\(\int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx\) [1366]

Optimal result

Integrand size = 22, antiderivative size = 52 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx=-\frac {1546 x}{81}+\frac {230 x^2}{27}-\frac {200 x^3}{81}-\frac {343}{1458 (2+3 x)^2}+\frac {3724}{729 (2+3 x)}+\frac {11599}{729} \log (2+3 x) \]

[Out]

-1546/81*x+230/27*x^2-200/81*x^3-343/1458/(2+3*x)^2+3724/729/(2+3*x)+11599/729*ln(2+3*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx=-\frac {200 x^3}{81}+\frac {230 x^2}{27}-\frac {1546 x}{81}+\frac {3724}{729 (3 x+2)}-\frac {343}{1458 (3 x+2)^2}+\frac {11599}{729} \log (3 x+2) \]

[In]

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

[Out]

(-1546*x)/81 + (230*x^2)/27 - (200*x^3)/81 - 343/(1458*(2 + 3*x)^2) + 3724/(729*(2 + 3*x)) + (11599*Log[2 + 3*
x])/729

Rule 90

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*} \text {integral}& = \int \left (-\frac {1546}{81}+\frac {460 x}{27}-\frac {200 x^2}{27}+\frac {343}{243 (2+3 x)^3}-\frac {3724}{243 (2+3 x)^2}+\frac {11599}{243 (2+3 x)}\right ) \, dx \\ & = -\frac {1546 x}{81}+\frac {230 x^2}{27}-\frac {200 x^3}{81}-\frac {343}{1458 (2+3 x)^2}+\frac {3724}{729 (2+3 x)}+\frac {11599}{729} \log (2+3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx=-\frac {258005+1171896 x+1531512 x^2+347436 x^3-205740 x^4+97200 x^5-69594 (2+3 x)^2 \log (20+30 x)}{4374 (2+3 x)^2} \]

[In]

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

[Out]

-1/4374*(258005 + 1171896*x + 1531512*x^2 + 347436*x^3 - 205740*x^4 + 97200*x^5 - 69594*(2 + 3*x)^2*Log[20 + 3
0*x])/(2 + 3*x)^2

Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {200 x^{3}}{81}+\frac {230 x^{2}}{27}-\frac {1546 x}{81}+\frac {\frac {3724 x}{243}+\frac {539}{54}}{\left (2+3 x \right )^{2}}+\frac {11599 \ln \left (2+3 x \right )}{729}\)	\(37\)
default	\(-\frac {1546 x}{81}+\frac {230 x^{2}}{27}-\frac {200 x^{3}}{81}-\frac {343}{1458 \left (2+3 x \right )^{2}}+\frac {3724}{729 \left (2+3 x \right )}+\frac {11599 \ln \left (2+3 x \right )}{729}\)	\(41\)
norman	\(\frac {-\frac {44209}{486} x -\frac {46963}{216} x^{2}-\frac {6434}{81} x^{3}+\frac {1270}{27} x^{4}-\frac {200}{9} x^{5}}{\left (2+3 x \right )^{2}}+\frac {11599 \ln \left (2+3 x \right )}{729}\)	\(42\)
parallelrisch	\(\frac {-129600 x^{5}+274320 x^{4}+835128 \ln \left (\frac {2}{3}+x \right ) x^{2}-463248 x^{3}+1113504 \ln \left (\frac {2}{3}+x \right ) x -1268001 x^{2}+371168 \ln \left (\frac {2}{3}+x \right )-530508 x}{5832 \left (2+3 x \right )^{2}}\)	\(56\)
meijerg	\(\frac {9 x \left (\frac {3 x}{2}+2\right )}{16 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {3 x^{2}}{2 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {47 x \left (\frac {27 x}{2}+6\right )}{108 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {11599 \ln \left (1+\frac {3 x}{2}\right )}{729}+\frac {23 x \left (9 x^{2}+27 x +12\right )}{18 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {4 x \left (-\frac {135}{8} x^{3}+45 x^{2}+135 x +60\right )}{27 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {400 x \left (\frac {81}{8} x^{4}-\frac {135}{8} x^{3}+45 x^{2}+135 x +60\right )}{729 \left (1+\frac {3 x}{2}\right )^{2}}\)	\(127\)

[In]

int((1-2*x)^3*(3+5*x)^2/(2+3*x)^3,x,method=_RETURNVERBOSE)

[Out]

-200/81*x^3+230/27*x^2-1546/81*x+9*(3724/2187*x+539/486)/(2+3*x)^2+11599/729*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx=-\frac {32400 \, x^{5} - 68580 \, x^{4} + 115812 \, x^{3} + 284256 \, x^{2} - 23198 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 88968 \, x - 14553}{1458 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^3,x, algorithm="fricas")

[Out]

-1/1458*(32400*x^5 - 68580*x^4 + 115812*x^3 + 284256*x^2 - 23198*(9*x^2 + 12*x + 4)*log(3*x + 2) + 88968*x - 1
4553)/(9*x^2 + 12*x + 4)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx=- \frac {200 x^{3}}{81} + \frac {230 x^{2}}{27} - \frac {1546 x}{81} - \frac {- 7448 x - 4851}{4374 x^{2} + 5832 x + 1944} + \frac {11599 \log {\left (3 x + 2 \right )}}{729} \]

[In]

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

[Out]

-200*x**3/81 + 230*x**2/27 - 1546*x/81 - (-7448*x - 4851)/(4374*x**2 + 5832*x + 1944) + 11599*log(3*x + 2)/729

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx=-\frac {200}{81} \, x^{3} + \frac {230}{27} \, x^{2} - \frac {1546}{81} \, x + \frac {49 \, {\left (152 \, x + 99\right )}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {11599}{729} \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^3,x, algorithm="maxima")

[Out]

-200/81*x^3 + 230/27*x^2 - 1546/81*x + 49/486*(152*x + 99)/(9*x^2 + 12*x + 4) + 11599/729*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx=-\frac {200}{81} \, x^{3} + \frac {230}{27} \, x^{2} - \frac {1546}{81} \, x + \frac {49 \, {\left (152 \, x + 99\right )}}{486 \, {\left (3 \, x + 2\right )}^{2}} + \frac {11599}{729} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^3,x, algorithm="giac")

[Out]

-200/81*x^3 + 230/27*x^2 - 1546/81*x + 49/486*(152*x + 99)/(3*x + 2)^2 + 11599/729*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {11599\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {1546\,x}{81}+\frac {\frac {3724\,x}{2187}+\frac {539}{486}}{x^2+\frac {4\,x}{3}+\frac {4}{9}}+\frac {230\,x^2}{27}-\frac {200\,x^3}{81} \]

[In]

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

[Out]

(11599*log(x + 2/3))/729 - (1546*x)/81 + ((3724*x)/2187 + 539/486)/((4*x)/3 + x^2 + 4/9) + (230*x^2)/27 - (200
*x^3)/81