\(\int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx\) [1325]

Optimal result

Integrand size = 22, antiderivative size = 59 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {1419 x}{3125}-\frac {4779 x^2}{6250}-\frac {72 x^3}{625}+\frac {81 x^4}{125}-\frac {121}{156250 (3+5 x)^2}-\frac {1408}{78125 (3+5 x)}+\frac {1202 \log (3+5 x)}{15625} \]

[Out]

1419/3125*x-4779/6250*x^2-72/625*x^3+81/125*x^4-121/156250/(3+5*x)^2-1408/78125/(3+5*x)+1202/15625*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, 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} \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {81 x^4}{125}-\frac {72 x^3}{625}-\frac {4779 x^2}{6250}+\frac {1419 x}{3125}-\frac {1408}{78125 (5 x+3)}-\frac {121}{156250 (5 x+3)^2}+\frac {1202 \log (5 x+3)}{15625} \]

[In]

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

[Out]

(1419*x)/3125 - (4779*x^2)/6250 - (72*x^3)/625 + (81*x^4)/125 - 121/(156250*(3 + 5*x)^2) - 1408/(78125*(3 + 5*
x)) + (1202*Log[3 + 5*x])/15625

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 {1419}{3125}-\frac {4779 x}{3125}-\frac {216 x^2}{625}+\frac {324 x^3}{125}+\frac {121}{15625 (3+5 x)^3}+\frac {1408}{15625 (3+5 x)^2}+\frac {1202}{3125 (3+5 x)}\right ) \, dx \\ & = \frac {1419 x}{3125}-\frac {4779 x^2}{6250}-\frac {72 x^3}{625}+\frac {81 x^4}{125}-\frac {121}{156250 (3+5 x)^2}-\frac {1408}{78125 (3+5 x)}+\frac {1202 \log (3+5 x)}{15625} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {121714+536320 x+553500 x^2-394500 x^3-523125 x^4+517500 x^5+506250 x^6+2404 (3+5 x)^2 \log (6 (3+5 x))}{31250 (3+5 x)^2} \]

[In]

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

[Out]

(121714 + 536320*x + 553500*x^2 - 394500*x^3 - 523125*x^4 + 517500*x^5 + 506250*x^6 + 2404*(3 + 5*x)^2*Log[6*(
3 + 5*x)])/(31250*(3 + 5*x)^2)

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {81 x^{4}}{125}-\frac {72 x^{3}}{625}-\frac {4779 x^{2}}{6250}+\frac {1419 x}{3125}+\frac {-\frac {1408 x}{15625}-\frac {8569}{156250}}{\left (3+5 x \right )^{2}}+\frac {1202 \ln \left (3+5 x \right )}{15625}\)	\(42\)
default	\(\frac {1419 x}{3125}-\frac {4779 x^{2}}{6250}-\frac {72 x^{3}}{625}+\frac {81 x^{4}}{125}-\frac {121}{156250 \left (3+5 x \right )^{2}}-\frac {1408}{78125 \left (3+5 x \right )}+\frac {1202 \ln \left (3+5 x \right )}{15625}\)	\(46\)
norman	\(\frac {\frac {39182}{9375} x +\frac {38773}{5625} x^{2}-\frac {1578}{125} x^{3}-\frac {837}{50} x^{4}+\frac {414}{25} x^{5}+\frac {81}{5} x^{6}}{\left (3+5 x \right )^{2}}+\frac {1202 \ln \left (3+5 x \right )}{15625}\)	\(47\)
parallelrisch	\(\frac {4556250 x^{6}+4657500 x^{5}-4708125 x^{4}+540900 \ln \left (x +\frac {3}{5}\right ) x^{2}-3550500 x^{3}+649080 \ln \left (x +\frac {3}{5}\right ) x +1938650 x^{2}+194724 \ln \left (x +\frac {3}{5}\right )+1175460 x}{281250 \left (3+5 x \right )^{2}}\)	\(61\)
meijerg	\(\frac {8 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {16 x^{2}}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {52 x \left (15 x +6\right )}{225 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {1202 \ln \left (1+\frac {5 x}{3}\right )}{15625}-\frac {66 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {243 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {162 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{625 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {2187 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{109375 \left (1+\frac {5 x}{3}\right )^{2}}\)	\(162\)

[In]

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

[Out]

81/125*x^4-72/625*x^3-4779/6250*x^2+1419/3125*x+25*(-1408/390625*x-8569/3906250)/(3+5*x)^2+1202/15625*ln(3+5*x
)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {2531250 \, x^{6} + 2587500 \, x^{5} - 2615625 \, x^{4} - 1972500 \, x^{3} + 1053225 \, x^{2} + 12020 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 624470 \, x - 8569}{156250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

[In]

integrate((1-2*x)^2*(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/156250*(2531250*x^6 + 2587500*x^5 - 2615625*x^4 - 1972500*x^3 + 1053225*x^2 + 12020*(25*x^2 + 30*x + 9)*log(
5*x + 3) + 624470*x - 8569)/(25*x^2 + 30*x + 9)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {81 x^{4}}{125} - \frac {72 x^{3}}{625} - \frac {4779 x^{2}}{6250} + \frac {1419 x}{3125} + \frac {- 14080 x - 8569}{3906250 x^{2} + 4687500 x + 1406250} + \frac {1202 \log {\left (5 x + 3 \right )}}{15625} \]

[In]

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

[Out]

81*x**4/125 - 72*x**3/625 - 4779*x**2/6250 + 1419*x/3125 + (-14080*x - 8569)/(3906250*x**2 + 4687500*x + 14062
50) + 1202*log(5*x + 3)/15625

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {81}{125} \, x^{4} - \frac {72}{625} \, x^{3} - \frac {4779}{6250} \, x^{2} + \frac {1419}{3125} \, x - \frac {11 \, {\left (1280 \, x + 779\right )}}{156250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {1202}{15625} \, \log \left (5 \, x + 3\right ) \]

[In]

integrate((1-2*x)^2*(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

81/125*x^4 - 72/625*x^3 - 4779/6250*x^2 + 1419/3125*x - 11/156250*(1280*x + 779)/(25*x^2 + 30*x + 9) + 1202/15
625*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {81}{125} \, x^{4} - \frac {72}{625} \, x^{3} - \frac {4779}{6250} \, x^{2} + \frac {1419}{3125} \, x - \frac {11 \, {\left (1280 \, x + 779\right )}}{156250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {1202}{15625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

[In]

integrate((1-2*x)^2*(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

81/125*x^4 - 72/625*x^3 - 4779/6250*x^2 + 1419/3125*x - 11/156250*(1280*x + 779)/(5*x + 3)^2 + 1202/15625*log(
abs(5*x + 3))

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {1419\,x}{3125}+\frac {1202\,\ln \left (x+\frac {3}{5}\right )}{15625}-\frac {\frac {1408\,x}{390625}+\frac {8569}{3906250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {4779\,x^2}{6250}-\frac {72\,x^3}{625}+\frac {81\,x^4}{125} \]

[In]

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

[Out]

(1419*x)/3125 + (1202*log(x + 3/5))/15625 - ((1408*x)/390625 + 8569/3906250)/((6*x)/5 + x^2 + 9/25) - (4779*x^
2)/6250 - (72*x^3)/625 + (81*x^4)/125