\(\int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx\) [1306]

Optimal result

Integrand size = 22, antiderivative size = 69 \[ \int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx=\frac {2777053 x}{390625}+\frac {463086 x^2}{78125}-\frac {48771 x^3}{3125}-\frac {157599 x^4}{6250}+\frac {28917 x^5}{3125}+\frac {4374 x^6}{125}+\frac {2916 x^7}{175}-\frac {121}{1953125 (3+5 x)}+\frac {2134 \log (3+5 x)}{1953125} \]

[Out]

2777053/390625*x+463086/78125*x^2-48771/3125*x^3-157599/6250*x^4+28917/3125*x^5+4374/125*x^6+2916/175*x^7-121/
1953125/(3+5*x)+2134/1953125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, 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} \[ \int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx=\frac {2916 x^7}{175}+\frac {4374 x^6}{125}+\frac {28917 x^5}{3125}-\frac {157599 x^4}{6250}-\frac {48771 x^3}{3125}+\frac {463086 x^2}{78125}+\frac {2777053 x}{390625}-\frac {121}{1953125 (5 x+3)}+\frac {2134 \log (5 x+3)}{1953125} \]

[In]

Int[((1 - 2*x)^2*(2 + 3*x)^6)/(3 + 5*x)^2,x]

[Out]

(2777053*x)/390625 + (463086*x^2)/78125 - (48771*x^3)/3125 - (157599*x^4)/6250 + (28917*x^5)/3125 + (4374*x^6)
/125 + (2916*x^7)/175 - 121/(1953125*(3 + 5*x)) + (2134*Log[3 + 5*x])/1953125

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 {2777053}{390625}+\frac {926172 x}{78125}-\frac {146313 x^2}{3125}-\frac {315198 x^3}{3125}+\frac {28917 x^4}{625}+\frac {26244 x^5}{125}+\frac {2916 x^6}{25}+\frac {121}{390625 (3+5 x)^2}+\frac {2134}{390625 (3+5 x)}\right ) \, dx \\ & = \frac {2777053 x}{390625}+\frac {463086 x^2}{78125}-\frac {48771 x^3}{3125}-\frac {157599 x^4}{6250}+\frac {28917 x^5}{3125}+\frac {4374 x^6}{125}+\frac {2916 x^7}{175}-\frac {121}{1953125 (3+5 x)}+\frac {2134 \log (3+5 x)}{1953125} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx=\frac {648854027+3997343145 x+7291044250 x^2-2349191250 x^3-21011090625 x^4-13442034375 x^5+20677781250 x^6+30754687500 x^7+11390625000 x^8+149380 (3+5 x) \log (6 (3+5 x))}{136718750 (3+5 x)} \]

[In]

Integrate[((1 - 2*x)^2*(2 + 3*x)^6)/(3 + 5*x)^2,x]

[Out]

(648854027 + 3997343145*x + 7291044250*x^2 - 2349191250*x^3 - 21011090625*x^4 - 13442034375*x^5 + 20677781250*
x^6 + 30754687500*x^7 + 11390625000*x^8 + 149380*(3 + 5*x)*Log[6*(3 + 5*x)])/(136718750*(3 + 5*x))

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.72

method	result	size
risch	\(\frac {2916 x^{7}}{175}+\frac {4374 x^{6}}{125}+\frac {28917 x^{5}}{3125}-\frac {157599 x^{4}}{6250}-\frac {48771 x^{3}}{3125}+\frac {463086 x^{2}}{78125}+\frac {2777053 x}{390625}-\frac {121}{9765625 \left (x +\frac {3}{5}\right )}+\frac {2134 \ln \left (3+5 x \right )}{1953125}\)	\(50\)
default	\(\frac {2777053 x}{390625}+\frac {463086 x^{2}}{78125}-\frac {48771 x^{3}}{3125}-\frac {157599 x^{4}}{6250}+\frac {28917 x^{5}}{3125}+\frac {4374 x^{6}}{125}+\frac {2916 x^{7}}{175}-\frac {121}{1953125 \left (3+5 x \right )}+\frac {2134 \ln \left (3+5 x \right )}{1953125}\)	\(52\)
norman	\(\frac {\frac {24993598}{1171875} x +\frac {4166311}{78125} x^{2}-\frac {268479}{15625} x^{3}-\frac {960507}{6250} x^{4}-\frac {614493}{6250} x^{5}+\frac {94527}{625} x^{6}+\frac {39366}{175} x^{7}+\frac {2916}{35} x^{8}}{3+5 x}+\frac {2134 \ln \left (3+5 x \right )}{1953125}\)	\(57\)
parallelrisch	\(\frac {6834375000 x^{8}[phone] x^{7}[phone] x^{6}-8065220625 x^{5}-12606654375 x^{4}-1409514750 x^{3}+448140 \ln \left (x +\frac {3}{5}\right ) x [phone] x^{2}+268884 \ln \left (x +\frac {3}{5}\right )[phone] x}{246093750+410156250 x}\)	\(62\)
meijerg	\(-\frac {128 x}{9 \left (1+\frac {5 x}{3}\right )}+\frac {2134 \ln \left (1+\frac {5 x}{3}\right )}{1953125}+\frac {112 x \left (5 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )}+\frac {1512 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )}-\frac {2268 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{625 \left (1+\frac {5 x}{3}\right )}-\frac {1701 x \left (-\frac {625}{27} x^{4}+\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )}+\frac {19683 x \left (\frac {43750}{243} x^{5}-\frac {4375}{27} x^{4}+\frac {4375}{27} x^{3}-\frac {1750}{9} x^{2}+350 x +420\right )}{31250 \left (1+\frac {5 x}{3}\right )}-\frac {177147 x \left (-\frac {312500}{729} x^{6}+\frac {87500}{243} x^{5}-\frac {8750}{27} x^{4}+\frac {8750}{27} x^{3}-\frac {3500}{9} x^{2}+700 x +840\right )}{781250 \left (1+\frac {5 x}{3}\right )}+\frac {236196 x \left (\frac {390625}{243} x^{7}-\frac {312500}{243} x^{6}+\frac {87500}{81} x^{5}-\frac {8750}{9} x^{4}+\frac {8750}{9} x^{3}-\frac {3500}{3} x^{2}+2100 x +2520\right )}{13671875 \left (1+\frac {5 x}{3}\right )}\)	\(230\)

