\(\int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx\) [1337]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {7}{(2+3 x)^7}+\frac {707}{6 (2+3 x)^6}+\frac {6934}{5 (2+3 x)^5}+\frac {28555}{2 (2+3 x)^4}+\frac {424975}{3 (2+3 x)^3}+\frac {2958125}{2 (2+3 x)^2}+\frac {19637500}{2+3 x}-\frac {378125}{2 (3+5 x)^2}+\frac {9212500}{3+5 x}-125825000 \log (2+3 x)[phone] \log (3+5 x) \]

[Out]

7/(2+3*x)^7+707/6/(2+3*x)^6+6934/5/(2+3*x)^5+28555/2/(2+3*x)^4+424975/3/(2+3*x)^3+2958125/2/(2+3*x)^2+19637500
/(2+3*x)-378125/2/(3+5*x)^2+9212500/(3+5*x)-125825000*ln(2+3*x)[phone]*ln(3+5*x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 110, 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} \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {19637500}{3 x+2}+\frac {9212500}{5 x+3}+\frac {2958125}{2 (3 x+2)^2}-\frac {378125}{2 (5 x+3)^2}+\frac {424975}{3 (3 x+2)^3}+\frac {28555}{2 (3 x+2)^4}+\frac {6934}{5 (3 x+2)^5}+\frac {707}{6 (3 x+2)^6}+\frac {7}{(3 x+2)^7}-125825000 \log (3 x+2)[phone] \log (5 x+3) \]

[In]

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

[Out]

7/(2 + 3*x)^7 + 707/(6*(2 + 3*x)^6) + 6934/(5*(2 + 3*x)^5) + 28555/(2*(2 + 3*x)^4) + 424975/(3*(2 + 3*x)^3) +
2958125/(2*(2 + 3*x)^2) + 19637500/(2 + 3*x) - 378125/(2*(3 + 5*x)^2) + 9212500/(3 + 5*x) - 125825000*Log[2 +
3*x] + 125825000*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 {147}{(2+3 x)^8}-\frac {2121}{(2+3 x)^7}-\frac {20802}{(2+3 x)^6}-\frac {171330}{(2+3 x)^5}-\frac {1274925}{(2+3 x)^4}-\frac {8874375}{(2+3 x)^3}-\frac {58912500}{(2+3 x)^2}-\frac {377475000}{2+3 x}+\frac {1890625}{(3+5 x)^3}-\frac {46062500}{(3+5 x)^2}+\frac {629125000}{3+5 x}\right ) \, dx \\ & = \frac {7}{(2+3 x)^7}+\frac {707}{6 (2+3 x)^6}+\frac {6934}{5 (2+3 x)^5}+\frac {28555}{2 (2+3 x)^4}+\frac {424975}{3 (2+3 x)^3}+\frac {2958125}{2 (2+3 x)^2}+\frac {19637500}{2+3 x}-\frac {378125}{2 (3+5 x)^2}+\frac {9212500}{3+5 x}-125825000 \log (2+3 x)[phone] \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {7}{(2+3 x)^7}+\frac {707}{6 (2+3 x)^6}+\frac {6934}{5 (2+3 x)^5}+\frac {28555}{2 (2+3 x)^4}+\frac {424975}{3 (2+3 x)^3}+\frac {2958125}{2 (2+3 x)^2}+\frac {19637500}{2+3 x}-\frac {378125}{2 (3+5 x)^2}+\frac {9212500}{3+5 x}-125825000 \log (5 (2+3 x))[phone] \log (3+5 x) \]

[In]

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

[Out]

7/(2 + 3*x)^7 + 707/(6*(2 + 3*x)^6) + 6934/(5*(2 + 3*x)^5) + 28555/(2*(2 + 3*x)^4) + 424975/(3*(2 + 3*x)^3) +
2958125/(2*(2 + 3*x)^2) + 19637500/(2 + 3*x) - 378125/(2*(3 + 5*x)^2) + 9212500/(3 + 5*x) - 125825000*Log[5*(2
+ 3*x)] + 125825000*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.66

