\(\int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx\) [1433]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {49}{3 (2+3 x)^7}+\frac {539}{2 (2+3 x)^6}+\frac {15708}{5 (2+3 x)^5}+\frac {64317}{2 (2+3 x)^4}+\frac {317845}{(2+3 x)^3}+\frac {6618975}{2 (2+3 x)^2}+\frac {43848750}{2+3 x}-\frac {831875}{2 (3+5 x)^2}+\frac {20418750}{3+5 x}-280500000 \log (2+3 x)[phone] \log (3+5 x) \]

[Out]

49/3/(2+3*x)^7+539/2/(2+3*x)^6+15708/5/(2+3*x)^5+64317/2/(2+3*x)^4+317845/(2+3*x)^3+6618975/2/(2+3*x)^2+438487
50/(2+3*x)-831875/2/(3+5*x)^2+20418750/(3+5*x)-280500000*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} \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {43848750}{3 x+2}+\frac {20418750}{5 x+3}+\frac {6618975}{2 (3 x+2)^2}-\frac {831875}{2 (5 x+3)^2}+\frac {317845}{(3 x+2)^3}+\frac {64317}{2 (3 x+2)^4}+\frac {15708}{5 (3 x+2)^5}+\frac {539}{2 (3 x+2)^6}+\frac {49}{3 (3 x+2)^7}-280500000 \log (3 x+2)[phone] \log (5 x+3) \]

[In]

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

[Out]

49/(3*(2 + 3*x)^7) + 539/(2*(2 + 3*x)^6) + 15708/(5*(2 + 3*x)^5) + 64317/(2*(2 + 3*x)^4) + 317845/(2 + 3*x)^3
+ 6618975/(2*(2 + 3*x)^2) + 43848750/(2 + 3*x) - 831875/(2*(3 + 5*x)^2) + 20418750/(3 + 5*x) - 280500000*Log[2
+ 3*x] + 280500000*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 {343}{(2+3 x)^8}-\frac {4851}{(2+3 x)^7}-\frac {47124}{(2+3 x)^6}-\frac {385902}{(2+3 x)^5}-\frac {2860605}{(2+3 x)^4}-\frac {19856925}{(2+3 x)^3}-\frac {131546250}{(2+3 x)^2}-\frac {841500000}{2+3 x}+\frac {4159375}{(3+5 x)^3}-\frac {102093750}{(3+5 x)^2}+\frac {1402500000}{3+5 x}\right ) \, dx \\ & = \frac {49}{3 (2+3 x)^7}+\frac {539}{2 (2+3 x)^6}+\frac {15708}{5 (2+3 x)^5}+\frac {64317}{2 (2+3 x)^4}+\frac {317845}{(2+3 x)^3}+\frac {6618975}{2 (2+3 x)^2}+\frac {43848750}{2+3 x}-\frac {831875}{2 (3+5 x)^2}+\frac {20418750}{3+5 x}-280500000 \log (2+3 x)[phone] \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {49}{3 (2+3 x)^7}+\frac {539}{2 (2+3 x)^6}+\frac {15708}{5 (2+3 x)^5}+\frac {64317}{2 (2+3 x)^4}+\frac {317845}{(2+3 x)^3}+\frac {6618975}{2 (2+3 x)^2}+\frac {43848750}{2+3 x}-\frac {831875}{2 (3+5 x)^2}+\frac {20418750}{3+5 x}-280500000 \log (5 (2+3 x))[phone] \log (3+5 x) \]

[In]

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

[Out]

49/(3*(2 + 3*x)^7) + 539/(2*(2 + 3*x)^6) + 15708/(5*(2 + 3*x)^5) + 64317/(2*(2 + 3*x)^4) + 317845/(2 + 3*x)^3
+ 6618975/(2*(2 + 3*x)^2) + 43848750/(2 + 3*x) - 831875/(2*(3 + 5*x)^2) + 20418750/(3 + 5*x) - 280500000*Log[5
*(2 + 3*x)] + 280500000*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.66

