\(\int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx\) [1320]

Optimal result

Integrand size = 22, antiderivative size = 97 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {7}{3 (2+3 x)^7}-\frac {77}{3 (2+3 x)^6}-\frac {1133}{5 (2+3 x)^5}-\frac {1870}{(2+3 x)^4}-\frac {46475}{3 (2+3 x)^3}-\frac {138875}{(2+3 x)^2}-\frac {1615625}{2+3 x}-\frac {378125}{3+5 x}+9212500 \log (2+3 x)-9212500 \log (3+5 x) \]

[Out]

-7/3/(2+3*x)^7-77/3/(2+3*x)^6-1133/5/(2+3*x)^5-1870/(2+3*x)^4-46475/3/(2+3*x)^3-138875/(2+3*x)^2-1615625/(2+3*
x)-378125/(3+5*x)+9212500*ln(2+3*x)-9212500*ln(3+5*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 97, 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} \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {1615625}{3 x+2}-\frac {378125}{5 x+3}-\frac {138875}{(3 x+2)^2}-\frac {46475}{3 (3 x+2)^3}-\frac {1870}{(3 x+2)^4}-\frac {1133}{5 (3 x+2)^5}-\frac {77}{3 (3 x+2)^6}-\frac {7}{3 (3 x+2)^7}+9212500 \log (3 x+2)-9212500 \log (5 x+3) \]

[In]

Int[(1 - 2*x)^2/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

-7/(3*(2 + 3*x)^7) - 77/(3*(2 + 3*x)^6) - 1133/(5*(2 + 3*x)^5) - 1870/(2 + 3*x)^4 - 46475/(3*(2 + 3*x)^3) - 13
8875/(2 + 3*x)^2 - 1615625/(2 + 3*x) - 378125/(3 + 5*x) + 9212500*Log[2 + 3*x] - 9212500*Log[3 + 5*x]

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 {49}{(2+3 x)^8}+\frac {462}{(2+3 x)^7}+\frac {3399}{(2+3 x)^6}+\frac {22440}{(2+3 x)^5}+\frac {139425}{(2+3 x)^4}+\frac {833250}{(2+3 x)^3}+\frac {4846875}{(2+3 x)^2}+\frac {27637500}{2+3 x}+\frac {1890625}{(3+5 x)^2}-\frac {46062500}{3+5 x}\right ) \, dx \\ & = -\frac {7}{3 (2+3 x)^7}-\frac {77}{3 (2+3 x)^6}-\frac {1133}{5 (2+3 x)^5}-\frac {1870}{(2+3 x)^4}-\frac {46475}{3 (2+3 x)^3}-\frac {138875}{(2+3 x)^2}-\frac {1615625}{2+3 x}-\frac {378125}{3+5 x}+9212500 \log (2+3 x)-9212500 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {7}{3 (2+3 x)^7}-\frac {77}{3 (2+3 x)^6}-\frac {1133}{5 (2+3 x)^5}-\frac {1870}{(2+3 x)^4}-\frac {46475}{3 (2+3 x)^3}-\frac {138875}{(2+3 x)^2}-\frac {1615625}{2+3 x}-\frac {378125}{3+5 x}+9212500 \log (5 (2+3 x))-9212500 \log (3+5 x) \]

[In]

Integrate[(1 - 2*x)^2/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

-7/(3*(2 + 3*x)^7) - 77/(3*(2 + 3*x)^6) - 1133/(5*(2 + 3*x)^5) - 1870/(2 + 3*x)^4 - 46475/(3*(2 + 3*x)^3) - 13
8875/(2 + 3*x)^2 - 1615625/(2 + 3*x) - 378125/(3 + 5*x) + 9212500*Log[5*(2 + 3*x)] - 9212500*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70

