3.1381 \(\int \frac{(1-2 x)^3 (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=55 \[ -\frac{200 x^5}{9}+\frac{775 x^4}{27}-\frac{190 x^3}{81}-\frac{5287 x^2}{486}+\frac{2287 x}{729}+\frac{343}{2187 (3 x+2)}+\frac{1813}{729} \log (3 x+2) \]

[Out]

(2287*x)/729 - (5287*x^2)/486 - (190*x^3)/81 + (775*x^4)/27 - (200*x^5)/9 + 343/(2187*(2 + 3*x)) + (1813*Log[2
+ 3*x])/729

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Rubi [A]  time = 0.0253455, antiderivative size = 55, 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} \[ -\frac{200 x^5}{9}+\frac{775 x^4}{27}-\frac{190 x^3}{81}-\frac{5287 x^2}{486}+\frac{2287 x}{729}+\frac{343}{2187 (3 x+2)}+\frac{1813}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2287*x)/729 - (5287*x^2)/486 - (190*x^3)/81 + (775*x^4)/27 - (200*x^5)/9 + 343/(2187*(2 + 3*x)) + (1813*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)^3}{(2+3 x)^2} \, dx &=\int \left (\frac{2287}{729}-\frac{5287 x}{243}-\frac{190 x^2}{27}+\frac{3100 x^3}{27}-\frac{1000 x^4}{9}-\frac{343}{729 (2+3 x)^2}+\frac{1813}{243 (2+3 x)}\right ) \, dx\\ &=\frac{2287 x}{729}-\frac{5287 x^2}{486}-\frac{190 x^3}{81}+\frac{775 x^4}{27}-\frac{200 x^5}{9}+\frac{343}{2187 (2+3 x)}+\frac{1813}{729} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0125155, size = 54, normalized size = 0.98 \[ -\frac{291600 x^6-182250 x^5-220320 x^4+163269 x^3+54000 x^2+3588 x-10878 (3 x+2) \log (3 x+2)+20002}{4374 (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(20002 + 3588*x + 54000*x^2 + 163269*x^3 - 220320*x^4 - 182250*x^5 + 291600*x^6 - 10878*(2 + 3*x)*Log[2 + 3*x
])/(4374*(2 + 3*x))

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Maple [A]  time = 0.007, size = 42, normalized size = 0.8 \begin{align*}{\frac{2287\,x}{729}}-{\frac{5287\,{x}^{2}}{486}}-{\frac{190\,{x}^{3}}{81}}+{\frac{775\,{x}^{4}}{27}}-{\frac{200\,{x}^{5}}{9}}+{\frac{343}{4374+6561\,x}}+{\frac{1813\,\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)^3/(2+3*x)^2,x)

[Out]

2287/729*x-5287/486*x^2-190/81*x^3+775/27*x^4-200/9*x^5+343/2187/(2+3*x)+1813/729*ln(2+3*x)

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Maxima [A]  time = 1.39411, size = 55, normalized size = 1. \begin{align*} -\frac{200}{9} \, x^{5} + \frac{775}{27} \, x^{4} - \frac{190}{81} \, x^{3} - \frac{5287}{486} \, x^{2} + \frac{2287}{729} \, x + \frac{343}{2187 \,{\left (3 \, x + 2\right )}} + \frac{1813}{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)^3/(2+3*x)^2,x, algorithm="maxima")

[Out]

-200/9*x^5 + 775/27*x^4 - 190/81*x^3 - 5287/486*x^2 + 2287/729*x + 343/2187/(3*x + 2) + 1813/729*log(3*x + 2)

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Fricas [A]  time = 1.2607, size = 176, normalized size = 3.2 \begin{align*} -\frac{291600 \, x^{6} - 182250 \, x^{5} - 220320 \, x^{4} + 163269 \, x^{3} + 54000 \, x^{2} - 10878 \,{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 27444 \, x - 686}{4374 \,{\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)^3/(2+3*x)^2,x, algorithm="fricas")

[Out]

-1/4374*(291600*x^6 - 182250*x^5 - 220320*x^4 + 163269*x^3 + 54000*x^2 - 10878*(3*x + 2)*log(3*x + 2) - 27444*
x - 686)/(3*x + 2)

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Sympy [A]  time = 0.107294, size = 48, normalized size = 0.87 \begin{align*} - \frac{200 x^{5}}{9} + \frac{775 x^{4}}{27} - \frac{190 x^{3}}{81} - \frac{5287 x^{2}}{486} + \frac{2287 x}{729} + \frac{1813 \log{\left (3 x + 2 \right )}}{729} + \frac{343}{6561 x + 4374} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200*x**5/9 + 775*x**4/27 - 190*x**3/81 - 5287*x**2/486 + 2287*x/729 + 1813*log(3*x + 2)/729 + 343/(6561*x + 4
374)

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Giac [A]  time = 2.71205, size = 101, normalized size = 1.84 \begin{align*} \frac{1}{4374} \,{\left (3 \, x + 2\right )}^{5}{\left (\frac{5550}{3 \, x + 2} - \frac{28780}{{\left (3 \, x + 2\right )}^{2}} + \frac{66193}{{\left (3 \, x + 2\right )}^{3}} - \frac{60438}{{\left (3 \, x + 2\right )}^{4}} - 400\right )} + \frac{343}{2187 \,{\left (3 \, x + 2\right )}} - \frac{1813}{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)^3/(2+3*x)^2,x, algorithm="giac")

[Out]

1/4374*(3*x + 2)^5*(5550/(3*x + 2) - 28780/(3*x + 2)^2 + 66193/(3*x + 2)^3 - 60438/(3*x + 2)^4 - 400) + 343/21
87/(3*x + 2) - 1813/729*log(1/3*abs(3*x + 2)/(3*x + 2)^2)