method	result	size
norman	\(\frac {458632125000 x^{8}[phone] x^{7}[phone] x^{3}[phone] x^{6}[phone] x^{4}[phone] x^{5}+\frac {1871049429619}{10} x +\frac {5013039895644}{5} x^{2}+\frac {76360244444}{5}}{\left (2+3 x \right )^{7} \left (3+5 x \right )^{2}}-125825000 \ln \left (2+3 x \right )[phone] \ln \left (3+5 x \right )\)	\(73\)
risch	\(\frac {458632125000 x^{8}[phone] x^{7}[phone] x^{3}[phone] x^{6}[phone] x^{4}[phone] x^{5}+\frac {1871049429619}{10} x +\frac {5013039895644}{5} x^{2}+\frac {76360244444}{5}}{\left (2+3 x \right )^{7} \left (3+5 x \right )^{2}}-125825000 \ln \left (2+3 x \right )[phone] \ln \left (3+5 x \right )\)	\(74\)
default	\(\frac {7}{\left (2+3 x \right )^{7}}+\frac {707}{6 \left (2+3 x \right )^{6}}+\frac {6934}{5 \left (2+3 x \right )^{5}}+\frac {28555}{2 \left (2+3 x \right )^{4}}+\frac {424975}{3 \left (2+3 x \right )^{3}}+\frac {2958125}{2 \left (2+3 x \right )^{2}}+\frac {19637500}{2+3 x}-\frac {378125}{2 \left (3+5 x \right )^{2}}+\frac {9212500}{3+5 x}-125825000 \ln \left (2+3 x \right )[phone] \ln \left (3+5 x \right )\)	\(99\)
parallelrisch	\(-\frac {139152383999040 x -70990907904000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+254474922240000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-11549647872000000 \ln \left (x +\frac {3}{5}\right ) x +70990907904000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+11549647872000000 \ln \left (\frac {2}{3}+x \right ) x +53464555978967952 x^{5}+65424350551835016 x^{6}+50024604299028372 x^{7}+9133163647984320 x^{3}+27955080458635680 x^{4}+1704616703996800 x^{2}+21851590967857440 x^{8}+4174996364975700 x^{9}+586266387840000000 \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )+606010806576000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-606010806576000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-834914304000000 \ln \left (x +\frac {3}{5}\right )+900220257216000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-254474922240000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-900220257216000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-586266387840000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+921314888928000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-921314888928000000 \ln \left (x +\frac {3}{5}\right ) x^{6}+39625815600000000 \ln \left (\frac {2}{3}+x \right ) x^{9}-39625815600000000 \ln \left (x +\frac {3}{5}\right ) x^{9}+232471451520000000 \ln \left (\frac {2}{3}+x \right ) x^{8}-232471451520000000 \ln \left (x +\frac {3}{5}\right ) x^{8}}{5760 \left (2+3 x \right )^{7} \left (3+5 x \right )^{2}}\)	\(231\)

[In]

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

[Out]

(458632125000*x^8+2400174787500*x^7+3069174545346*x^3+5494073130000*x^6+5870529249480*x^4+7184568494250*x^5+18
71049429619/10*x+5013039895644/5*x^2+76360244444/5)/(2+3*x)^7/(3+5*x)^2-125825000*ln(2+3*x)[phone]*ln(3+5*x
)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.77 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {4586321250000 \, x^{8} + 24001747875000 \, x^{7} + 54940731300000 \, x^{6} + 71845684942500 \, x^{5} + 58705292494800 \, x^{4} + 30691745453460 \, x^{3} + 10026079791288 \, x^{2} + 1258250000 \, {\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )} \log \left (5 \, x + 3\right ) - 1258250000 \, {\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )} \log \left (3 \, x + 2\right ) + 1871049429619 \, x + 152720488888}{10 \, {\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )}} \]

[In]

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

[Out]

1/10*(4586321250000*x^8 + 24001747875000*x^7 + 54940731300000*x^6 + 71845684942500*x^5 + 58705292494800*x^4 +
30691745453460*x^3 + 10026079791288*x^2 + 1258250000*(54675*x^9 + 320760*x^8 + 836163*x^7 + 1271214*x^6 + 1242
108*x^5 + 808920*x^4 + 351120*x^3 + 97952*x^2 + 15936*x + 1152)*log(5*x + 3) - 1258250000*(54675*x^9 + 320760*
x^8 + 836163*x^7 + 1271214*x^6 + 1242108*x^5 + 808920*x^4 + 351120*x^3 + 97952*x^2 + 15936*x + 1152)*log(3*x +
2) + 1871049429619*x + 152720488888)/(54675*x^9 + 320760*x^8 + 836163*x^7 + 1271214*x^6 + 1242108*x^5 + 80892
0*x^4 + 351120*x^3 + 97952*x^2 + 15936*x + 1152)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {4586321250000 x^{8} + 24001747875000 x^{7} + 54940731300000 x^{6} + 71845684942500 x^{5} + 58705292494800 x^{4} + 30691745453460 x^{3} + 10026079791288 x^{2} + 1871049429619 x + 152720488888}{546750 x^{9} + 3207600 x^{8} + 8361630 x^{7} + 12712140 x^{6} + 12421080 x^{5} + 8089200 x^{4} + 3511200 x^{3} + 979520 x^{2} + 159360 x + 11520} + 125825000 \log {\left (x + \frac {3}{5} \right )} - 125825000 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