method	result	size
norman	\(\frac {-68137326675 x^{4}-61781420250 x^{5}-45081841167 x^{3}-31117061250 x^{6}-6715912500 x^{7}-\frac {89469521196}{5} x^{2}-\frac {59178013234}{15} x -\frac {1863616801}{5}}{\left (2+3 x \right )^{7} \left (3+5 x \right )}+9212500 \ln \left (2+3 x \right )-9212500 \ln \left (3+5 x \right )\)	\(68\)
risch	\(\frac {-68137326675 x^{4}-61781420250 x^{5}-45081841167 x^{3}-31117061250 x^{6}-6715912500 x^{7}-\frac {89469521196}{5} x^{2}-\frac {59178013234}{15} x -\frac {1863616801}{5}}{\left (2+3 x \right )^{7} \left (3+5 x \right )}+9212500 \ln \left (2+3 x \right )-9212500 \ln \left (3+5 x \right )\)	\(69\)
default	\(-\frac {7}{3 \left (2+3 x \right )^{7}}-\frac {77}{3 \left (2+3 x \right )^{6}}-\frac {1133}{5 \left (2+3 x \right )^{5}}-\frac {1870}{\left (2+3 x \right )^{4}}-\frac {46475}{3 \left (2+3 x \right )^{3}}-\frac {138875}{\left (2+3 x \right )^{2}}-\frac {1615625}{2+3 x}-\frac {378125}{3+5 x}+9212500 \ln \left (2+3 x \right )-9212500 \ln \left (3+5 x \right )\)	\(90\)
parallelrisch	\(\frac {1132032000320 x -439794432000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+1337212800000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-82638336000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] x^{8}+2540704320000000 \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )+1018669608000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-1018669608000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-6792192000000 \ln \left (x +\frac {3}{5}\right )+3088961568000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-1337212800000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-3088961568000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-2540704320000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+2346808464000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-2346808464000000 \ln \left (x +\frac {3}{5}\right ) x^{6}[phone] \ln \left (\frac {2}{3}+x \right ) x^{8}-193418280000000 \ln \left (x +\frac {3}{5}\right ) x^{8}}{1920 \left (2+3 x \right )^{7} \left (3+5 x \right )}\)	\(208\)

[In]

int((1-2*x)^2/(2+3*x)^8/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

(-68137326675*x^4-61781420250*x^5-45081841167*x^3-31117061250*x^6-6715912500*x^7-89469521196/5*x^2-59178013234
/15*x-1863616801/5)/(2+3*x)^7/(3+5*x)+9212500*ln(2+3*x)-9212500*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {100738687500 \, x^{7} + 466755918750 \, x^{6} + 926721303750 \, x^{5} + 1022059900125 \, x^{4} + 676227617505 \, x^{3} + 268408563588 \, x^{2} + 138187500 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (5 \, x + 3\right ) - 138187500 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (3 \, x + 2\right ) + 59178013234 \, x + 5590850403}{15 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} \]

[In]

integrate((1-2*x)^2/(2+3*x)^8/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/15*(100738687500*x^7 + 466755918750*x^6 + 926721303750*x^5 + 1022059900125*x^4 + 676227617505*x^3 + 2684085
63588*x^2 + 138187500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 +
4672*x + 384)*log(5*x + 3) - 138187500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x
^3 + 24864*x^2 + 4672*x + 384)*log(3*x + 2) + 59178013234*x + 5590850403)/(10935*x^8 + 57591*x^7 + 132678*x^6
+ 174636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=\frac {- 100738687500 x^{7} - 466755918750 x^{6} - 926721303750 x^{5} - 1022059900125 x^{4} - 676227617505 x^{3} - 268408563588 x^{2} - 59178013234 x - 5590850403}{164025 x^{8} + 863865 x^{7} + 1990170 x^{6} + 2619540 x^{5} + 2154600 x^{4} + 1134000 x^{3} + 372960 x^{2} + 70080 x + 5760} - 9212500 \log {\left (x + \frac {3}{5} \right )} + 9212500 \log {\left (x + \frac {2}{3} \right )} \]

