\(\int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx\) [1224]

Optimal result

Integrand size = 20, antiderivative size = 59 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx=\frac {3636 x}{3125}+\frac {1971 x^2}{6250}-\frac {837 x^3}{625}-\frac {243 x^4}{250}-\frac {11}{156250 (3+5 x)^2}-\frac {163}{78125 (3+5 x)}+\frac {192 \log (3+5 x)}{15625} \]

[Out]

3636/3125*x+1971/6250*x^2-837/625*x^3-243/250*x^4-11/156250/(3+5*x)^2-163/78125/(3+5*x)+192/15625*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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx=-\frac {243 x^4}{250}-\frac {837 x^3}{625}+\frac {1971 x^2}{6250}+\frac {3636 x}{3125}-\frac {163}{78125 (5 x+3)}-\frac {11}{156250 (5 x+3)^2}+\frac {192 \log (5 x+3)}{15625} \]

[In]

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

[Out]

(3636*x)/3125 + (1971*x^2)/6250 - (837*x^3)/625 - (243*x^4)/250 - 11/(156250*(3 + 5*x)^2) - 163/(78125*(3 + 5*
x)) + (192*Log[3 + 5*x])/15625

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3636}{3125}+\frac {1971 x}{3125}-\frac {2511 x^2}{625}-\frac {486 x^3}{125}+\frac {11}{15625 (3+5 x)^3}+\frac {163}{15625 (3+5 x)^2}+\frac {192}{3125 (3+5 x)}\right ) \, dx \\ & = \frac {3636 x}{3125}+\frac {1971 x^2}{6250}-\frac {837 x^3}{625}-\frac {243 x^4}{250}-\frac {11}{156250 (3+5 x)^2}-\frac {163}{78125 (3+5 x)}+\frac {192 \log (3+5 x)}{15625} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx=\frac {604711+3653570 x+7579975 x^2+4140000 x^3-6412500 x^4-9787500 x^5-3796875 x^6+1920 (3+5 x)^2 \log (-3 (3+5 x))}{156250 (3+5 x)^2} \]

[In]

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

[Out]

(604711 + 3653570*x + 7579975*x^2 + 4140000*x^3 - 6412500*x^4 - 9787500*x^5 - 3796875*x^6 + 1920*(3 + 5*x)^2*L
og[-3*(3 + 5*x)])/(156250*(3 + 5*x)^2)

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {243 x^{4}}{250}-\frac {837 x^{3}}{625}+\frac {1971 x^{2}}{6250}+\frac {3636 x}{3125}+\frac {-\frac {163 x}{15625}-\frac {989}{156250}}{\left (3+5 x \right )^{2}}+\frac {192 \ln \left (3+5 x \right )}{15625}\)	\(42\)
default	\(\frac {3636 x}{3125}+\frac {1971 x^{2}}{6250}-\frac {837 x^{3}}{625}-\frac {243 x^{4}}{250}-\frac {11}{156250 \left (3+5 x \right )^{2}}-\frac {163}{78125 \left (3+5 x \right )}+\frac {192 \ln \left (3+5 x \right )}{15625}\)	\(46\)
norman	\(\frac {\frac {98272}{9375} x +\frac {212408}{5625} x^{2}+\frac {3312}{125} x^{3}-\frac {1026}{25} x^{4}-\frac {1566}{25} x^{5}-\frac {243}{10} x^{6}}{\left (3+5 x \right )^{2}}+\frac {192 \ln \left (3+5 x \right )}{15625}\)	\(47\)
parallelrisch	\(\frac {-6834375 x^{6}-17617500 x^{5}-11542500 x^{4}+86400 \ln \left (x +\frac {3}{5}\right ) x^{2}+7452000 x^{3}+103680 \ln \left (x +\frac {3}{5}\right ) x [phone] x^{2}+31104 \ln \left (x +\frac {3}{5}\right )+2948160 x}{281250 \left (3+5 x \right )^{2}}\)	\(61\)
meijerg	\(\frac {16 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {88 x^{2}}{27 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {8 x \left (15 x +6\right )}{15 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {192 \ln \left (1+\frac {5 x}{3}\right )}{15625}-\frac {18 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{25 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {81 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {4131 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {6561 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{218750 \left (1+\frac {5 x}{3}\right )^{2}}\)	\(162\)

[In]

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

[Out]

-243/250*x^4-837/625*x^3+1971/6250*x^2+3636/3125*x+25*(-163/390625*x-989/3906250)/(3+5*x)^2+192/15625*ln(3+5*x
)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx=-\frac {3796875 \, x^{6} + 9787500 \, x^{5} + 6412500 \, x^{4} - 4140000 \, x^{3} - 5897475 \, x^{2} - 1920 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 1634570 \, x + 989}{156250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

[In]

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

[Out]

-1/156250*(3796875*x^6 + 9787500*x^5 + 6412500*x^4 - 4140000*x^3 - 5897475*x^2 - 1920*(25*x^2 + 30*x + 9)*log(
5*x + 3) - 1634570*x + 989)/(25*x^2 + 30*x + 9)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx=- \frac {243 x^{4}}{250} - \frac {837 x^{3}}{625} + \frac {1971 x^{2}}{6250} + \frac {3636 x}{3125} - \frac {1630 x + 989}{3906250 x^{2} + 4687500 x + 1406250} + \frac {192 \log {\left (5 x + 3 \right )}}{15625} \]

[In]

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

[Out]

-243*x**4/250 - 837*x**3/625 + 1971*x**2/6250 + 3636*x/3125 - (1630*x + 989)/(3906250*x**2 + 4687500*x + 14062
50) + 192*log(5*x + 3)/15625

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx=-\frac {243}{250} \, x^{4} - \frac {837}{625} \, x^{3} + \frac {1971}{6250} \, x^{2} + \frac {3636}{3125} \, x - \frac {1630 \, x + 989}{156250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {192}{15625} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

-243/250*x^4 - 837/625*x^3 + 1971/6250*x^2 + 3636/3125*x - 1/156250*(1630*x + 989)/(25*x^2 + 30*x + 9) + 192/1
5625*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx=-\frac {243}{250} \, x^{4} - \frac {837}{625} \, x^{3} + \frac {1971}{6250} \, x^{2} + \frac {3636}{3125} \, x - \frac {1630 \, x + 989}{156250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {192}{15625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

[In]

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

[Out]

-243/250*x^4 - 837/625*x^3 + 1971/6250*x^2 + 3636/3125*x - 1/156250*(1630*x + 989)/(5*x + 3)^2 + 192/15625*log
(abs(5*x + 3))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx=\frac {3636\,x}{3125}+\frac {192\,\ln \left (x+\frac {3}{5}\right )}{15625}-\frac {\frac {163\,x}{390625}+\frac {989}{3906250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\frac {1971\,x^2}{6250}-\frac {837\,x^3}{625}-\frac {243\,x^4}{250} \]

[In]

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

[Out]

(3636*x)/3125 + (192*log(x + 3/5))/15625 - ((163*x)/390625 + 989/3906250)/((6*x)/5 + x^2 + 9/25) + (1971*x^2)/
6250 - (837*x^3)/625 - (243*x^4)/250