method	result	size
norman	\(\frac {1022422500000 x^{8}[phone] x^{7}[phone] x^{3}[phone] x^{6}[phone] x^{4}[phone] x^{5}+\frac {4171105622953}{10} x +\frac {33526509614024}{15} x^{2}+\frac {170228877938}{5}}{\left (2+3 x \right )^{7} \left (3+5 x \right )^{2}}-280500000 \ln \left (2+3 x \right )[phone] \ln \left (3+5 x \right )\)	\(73\)
risch	\(\frac {1022422500000 x^{8}[phone] x^{7}[phone] x^{3}[phone] x^{6}[phone] x^{4}[phone] x^{5}+\frac {4171105622953}{10} x +\frac {33526509614024}{15} x^{2}+\frac {170228877938}{5}}{\left (2+3 x \right )^{7} \left (3+5 x \right )^{2}}-280500000 \ln \left (2+3 x \right )[phone] \ln \left (3+5 x \right )\)	\(74\)
default	\(\frac {49}{3 \left (2+3 x \right )^{7}}+\frac {539}{2 \left (2+3 x \right )^{6}}+\frac {15708}{5 \left (2+3 x \right )^{5}}+\frac {64317}{2 \left (2+3 x \right )^{4}}+\frac {317845}{\left (2+3 x \right )^{3}}+\frac {6618975}{2 \left (2+3 x \right )^{2}}+\frac {43848750}{2+3 x}-\frac {831875}{2 \left (3+5 x \right )^{2}}+\frac {20418750}{3+5 x}-280500000 \ln \left (2+3 x \right )[phone] \ln \left (3+5 x \right )\)	\(99\)
parallelrisch	\(-\frac {310210559999040 x -158259087360000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+567297561600000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-25747476480000000 \ln \left (x +\frac {3}{5}\right ) x +158259087360000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+25747476480000000 \ln \left (\frac {2}{3}+x \right ) x +119187823978613304 x^{5}+145849635047076732 x^{6}+111519185423271894 x^{7}+20360440319984160 x^{3}+62319889279974960 x^{4}+3800079359997760 x^{2}+48713461287392880 x^{8}+9307263901260150 x^{9}+1306955865600000000 \ln \left (\frac {2}{3}+x \right ) x^{4}+1861263360000000 \ln \left (\frac {2}{3}+x \right )+1350971835840000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-1350971835840000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-1861263360000000 \ln \left (x +\frac {3}{5}\right )+2006849053440000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-567297561600000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-2006849053440000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-1306955865600000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+2053875035520000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-2053875035520000000 \ln \left (x +\frac {3}{5}\right ) x^{6}+88337304000000000 \ln \left (\frac {2}{3}+x \right ) x^{9}-88337304000000000 \ln \left (x +\frac {3}{5}\right ) x^{9}+518245516800000000 \ln \left (\frac {2}{3}+x \right ) x^{8}-518245516800000000 \ln \left (x +\frac {3}{5}\right ) x^{8}}{5760 \left (2+3 x \right )^{7} \left (3+5 x \right )^{2}}\)	\(231\)

[In]

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

[Out]

(1022422500000*x^8+5350677750000*x^7+6842070017640*x^3+12247864200000*x^6+13087092823200*x^4+16016463045000*x^
5+4171105622953/10*x+33526509614024/15*x^2+170228877938/5)/(2+3*x)^7/(3+5*x)^2-280500000*ln(2+3*x)[phone]*l
n(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.77 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {30672675000000 \, x^{8} + 160520332500000 \, x^{7} + 367435926000000 \, x^{6} + 480493891350000 \, x^{5} + 392612784696000 \, x^{4} + 205262100529200 \, x^{3} + 67053019228048 \, x^{2} + 8415000000 \, {\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )} \log \left (5 \, x + 3\right ) - 8415000000 \, {\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )} \log \left (3 \, x + 2\right ) + 12513316868859 \, x + 1021373267628}{30 \, {\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )}} \]

[In]

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

[Out]

