^2+148152 x-11172 (3 x+2) \log (30 x+20)+69863}{2187 (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(69863 + 148152*x + 12015*x^2 - 23139*x^3 + 55080*x^4 - 36450*x^5 - 11172*(2 + 3*x)*Log[20 + 30*x])/(2187*(2 +
3*x))

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Maple [A]  time = 0.006, size = 37, normalized size = 0.8 \begin{align*}{\frac{2323\,x}{243}}-{\frac{313\,{x}^{2}}{27}}+{\frac{980\,{x}^{3}}{81}}-{\frac{50\,{x}^{4}}{9}}-{\frac{343}{1458+2187\,x}}-{\frac{3724\,\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)^2,x)

[Out]

2323/243*x-313/27*x^2+980/81*x^3-50/9*x^4-343/729/(2+3*x)-3724/729*ln(2+3*x)

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Maxima [A]  time = 1.02432, size = 49, normalized size = 1.02 \begin{align*} -\frac{50}{9} \, x^{4} + \frac{980}{81} \, x^{3} - \frac{313}{27} \, x^{2} + \frac{2323}{243} \, x - \frac{343}{729 \,{\left (3 \, x + 2\right )}} - \frac{3724}{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)^2,x, algorithm="maxima")

[Out]

-50/9*x^4 + 980/81*x^3 - 313/27*x^2 + 2323/243*x - 343/729/(3*x + 2) - 3724/729*log(3*x + 2)

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Fricas [A]  time = 1.32008, size = 149, normalized size = 3.1 \begin{align*} -\frac{12150 \, x^{5} - 18360 \, x^{4} + 7713 \, x^{3} - 4005 \, x^{2} + 3724 \,{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 13938 \, x + 343}{729 \,{\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)^2,x, algorithm="fricas")

[Out]

-1/729*(12150*x^5 - 18360*x^4 + 7713*x^3 - 4005*x^2 + 3724*(3*x + 2)*log(3*x + 2) - 13938*x + 343)/(3*x + 2)

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Sympy [A]  time = 0.106292, size = 41, normalized size = 0.85 \begin{align*} - \frac{50 x^{4}}{9} + \frac{980 x^{3}}{81} - \frac{313 x^{2}}{27} + \frac{2323 x}{243} - \frac{3724 \log{\left (3 x + 2 \right )}}{729} - \frac{343}{2187 x + 1458} \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)**2,x)

[Out]

-50*x**4/9 + 980*x**3/81 - 313*x**2/27 + 2323*x/243 - 3724*log(3*x + 2)/729 - 343/(2187*x + 1458)

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Giac [A]  time = 2.51673, size = 89, normalized size = 1.85 \begin{align*} \frac{1}{2187} \,{\left (3 \, x + 2\right )}^{4}{\left (\frac{2180}{3 \, x + 2} - \frac{12297}{{\left (3 \, x + 2\right )}^{2}} + \frac{34797}{{\left (3 \, x + 2\right )}^{3}} - 150\right )} - \frac{343}{729 \,{\left (3 \, x + 2\right )}} + \frac{3724}{729} \, \log \left (\frac{{\left | 3 \, x + 2 \right |}}{3 \,{\left (3 \, x + 2\right )}^{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)^2,x, algorithm="giac")

[Out]

1/2187*(3*x + 2)^4*(2180/(3*x + 2) - 12297/(3*x + 2)^2 + 34797/(3*x + 2)^3 - 150) - 343/729/(3*x + 2) + 3724/7
29*log(1/3*abs(3*x + 2)/(3*x + 2)^2)