(4586321250000*x**8 + 24001747875000*x**7 + 54940731300000*x**6 + 71845684942500*x**5 + 58705292494800*x**4 +
30691745453460*x**3 + 10026079791288*x**2 + 1871049429619*x + 152720488888)/(546750*x**9 + 3207600*x**8 + 8361
630*x**7 + 12712140*x**6 + 12421080*x**5 + 8089200*x**4 + 3511200*x**3 + 979520*x**2 + 159360*x + 11520) + 125
825000*log(x + 3/5) - 125825000*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {4586321250000 \, x^{8} + 24001747875000 \, x^{7} + 54940731300000 \, x^{6} + 71845684942500 \, x^{5} + 58705292494800 \, x^{4} + 30691745453460 \, x^{3} + 10026079791288 \, x^{2} + 1871049429619 \, x + 152720488888}{10 \, {\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )}} + 125825000 \, \log \left (5 \, x + 3\right ) - 125825000 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/10*(4586321250000*x^8 + 24001747875000*x^7 + 54940731300000*x^6 + 71845684942500*x^5 + 58705292494800*x^4 +
30691745453460*x^3 + 10026079791288*x^2 + 1871049429619*x + 152720488888)/(54675*x^9 + 320760*x^8 + 836163*x^7
+ 1271214*x^6 + 1242108*x^5 + 808920*x^4 + 351120*x^3 + 97952*x^2 + 15936*x + 1152) + 125825000*log(5*x + 3)
- 125825000*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {4586321250000 \, x^{8} + 24001747875000 \, x^{7} + 54940731300000 \, x^{6} + 71845684942500 \, x^{5} + 58705292494800 \, x^{4} + 30691745453460 \, x^{3} + 10026079791288 \, x^{2} + 1871049429619 \, x + 152720488888}{10 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{7}} + 125825000 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 125825000 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

1/10*(4586321250000*x^8 + 24001747875000*x^7 + 54940731300000*x^6 + 71845684942500*x^5 + 58705292494800*x^4 +
30691745453460*x^3 + 10026079791288*x^2 + 1871049429619*x + 152720488888)/((5*x + 3)^2*(3*x + 2)^7) + 12582500
0*log(abs(5*x + 3)) - 125825000*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {\frac {25165000\,x^8}{3}+\frac {395090500\,x^7}{9}+\frac {8139367600\,x^6}{81}+\frac {31931415530\,x^5}{243}+\frac {43485401848\,x^4}{405}+\frac {1023058181782\,x^3}{18225}+\frac {1671013298548\,x^2}{91125}+\frac {1871049429619\,x}{546750}+\frac {76360244444}{273375}}{x^9+\frac {88\,x^8}{15}+\frac {1147\,x^7}{75}+\frac {15694\,x^6}{675}+\frac {46004\,x^5}{2025}+\frac {5992\,x^4}{405}+\frac {23408\,x^3}{3645}+\frac {97952\,x^2}{54675}+\frac {5312\,x}{18225}+\frac {128}{6075}}-251650000\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

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

[Out]

((1871049429619*x)/546750 + (1671013298548*x^2)/91125 + (1023058181782*x^3)/18225 + (43485401848*x^4)/405 + (3
1931415530*x^5)/243 + (8139367600*x^6)/81 + (395090500*x^7)/9 + (25165000*x^8)/3 + 76360244444/273375)/((5312*
x)/18225 + (97952*x^2)/54675 + (23408*x^3)/3645 + (5992*x^4)/405 + (46004*x^5)/2025 + (15694*x^6)/675 + (1147*
x^7)/75 + (88*x^8)/15 + x^9 + 128/6075) - 251650000*atanh(30*x + 19)