[In]

integrate((1-2*x)**2/(2+3*x)**8/(3+5*x)**2,x)

[Out]

(-100738687500*x**7 - 466755918750*x**6 - 926721303750*x**5 - 1022059900125*x**4 - 676227617505*x**3 - 2684085
63588*x**2 - 59178013234*x - 5590850403)/(164025*x**8 + 863865*x**7 + 1990170*x**6 + 2619540*x**5 + 2154600*x*
*4 + 1134000*x**3 + 372960*x**2 + 70080*x + 5760) - 9212500*log(x + 3/5) + 9212500*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {100738687500 \, x^{7} + 466755918750 \, x^{6} + 926721303750 \, x^{5} + 1022059900125 \, x^{4} + 676227617505 \, x^{3} + 268408563588 \, x^{2} + 59178013234 \, x + 5590850403}{15 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} - 9212500 \, \log \left (5 \, x + 3\right ) + 9212500 \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^2/(2+3*x)^8/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/15*(100738687500*x^7 + 466755918750*x^6 + 926721303750*x^5 + 1022059900125*x^4 + 676227617505*x^3 + 2684085
63588*x^2 + 59178013234*x + 5590850403)/(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*
x^3 + 24864*x^2 + 4672*x + 384) - 9212500*log(5*x + 3) + 9212500*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {378125}{5 \, x + 3} + \frac {625 \, {\left (\frac {120779019}{5 \, x + 3} + \frac {110006829}{{\left (5 \, x + 3\right )}^{2}} + \frac {54129465}{{\left (5 \, x + 3\right )}^{3}} + \frac {15246900}{{\left (5 \, x + 3\right )}^{4}} + \frac {2349450}{{\left (5 \, x + 3\right )}^{5}} + \frac {157100}{{\left (5 \, x + 3\right )}^{6}} + 55800576\right )}}{{\left (\frac {1}{5 \, x + 3} + 3\right )}^{7}} + 9212500 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

integrate((1-2*x)^2/(2+3*x)^8/(3+5*x)^2,x, algorithm="giac")

[Out]

-378125/(5*x + 3) + 625*(120779019/(5*x + 3) + 110006829/(5*x + 3)^2 + 54129465/(5*x + 3)^3 + 15246900/(5*x +
3)^4 + 2349450/(5*x + 3)^5 + 157100/(5*x + 3)^6 + 55800576)/(1/(5*x + 3) + 3)^7 + 9212500*log(abs(-1/(5*x + 3)
- 3))

Mupad [B] (verification not implemented)

Time = 1.37 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx=18425000\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {1842500\,x^7}{3}+\frac {25610750\,x^6}{9}+\frac {457640150\,x^5}{81}+\frac {1514162815\,x^4}{243}+\frac {1669697821\,x^3}{405}+\frac {29823173732\,x^2}{18225}+\frac {59178013234\,x}{164025}+\frac {1863616801}{54675}}{x^8+\frac {79\,x^7}{15}+\frac {182\,x^6}{15}+\frac {2156\,x^5}{135}+\frac {1064\,x^4}{81}+\frac {560\,x^3}{81}+\frac {8288\,x^2}{3645}+\frac {4672\,x}{10935}+\frac {128}{3645}} \]

[In]

int((2*x - 1)^2/((3*x + 2)^8*(5*x + 3)^2),x)

[Out]

18425000*atanh(30*x + 19) - ((59178013234*x)/164025 + (29823173732*x^2)/18225 + (1669697821*x^3)/405 + (151416
2815*x^4)/243 + (457640150*x^5)/81 + (25610750*x^6)/9 + (1842500*x^7)/3 + 1863616801/54675)/((4672*x)/10935 +
(8288*x^2)/3645 + (560*x^3)/81 + (1064*x^4)/81 + (2156*x^5)/135 + (182*x^6)/15 + (79*x^7)/15 + x^8 + 128/3645)