3.1384 \(\int \frac{(1-2 x)^3 (3+5 x)^3}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=67 \[ -\frac{500 x^2}{243}+\frac{9100 x}{729}-\frac{66193}{2187 (3 x+2)}+\frac{10073}{1458 (3 x+2)^2}-\frac{1813}{2187 (3 x+2)^3}+\frac{343}{8748 (3 x+2)^4}-\frac{14390}{729} \log (3 x+2) \]

[Out]

(9100*x)/729 - (500*x^2)/243 + 343/(8748*(2 + 3*x)^4) - 1813/(2187*(2 + 3*x)^3) + 10073/(1458*(2 + 3*x)^2) - 6
6193/(2187*(2 + 3*x)) - (14390*Log[2 + 3*x])/729

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Rubi [A]  time = 0.0303992, antiderivative size = 67, 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} \[ -\frac{500 x^2}{243}+\frac{9100 x}{729}-\frac{66193}{2187 (3 x+2)}+\frac{10073}{1458 (3 x+2)^2}-\frac{1813}{2187 (3 x+2)^3}+\frac{343}{8748 (3 x+2)^4}-\frac{14390}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9100*x)/729 - (500*x^2)/243 + 343/(8748*(2 + 3*x)^4) - 1813/(2187*(2 + 3*x)^3) + 10073/(1458*(2 + 3*x)^2) - 6
6193/(2187*(2 + 3*x)) - (14390*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)^5} \, dx &=\int \left (\frac{9100}{729}-\frac{1000 x}{243}-\frac{343}{729 (2+3 x)^5}+\frac{1813}{243 (2+3 x)^4}-\frac{10073}{243 (2+3 x)^3}+\frac{66193}{729 (2+3 x)^2}-\frac{14390}{243 (2+3 x)}\right ) \, dx\\ &=\frac{9100 x}{729}-\frac{500 x^2}{243}+\frac{343}{8748 (2+3 x)^4}-\frac{1813}{2187 (2+3 x)^3}+\frac{10073}{1458 (2+3 x)^2}-\frac{66193}{2187 (2+3 x)}-\frac{14390}{729} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0171684, size = 56, normalized size = 0.84 \[ \frac{-1458000 x^6+4957200 x^5+26244000 x^4+32163156 x^3+13894254 x^2+675708 x-172680 (3 x+2)^4 \log (3 x+2)-597785}{8748 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-597785 + 675708*x + 13894254*x^2 + 32163156*x^3 + 26244000*x^4 + 4957200*x^5 - 1458000*x^6 - 172680*(2 + 3*x
)^4*Log[2 + 3*x])/(8748*(2 + 3*x)^4)

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Maple [A]  time = 0.007, size = 54, normalized size = 0.8 \begin{align*}{\frac{9100\,x}{729}}-{\frac{500\,{x}^{2}}{243}}+{\frac{343}{8748\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{1813}{2187\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{10073}{1458\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{66193}{4374+6561\,x}}-{\frac{14390\,\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)^5,x)

[Out]

9100/729*x-500/243*x^2+343/8748/(2+3*x)^4-1813/2187/(2+3*x)^3+10073/1458/(2+3*x)^2-66193/2187/(2+3*x)-14390/72
9*ln(2+3*x)

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Maxima [A]  time = 2.46738, size = 76, normalized size = 1.13 \begin{align*} -\frac{500}{243} \, x^{2} + \frac{9100}{729} \, x - \frac{2382948 \, x^{3} + 4584582 \, x^{2} + 2942764 \, x + 630195}{2916 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{14390}{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)^5,x, algorithm="maxima")

[Out]

-500/243*x^2 + 9100/729*x - 1/2916*(2382948*x^3 + 4584582*x^2 + 2942764*x + 630195)/(81*x^4 + 216*x^3 + 216*x^
2 + 96*x + 16) - 14390/729*log(3*x + 2)

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Fricas [A]  time = 1.35123, size = 273, normalized size = 4.07 \begin{align*} -\frac{486000 \, x^{6} - 1652400 \, x^{5} - 6566400 \, x^{4} - 4903452 \, x^{3} + 1186182 \, x^{2} + 57560 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 2360364 \, x + 630195}{2916 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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)^5,x, algorithm="fricas")

[Out]

-1/2916*(486000*x^6 - 1652400*x^5 - 6566400*x^4 - 4903452*x^3 + 1186182*x^2 + 57560*(81*x^4 + 216*x^3 + 216*x^
2 + 96*x + 16)*log(3*x + 2) + 2360364*x + 630195)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]  time = 0.148324, size = 56, normalized size = 0.84 \begin{align*} - \frac{500 x^{2}}{243} + \frac{9100 x}{729} - \frac{2382948 x^{3} + 4584582 x^{2} + 2942764 x + 630195}{236196 x^{4} + 629856 x^{3} + 629856 x^{2} + 279936 x + 46656} - \frac{14390 \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)**3/(2+3*x)**5,x)

[Out]

-500*x**2/243 + 9100*x/729 - (2382948*x**3 + 4584582*x**2 + 2942764*x + 630195)/(236196*x**4 + 629856*x**3 + 6
29856*x**2 + 279936*x + 46656) - 14390*log(3*x + 2)/729

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Giac [A]  time = 3.08148, size = 101, normalized size = 1.51 \begin{align*} \frac{100}{2187} \,{\left (3 \, x + 2\right )}^{2}{\left (\frac{111}{3 \, x + 2} - 5\right )} - \frac{66193}{2187 \,{\left (3 \, x + 2\right )}} + \frac{10073}{1458 \,{\left (3 \, x + 2\right )}^{2}} - \frac{1813}{2187 \,{\left (3 \, x + 2\right )}^{3}} + \frac{343}{8748 \,{\left (3 \, x + 2\right )}^{4}} + \frac{14390}{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)^5,x, algorithm="giac")

[Out]

100/2187*(3*x + 2)^2*(111/(3*x + 2) - 5) - 66193/2187/(3*x + 2) + 10073/1458/(3*x + 2)^2 - 1813/2187/(3*x + 2)
^3 + 343/8748/(3*x + 2)^4 + 14390/729*log(1/3*abs(3*x + 2)/(3*x + 2)^2)