\(\int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx\) [1408]

Optimal result

Integrand size = 20, antiderivative size = 41 \[ \int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx=-\frac {1098 x}{625}+\frac {122 x^2}{125}-\frac {8 x^3}{25}-\frac {1331}{3125 (3+5 x)}+\frac {3267 \log (3+5 x)}{3125} \]

[Out]

-1098/625*x+122/125*x^2-8/25*x^3-1331/3125/(3+5*x)+3267/3125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, 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} \[ \int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx=-\frac {8 x^3}{25}+\frac {122 x^2}{125}-\frac {1098 x}{625}-\frac {1331}{3125 (5 x+3)}+\frac {3267 \log (5 x+3)}{3125} \]

[In]

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

[Out]

(-1098*x)/625 + (122*x^2)/125 - (8*x^3)/25 - 1331/(3125*(3 + 5*x)) + (3267*Log[3 + 5*x])/3125

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 {1098}{625}+\frac {244 x}{125}-\frac {24 x^2}{25}+\frac {1331}{625 (3+5 x)^2}+\frac {3267}{625 (3+5 x)}\right ) \, dx \\ & = -\frac {1098 x}{625}+\frac {122 x^2}{125}-\frac {8 x^3}{25}-\frac {1331}{3125 (3+5 x)}+\frac {3267 \log (3+5 x)}{3125} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx=\frac {9983-11865 x-36600 x^2+24500 x^3-10000 x^4+6534 (3+5 x) \log (6+10 x)}{6250 (3+5 x)} \]

[In]

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

[Out]

(9983 - 11865*x - 36600*x^2 + 24500*x^3 - 10000*x^4 + 6534*(3 + 5*x)*Log[6 + 10*x])/(6250*(3 + 5*x))

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {8 x^{3}}{25}+\frac {122 x^{2}}{125}-\frac {1098 x}{625}-\frac {1331}{15625 \left (x +\frac {3}{5}\right )}+\frac {3267 \ln \left (3+5 x \right )}{3125}\)	\(30\)
default	\(-\frac {1098 x}{625}+\frac {122 x^{2}}{125}-\frac {8 x^{3}}{25}-\frac {1331}{3125 \left (3+5 x \right )}+\frac {3267 \ln \left (3+5 x \right )}{3125}\)	\(32\)
norman	\(\frac {-\frac {8551}{1875} x -\frac {732}{125} x^{2}+\frac {98}{25} x^{3}-\frac {8}{5} x^{4}}{3+5 x}+\frac {3267 \ln \left (3+5 x \right )}{3125}\)	\(37\)
parallelrisch	\(\frac {-15000 x^{4}+36750 x^{3}+49005 \ln \left (x +\frac {3}{5}\right ) x -54900 x^{2}+29403 \ln \left (x +\frac {3}{5}\right )-42755 x}{28125+46875 x}\)	\(42\)
meijerg	\(\frac {37 x}{45 \left (1+\frac {5 x}{3}\right )}+\frac {3267 \ln \left (1+\frac {5 x}{3}\right )}{3125}+\frac {2 x \left (5 x +6\right )}{25 \left (1+\frac {5 x}{3}\right )}-\frac {3 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{25 \left (1+\frac {5 x}{3}\right )}-\frac {72 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )}\)	\(80\)

[In]

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

[Out]

-8/25*x^3+122/125*x^2-1098/625*x-1331/15625/(x+3/5)+3267/3125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx=-\frac {5000 \, x^{4} - 12250 \, x^{3} + 18300 \, x^{2} - 3267 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 16470 \, x + 1331}{3125 \, {\left (5 \, x + 3\right )}} \]

[In]

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

[Out]

-1/3125*(5000*x^4 - 12250*x^3 + 18300*x^2 - 3267*(5*x + 3)*log(5*x + 3) + 16470*x + 1331)/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx=- \frac {8 x^{3}}{25} + \frac {122 x^{2}}{125} - \frac {1098 x}{625} + \frac {3267 \log {\left (5 x + 3 \right )}}{3125} - \frac {1331}{15625 x + 9375} \]

[In]

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

[Out]

-8*x**3/25 + 122*x**2/125 - 1098*x/625 + 3267*log(5*x + 3)/3125 - 1331/(15625*x + 9375)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx=-\frac {8}{25} \, x^{3} + \frac {122}{125} \, x^{2} - \frac {1098}{625} \, x - \frac {1331}{3125 \, {\left (5 \, x + 3\right )}} + \frac {3267}{3125} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

-8/25*x^3 + 122/125*x^2 - 1098/625*x - 1331/3125/(5*x + 3) + 3267/3125*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.39 \[ \int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx=\frac {2}{3125} \, {\left (5 \, x + 3\right )}^{3} {\left (\frac {97}{5 \, x + 3} - \frac {1023}{{\left (5 \, x + 3\right )}^{2}} - 4\right )} - \frac {1331}{3125 \, {\left (5 \, x + 3\right )}} - \frac {3267}{3125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

[In]

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

[Out]

2/3125*(5*x + 3)^3*(97/(5*x + 3) - 1023/(5*x + 3)^2 - 4) - 1331/3125/(5*x + 3) - 3267/3125*log(1/5*abs(5*x + 3
)/(5*x + 3)^2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx=\frac {3267\,\ln \left (x+\frac {3}{5}\right )}{3125}-\frac {1098\,x}{625}-\frac {1331}{15625\,\left (x+\frac {3}{5}\right )}+\frac {122\,x^2}{125}-\frac {8\,x^3}{25} \]

[In]

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

[Out]

(3267*log(x + 3/5))/3125 - (1098*x)/625 - 1331/(15625*(x + 3/5)) + (122*x^2)/125 - (8*x^3)/25