1/30*(30672675000000*x^8 + 160520332500000*x^7 + 367435926000000*x^6 + 480493891350000*x^5 + 392612784696000*x
^4 + 205262100529200*x^3 + 67053019228048*x^2 + 8415000000*(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(5*x + 3) - 8415000000*(54675*x^9 + 3
20760*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) + 12513316868859*x + 1021373267628)/(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)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx=- \frac {- 30672675000000 x^{8} - 160520332500000 x^{7} - 367435926000000 x^{6} - 480493891350000 x^{5} - 392612784696000 x^{4} - 205262100529200 x^{3} - 67053019228048 x^{2} - 12513316868859 x - 1021373267628}{1640250 x^{9} + 9622800 x^{8} + 25084890 x^{7} + 38136420 x^{6} + 37263240 x^{5} + 24267600 x^{4} + 10533600 x^{3} + 2938560 x^{2} + 478080 x + 34560} + 280500000 \log {\left (x + \frac {3}{5} \right )} - 280500000 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

-(-30672675000000*x**8 - 160520332500000*x**7 - 367435926000000*x**6 - 480493891350000*x**5 - 392612784696000*
x**4 - 205262100529200*x**3 - 67053019228048*x**2 - 12513316868859*x - 1021373267628)/(1640250*x**9 + 9622800*
x**8 + 25084890*x**7 + 38136420*x**6 + 37263240*x**5 + 24267600*x**4 + 10533600*x**3 + 2938560*x**2 + 478080*x
+ 34560) + 280500000*log(x + 3/5) - 280500000*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {30672675000000 \, x^{8} + 160520332500000 \, x^{7} + 367435926000000 \, x^{6} + 480493891350000 \, x^{5} + 392612784696000 \, x^{4} + 205262100529200 \, x^{3} + 67053019228048 \, x^{2} + 12513316868859 \, x + 1021373267628}{30 \, {\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )}} + 280500000 \, \log \left (5 \, x + 3\right ) - 280500000 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/30*(30672675000000*x^8 + 160520332500000*x^7 + 367435926000000*x^6 + 480493891350000*x^5 + 392612784696000*x
^4 + 205262100529200*x^3 + 67053019228048*x^2 + 12513316868859*x + 1021373267628)/(54675*x^9 + 320760*x^8 + 83
6163*x^7 + 1271214*x^6 + 1242108*x^5 + 808920*x^4 + 351120*x^3 + 97952*x^2 + 15936*x + 1152) + 280500000*log(5
*x + 3) - 280500000*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {30672675000000 \, x^{8} + 160520332500000 \, x^{7} + 367435926000000 \, x^{6} + 480493891350000 \, x^{5} + 392612784696000 \, x^{4} + 205262100529200 \, x^{3} + 67053019228048 \, x^{2} + 12513316868859 \, x + 1021373267628}{30 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{7}} + 280500000 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 280500000 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

1/30*(30672675000000*x^8 + 160520332500000*x^7 + 367435926000000*x^6 + 480493891350000*x^5 + 392612784696000*x
^4 + 205262100529200*x^3 + 67053019228048*x^2 + 12513316868859*x + 1021373267628)/((5*x + 3)^2*(3*x + 2)^7) +
280500000*log(abs(5*x + 3)) - 280500000*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx=\frac {18700000\,x^8+\frac {293590000\,x^7}{3}+\frac {6048328000\,x^6}{27}+\frac {23728093400\,x^5}{81}+\frac {6462761888\,x^4}{27}+\frac {152046000392\,x^3}{1215}+\frac {33526509614024\,x^2}{820125}+\frac {4171105622953\,x}{546750}+\frac {170228877938}{273375}}{x^9+\frac {88\,x^8}{15}+\frac {1147\,x^7}{75}+\frac {15694\,x^6}{675}+\frac {46004\,x^5}{2025}+\frac {5992\,x^4}{405}+\frac {23408\,x^3}{3645}+\frac {97952\,x^2}{54675}+\frac {5312\,x}{18225}+\frac {128}{6075}}-561000000\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

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

[Out]

((4171105622953*x)/546750 + (33526509614024*x^2)/820125 + (152046000392*x^3)/1215 + (6462761888*x^4)/27 + (237
28093400*x^5)/81 + (6048328000*x^6)/27 + (293590000*x^7)/3 + 18700000*x^8 + 170228877938/273375)/((5312*x)/182
25 + (97952*x^2)/54675 + (23408*x^3)/3645 + (5992*x^4)/405 + (46004*x^5)/2025 + (15694*x^6)/675 + (1147*x^7)/7
5 + (88*x^8)/15 + x^9 + 128/6075) - 561000000*atanh(30*x + 19)