3.1383 \(\int \frac{(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=63 \[ -\frac{1000 x^3}{243}+\frac{3550 x^2}{243}-\frac{24970 x}{729}+\frac{10073}{729 (3 x+2)}-\frac{1813}{1458 (3 x+2)^2}+\frac{343}{6561 (3 x+2)^3}+\frac{66193 \log (3 x+2)}{2187} \]

[Out]

(-24970*x)/729 + (3550*x^2)/243 - (1000*x^3)/243 + 343/(6561*(2 + 3*x)^3) - 1813/(1458*(2 + 3*x)^2) + 10073/(7
29*(2 + 3*x)) + (66193*Log[2 + 3*x])/2187

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Rubi [A]  time = 0.0301268, antiderivative size = 63, 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} \[ -\frac{1000 x^3}{243}+\frac{3550 x^2}{243}-\frac{24970 x}{729}+\frac{10073}{729 (3 x+2)}-\frac{1813}{1458 (3 x+2)^2}+\frac{343}{6561 (3 x+2)^3}+\frac{66193 \log (3 x+2)}{2187} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

(-24970*x)/729 + (3550*x^2)/243 - (1000*x^3)/243 + 343/(6561*(2 + 3*x)^3) - 1813/(1458*(2 + 3*x)^2) + 10073/(7
29*(2 + 3*x)) + (66193*Log[2 + 3*x])/2187

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)^4} \, dx &=\int \left (-\frac{24970}{729}+\frac{7100 x}{243}-\frac{1000 x^2}{81}-\frac{343}{729 (2+3 x)^4}+\frac{1813}{243 (2+3 x)^3}-\frac{10073}{243 (2+3 x)^2}+\frac{66193}{729 (2+3 x)}\right ) \, dx\\ &=-\frac{24970 x}{729}+\frac{3550 x^2}{243}-\frac{1000 x^3}{243}+\frac{343}{6561 (2+3 x)^3}-\frac{1813}{1458 (2+3 x)^2}+\frac{10073}{729 (2+3 x)}+\frac{66193 \log (2+3 x)}{2187}\\ \end{align*}

Mathematica [A]  time = 0.0181294, size = 52, normalized size = 0.83 \[ \frac{132386 \log (3 x+2)-\frac{3 \left (162000 x^6-251100 x^5+414180 x^4+3180480 x^3+3851166 x^2+1766567 x+279268\right )}{(3 x+2)^3}}{4374} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

((-3*(279268 + 1766567*x + 3851166*x^2 + 3180480*x^3 + 414180*x^4 - 251100*x^5 + 162000*x^6))/(2 + 3*x)^3 + 13
2386*Log[2 + 3*x])/4374

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Maple [A]  time = 0.005, size = 50, normalized size = 0.8 \begin{align*} -{\frac{24970\,x}{729}}+{\frac{3550\,{x}^{2}}{243}}-{\frac{1000\,{x}^{3}}{243}}+{\frac{343}{6561\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{1813}{1458\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{10073}{1458+2187\,x}}+{\frac{66193\,\ln \left ( 2+3\,x \right ) }{2187}} \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)^4,x)

[Out]

-24970/729*x+3550/243*x^2-1000/243*x^3+343/6561/(2+3*x)^3-1813/1458/(2+3*x)^2+10073/729/(2+3*x)+66193/2187*ln(
2+3*x)

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Maxima [A]  time = 1.04477, size = 69, normalized size = 1.1 \begin{align*} -\frac{1000}{243} \, x^{3} + \frac{3550}{243} \, x^{2} - \frac{24970}{729} \, x + \frac{7 \,{\left (233118 \, x^{2} + 303831 \, x + 99044\right )}}{13122 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{66193}{2187} \, \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)^4,x, algorithm="maxima")

[Out]

-1000/243*x^3 + 3550/243*x^2 - 24970/729*x + 7/13122*(233118*x^2 + 303831*x + 99044)/(27*x^3 + 54*x^2 + 36*x +
8) + 66193/2187*log(3*x + 2)

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Fricas [A]  time = 1.31263, size = 247, normalized size = 3.92 \begin{align*} -\frac{1458000 \, x^{6} - 2259900 \, x^{5} + 3727620 \, x^{4} + 17801640 \, x^{3} + 13015134 \, x^{2} - 397158 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 1468863 \, x - 693308}{13122 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

-1/13122*(1458000*x^6 - 2259900*x^5 + 3727620*x^4 + 17801640*x^3 + 13015134*x^2 - 397158*(27*x^3 + 54*x^2 + 36
*x + 8)*log(3*x + 2) + 1468863*x - 693308)/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]  time = 0.135933, size = 53, normalized size = 0.84 \begin{align*} - \frac{1000 x^{3}}{243} + \frac{3550 x^{2}}{243} - \frac{24970 x}{729} + \frac{1631826 x^{2} + 2126817 x + 693308}{354294 x^{3} + 708588 x^{2} + 472392 x + 104976} + \frac{66193 \log{\left (3 x + 2 \right )}}{2187} \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)**4,x)

[Out]

-1000*x**3/243 + 3550*x**2/243 - 24970*x/729 + (1631826*x**2 + 2126817*x + 693308)/(354294*x**3 + 708588*x**2
+ 472392*x + 104976) + 66193*log(3*x + 2)/2187

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Giac [A]  time = 2.11177, size = 57, normalized size = 0.9 \begin{align*} -\frac{1000}{243} \, x^{3} + \frac{3550}{243} \, x^{2} - \frac{24970}{729} \, x + \frac{7 \,{\left (233118 \, x^{2} + 303831 \, x + 99044\right )}}{13122 \,{\left (3 \, x + 2\right )}^{3}} + \frac{66193}{2187} \, \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)^3/(2+3*x)^4,x, algorithm="giac")

[Out]

-1000/243*x^3 + 3550/243*x^2 - 24970/729*x + 7/13122*(233118*x^2 + 303831*x + 99044)/(3*x + 2)^3 + 66193/2187*
log(abs(3*x + 2))