\(\int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx\) [1308]

Optimal result

Integrand size = 22, antiderivative size = 55 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {5459 x}{3125}-\frac {3621 x^2}{3125}-\frac {1809 x^3}{625}+\frac {189 x^4}{125}+\frac {324 x^5}{125}-\frac {121}{78125 (3+5 x)}+\frac {1408 \log (3+5 x)}{78125} \]

[Out]

5459/3125*x-3621/3125*x^2-1809/625*x^3+189/125*x^4+324/125*x^5-121/78125/(3+5*x)+1408/78125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, 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} \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {324 x^5}{125}+\frac {189 x^4}{125}-\frac {1809 x^3}{625}-\frac {3621 x^2}{3125}+\frac {5459 x}{3125}-\frac {121}{78125 (5 x+3)}+\frac {1408 \log (5 x+3)}{78125} \]

[In]

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

[Out]

(5459*x)/3125 - (3621*x^2)/3125 - (1809*x^3)/625 + (189*x^4)/125 + (324*x^5)/125 - 121/(78125*(3 + 5*x)) + (14
08*Log[3 + 5*x])/78125

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 {5459}{3125}-\frac {7242 x}{3125}-\frac {5427 x^2}{625}+\frac {756 x^3}{125}+\frac {324 x^4}{25}+\frac {121}{15625 (3+5 x)^2}+\frac {1408}{15625 (3+5 x)}\right ) \, dx \\ & = \frac {5459 x}{3125}-\frac {3621 x^2}{3125}-\frac {1809 x^3}{625}+\frac {189 x^4}{125}+\frac {324 x^5}{125}-\frac {121}{78125 (3+5 x)}+\frac {1408 \log (3+5 x)}{78125} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {990421+3698835 x+2054000 x^2-5655000 x^3-3881250 x^4+5990625 x^5+5062500 x^6+7040 (3+5 x) \log (6 (3+5 x))}{390625 (3+5 x)} \]

[In]

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

[Out]

(990421 + 3698835*x + 2054000*x^2 - 5655000*x^3 - 3881250*x^4 + 5990625*x^5 + 5062500*x^6 + 7040*(3 + 5*x)*Log
[6*(3 + 5*x)])/(390625*(3 + 5*x))

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73

method	result	size
risch	\(\frac {324 x^{5}}{125}+\frac {189 x^{4}}{125}-\frac {1809 x^{3}}{625}-\frac {3621 x^{2}}{3125}+\frac {5459 x}{3125}-\frac {121}{390625 \left (x +\frac {3}{5}\right )}+\frac {1408 \ln \left (3+5 x \right )}{78125}\)	\(40\)
default	\(\frac {5459 x}{3125}-\frac {3621 x^{2}}{3125}-\frac {1809 x^{3}}{625}+\frac {189 x^{4}}{125}+\frac {324 x^{5}}{125}-\frac {121}{78125 \left (3+5 x \right )}+\frac {1408 \ln \left (3+5 x \right )}{78125}\)	\(42\)
norman	\(\frac {\frac {245776}{46875} x +\frac {16432}{3125} x^{2}-\frac {9048}{625} x^{3}-\frac {1242}{125} x^{4}+\frac {1917}{125} x^{5}+\frac {324}{25} x^{6}}{3+5 x}+\frac {1408 \ln \left (3+5 x \right )}{78125}\)	\(47\)
parallelrisch	\(\frac {3037500 x^{6}+3594375 x^{5}-2328750 x^{4}-3393000 x^{3}+21120 \ln \left (x +\frac {3}{5}\right ) x +1232400 x^{2}+12672 \ln \left (x +\frac {3}{5}\right )+1228880 x}{703125+1171875 x}\)	\(52\)
meijerg	\(-\frac {16 x}{45 \left (1+\frac {5 x}{3}\right )}+\frac {1408 \ln \left (1+\frac {5 x}{3}\right )}{78125}-\frac {104 x \left (5 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )}+\frac {198 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )}+\frac {243 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )}-\frac {243 x \left (-\frac {625}{27} x^{4}+\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{625 \left (1+\frac {5 x}{3}\right )}+\frac {13122 x \left (\frac {43750}{243} x^{5}-\frac {4375}{27} x^{4}+\frac {4375}{27} x^{3}-\frac {1750}{9} x^{2}+350 x +420\right )}{546875 \left (1+\frac {5 x}{3}\right )}\)	\(145\)

[In]

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

[Out]

324/125*x^5+189/125*x^4-1809/625*x^3-3621/3125*x^2+5459/3125*x-121/390625/(x+3/5)+1408/78125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {1012500 \, x^{6} + 1198125 \, x^{5} - 776250 \, x^{4} - 1131000 \, x^{3} + 410800 \, x^{2} + 1408 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 409425 \, x - 121}{78125 \, {\left (5 \, x + 3\right )}} \]

[In]

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

[Out]

1/78125*(1012500*x^6 + 1198125*x^5 - 776250*x^4 - 1131000*x^3 + 410800*x^2 + 1408*(5*x + 3)*log(5*x + 3) + 409
425*x - 121)/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {324 x^{5}}{125} + \frac {189 x^{4}}{125} - \frac {1809 x^{3}}{625} - \frac {3621 x^{2}}{3125} + \frac {5459 x}{3125} + \frac {1408 \log {\left (5 x + 3 \right )}}{78125} - \frac {121}{390625 x + 234375} \]

[In]

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

[Out]

324*x**5/125 + 189*x**4/125 - 1809*x**3/625 - 3621*x**2/3125 + 5459*x/3125 + 1408*log(5*x + 3)/78125 - 121/(39
0625*x + 234375)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {324}{125} \, x^{5} + \frac {189}{125} \, x^{4} - \frac {1809}{625} \, x^{3} - \frac {3621}{3125} \, x^{2} + \frac {5459}{3125} \, x - \frac {121}{78125 \, {\left (5 \, x + 3\right )}} + \frac {1408}{78125} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

324/125*x^5 + 189/125*x^4 - 1809/625*x^3 - 3621/3125*x^2 + 5459/3125*x - 121/78125/(5*x + 3) + 1408/78125*log(
5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.36 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {1}{390625} \, {\left (5 \, x + 3\right )}^{5} {\left (\frac {3915}{5 \, x + 3} - \frac {8775}{{\left (5 \, x + 3\right )}^{2}} - \frac {26850}{{\left (5 \, x + 3\right )}^{3}} - \frac {30050}{{\left (5 \, x + 3\right )}^{4}} - 324\right )} - \frac {121}{78125 \, {\left (5 \, x + 3\right )}} - \frac {1408}{78125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

[In]

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

[Out]

-1/390625*(5*x + 3)^5*(3915/(5*x + 3) - 8775/(5*x + 3)^2 - 26850/(5*x + 3)^3 - 30050/(5*x + 3)^4 - 324) - 121/
78125/(5*x + 3) - 1408/78125*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {5459\,x}{3125}+\frac {1408\,\ln \left (x+\frac {3}{5}\right )}{78125}-\frac {121}{390625\,\left (x+\frac {3}{5}\right )}-\frac {3621\,x^2}{3125}-\frac {1809\,x^3}{625}+\frac {189\,x^4}{125}+\frac {324\,x^5}{125} \]

[In]

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

[Out]

(5459*x)/3125 + (1408*log(x + 3/5))/78125 - 121/(390625*(x + 3/5)) - (3621*x^2)/3125 - (1809*x^3)/625 + (189*x
^4)/125 + (324*x^5)/125