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

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 = 0.03, size = 52, normalized size = 0.83 \begin {gather*} \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} \end {gather*}

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.33, size = 72, normalized size = 1.14 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.85, size = 42, normalized size = 0.67 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 50, normalized size = 0.79 \begin {gather*} -\frac {1000 x^{3}}{243}+\frac {3550 x^{2}}{243}-\frac {24970 x}{729}+\frac {66193 \ln \left (3 x +2\right )}{2187}+\frac {343}{6561 \left (3 x +2\right )^{3}}-\frac {1813}{1458 \left (3 x +2\right )^{2}}+\frac {10073}{729 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(5*x+3)^3/(3*x+2)^4,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 51, normalized size = 0.81 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 46, normalized size = 0.73 \begin {gather*} \frac {66193\,\ln \left (x+\frac {2}{3}\right )}{2187}-\frac {24970\,x}{729}+\frac {\frac {10073\,x^2}{2187}+\frac {26257\,x}{4374}+\frac {346654}{177147}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}}+\frac {3550\,x^2}{243}-\frac {1000\,x^3}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)^3*(5*x + 3)^3)/(3*x + 2)^4,x)

[Out]

(66193*log(x + 2/3))/2187 - (24970*x)/729 + ((26257*x)/4374 + (10073*x^2)/2187 + 346654/177147)/((4*x)/3 + 2*x
^2 + x^3 + 8/27) + (3550*x^2)/243 - (1000*x^3)/243

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 54, normalized size = 0.86 \begin {gather*} - \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 {gather*}

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

________________________________________________________________________________________