[In]

int((1-2*x)^2*(2+3*x)^6/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

2916/175*x^7+4374/125*x^6+28917/3125*x^5-157599/6250*x^4-48771/3125*x^3+463086/78125*x^2+2777053/390625*x-121/
9765625/(x+3/5)+2134/1953125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx=\frac {2278125000 \, x^{8} + 6150937500 \, x^{7} + 4135556250 \, x^{6} - 2688406875 \, x^{5} - 4202218125 \, x^{4} - 469838250 \, x^{3} + 1458208850 \, x^{2} + 29876 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 583181130 \, x - 1694}{27343750 \, {\left (5 \, x + 3\right )}} \]

[In]

integrate((1-2*x)^2*(2+3*x)^6/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/27343750*(2278125000*x^8 + 6150937500*x^7 + 4135556250*x^6 - 2688406875*x^5 - 4202218125*x^4 - 469838250*x^3
+ 1458208850*x^2 + 29876*(5*x + 3)*log(5*x + 3) + 583181130*x - 1694)/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx=\frac {2916 x^{7}}{175} + \frac {4374 x^{6}}{125} + \frac {28917 x^{5}}{3125} - \frac {157599 x^{4}}{6250} - \frac {48771 x^{3}}{3125} + \frac {463086 x^{2}}{78125} + \frac {2777053 x}{390625} + \frac {2134 \log {\left (5 x + 3 \right )}}{1953125} - \frac {121}{9765625 x + 5859375} \]

[In]

integrate((1-2*x)**2*(2+3*x)**6/(3+5*x)**2,x)

[Out]

2916*x**7/175 + 4374*x**6/125 + 28917*x**5/3125 - 157599*x**4/6250 - 48771*x**3/3125 + 463086*x**2/78125 + 277
7053*x/390625 + 2134*log(5*x + 3)/1953125 - 121/(9765625*x + 5859375)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx=\frac {2916}{175} \, x^{7} + \frac {4374}{125} \, x^{6} + \frac {28917}{3125} \, x^{5} - \frac {157599}{6250} \, x^{4} - \frac {48771}{3125} \, x^{3} + \frac {463086}{78125} \, x^{2} + \frac {2777053}{390625} \, x - \frac {121}{1953125 \, {\left (5 \, x + 3\right )}} + \frac {2134}{1953125} \, \log \left (5 \, x + 3\right ) \]

[In]

integrate((1-2*x)^2*(2+3*x)^6/(3+5*x)^2,x, algorithm="maxima")

[Out]

2916/175*x^7 + 4374/125*x^6 + 28917/3125*x^5 - 157599/6250*x^4 - 48771/3125*x^3 + 463086/78125*x^2 + 2777053/3
90625*x - 121/1953125/(5*x + 3) + 2134/1953125*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.35 \[ \int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx=-\frac {1}{136718750} \, {\left (5 \, x + 3\right )}^{7} {\left (\frac {306180}{5 \, x + 3} - \frac {404838}{{\left (5 \, x + 3\right )}^{2}} - \frac {2189565}{{\left (5 \, x + 3\right )}^{3}} - \frac {2888550}{{\left (5 \, x + 3\right )}^{4}} - \frac {2081520}{{\left (5 \, x + 3\right )}^{5}} - \frac {1088290}{{\left (5 \, x + 3\right )}^{6}} - 29160\right )} - \frac {121}{1953125 \, {\left (5 \, x + 3\right )}} - \frac {2134}{1953125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

[In]

integrate((1-2*x)^2*(2+3*x)^6/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/136718750*(5*x + 3)^7*(306180/(5*x + 3) - 404838/(5*x + 3)^2 - 2189565/(5*x + 3)^3 - 2888550/(5*x + 3)^4 -
2081520/(5*x + 3)^5 - 1088290/(5*x + 3)^6 - 29160) - 121/1953125/(5*x + 3) - 2134/1953125*log(1/5*abs(5*x + 3)
/(5*x + 3)^2)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx=\frac {2777053\,x}{390625}+\frac {2134\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {121}{9765625\,\left (x+\frac {3}{5}\right )}+\frac {463086\,x^2}{78125}-\frac {48771\,x^3}{3125}-\frac {157599\,x^4}{6250}+\frac {28917\,x^5}{3125}+\frac {4374\,x^6}{125}+\frac {2916\,x^7}{175} \]

[In]

int(((2*x - 1)^2*(3*x + 2)^6)/(5*x + 3)^2,x)

[Out]

(2777053*x)/390625 + (2134*log(x + 3/5))/1953125 - 121/(9765625*(x + 3/5)) + (463086*x^2)/78125 - (48771*x^3)/
3125 - (157599*x^4)/6250 + (28917*x^5)/3125 + (4374*x^6)/125 + (2916*x^7)/175