3.1416 \(\int \frac{(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx\)

Optimal. Leaf size=110 \[ \frac{43848750}{3 x+2}+\frac{20418750}{5 x+3}+\frac{6618975}{2 (3 x+2)^2}-\frac{831875}{2 (5 x+3)^2}+\frac{317845}{(3 x+2)^3}+\frac{64317}{2 (3 x+2)^4}+\frac{15708}{5 (3 x+2)^5}+\frac{539}{2 (3 x+2)^6}+\frac{49}{3 (3 x+2)^7}-280500000 \log (3 x+2)[phone] \log (5 x+3) \]

[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) - 8
31875/(2*(3 + 5*x)^2) + 20418750/(3 + 5*x) - 280500000*Log[2 + 3*x] + 280500000*
Log[3 + 5*x]

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Rubi [A]  time = 0.141336, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{43848750}{3 x+2}+\frac{20418750}{5 x+3}+\frac{6618975}{2 (3 x+2)^2}-\frac{831875}{2 (5 x+3)^2}+\frac{317845}{(3 x+2)^3}+\frac{64317}{2 (3 x+2)^4}+\frac{15708}{5 (3 x+2)^5}+\frac{539}{2 (3 x+2)^6}+\frac{49}{3 (3 x+2)^7}-280500000 \log (3 x+2)[phone] \log (5 x+3) \]

Antiderivative was successfully verified.

[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) - 8
31875/(2*(3 + 5*x)^2) + 20418750/(3 + 5*x) - 280500000*Log[2 + 3*x] + 280500000*
Log[3 + 5*x]

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Rubi in Sympy [A]  time = 17.2317, size = 99, normalized size = 0.9 \[ - 280500000 \log{\left (3 x + 2 \right )} + 280500000 \log{\left (5 x + 3 \right )} + \frac{20418750}{5 x + 3} - \frac{831875}{2 \left (5 x + 3\right )^{2}} + \frac{43848750}{3 x + 2} + \frac{6618975}{2 \left (3 x + 2\right )^{2}} + \frac{317845}{\left (3 x + 2\right )^{3}} + \frac{64317}{2 \left (3 x + 2\right )^{4}} + \frac{15708}{5 \left (3 x + 2\right )^{5}} + \frac{539}{2 \left (3 x + 2\right )^{6}} + \frac{49}{3 \left (3 x + 2\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**3/(2+3*x)**8/(3+5*x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.205873, size = 112, normalized size = 1.02 \[ \frac{43848750}{3 x+2}+\frac{20418750}{5 x+3}+\frac{6618975}{2 (3 x+2)^2}-\frac{831875}{2 (5 x+3)^2}+\frac{317845}{(3 x+2)^3}+\frac{64317}{2 (3 x+2)^4}+\frac{15708}{5 (3 x+2)^5}+\frac{539}{2 (3 x+2)^6}+\frac{49}{3 (3 x+2)^7}-280500000 \log (5 (3 x+2))[phone] \log (5 x+3) \]

Antiderivative was successfully verified.

[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) - 8
31875/(2*(3 + 5*x)^2) + 20418750/(3 + 5*x) - 280500000*Log[5*(2 + 3*x)] + 280500
000*Log[3 + 5*x]

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Maple [A]  time = 0.016, size = 99, normalized size = 0.9 \[{\frac{49}{3\, \left ( 2+3\,x \right ) ^{7}}}+{\frac{539}{2\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{15708}{5\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{64317}{2\, \left ( 2+3\,x \right ) ^{4}}}+317845\, \left ( 2+3\,x \right ) ^{-3}+{\frac{6618975}{2\, \left ( 2+3\,x \right ) ^{2}}}[phone]\, \left ( 2+3\,x \right ) ^{-1}-{\frac{831875}{2\, \left ( 3+5\,x \right ) ^{2}}}[phone]\, \left ( 3+5\,x \right ) ^{-1}-280500000\,\ln \left ( 2+3\,x \right ) [phone]\,\ln \left ( 3+5\,x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)-280
500000*ln(2+3*x)[phone]*ln(3+5*x)

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Maxima [A]  time = 1.34831, size = 143, normalized size = 1.3 \[ \frac{30672675000000 \, x^{8} + 160520332500000 \, x^{7} + 367435926000000 \, x^{6} + 480493891350000 \, x^{5} + 392612784696000 \, x^{4} + 205262100529200 \, x^{3} + 67053019228048 \, x^{2} + 12513316868859 \, x + 1021373267628}{30 \,{\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )}} + 280500000 \, \log \left (5 \, x + 3\right ) - 280500000 \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(30672675000000*x^8 + 160520332500000*x^7 + 367435926000000*x^6 + 480493891
350000*x^5 + 392612784696000*x^4 + 205262100529200*x^3 + 67053019228048*x^2 + 12
513316868859*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) + 28050
0000*log(5*x + 3) - 280500000*log(3*x + 2)

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Fricas [A]  time = 0.210544, size = 263, normalized size = 2.39 \[ \frac{30672675000000 \, x^{8} + 160520332500000 \, x^{7} + 367435926000000 \, x^{6} + 480493891350000 \, x^{5} + 392612784696000 \, x^{4} + 205262100529200 \, x^{3} + 67053019228048 \, x^{2} + 8415000000 \,{\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )} \log \left (5 \, x + 3\right ) - 8415000000 \,{\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )} \log \left (3 \, x + 2\right ) + 12513316868859 \, x + 1021373267628}{30 \,{\left (54675 \, x^{9} + 320760 \, x^{8} + 836163 \, x^{7} + 1271214 \, x^{6} + 1242108 \, x^{5} + 808920 \, x^{4} + 351120 \, x^{3} + 97952 \, x^{2} + 15936 \, x + 1152\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(30672675000000*x^8 + 160520332500000*x^7 + 367435926000000*x^6 + 480493891
350000*x^5 + 392612784696000*x^4 + 205262100529200*x^3 + 67053019228048*x^2 + 84
15000000*(54675*x^9 + 320760*x^8 + 836163*x^7 + 1271214*x^6 + 1242108*x^5 + 8089
20*x^4 + 351120*x^3 + 97952*x^2 + 15936*x + 1152)*log(5*x + 3) - 8415000000*(546
75*x^9 + 320760*x^8 + 836163*x^7 + 1271214*x^6 + 1242108*x^5 + 808920*x^4 + 3511
20*x^3 + 97952*x^2 + 15936*x + 1152)*log(3*x + 2) + 12513316868859*x + 102137326
7628)/(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)

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Sympy [A]  time = 0.796429, size = 102, normalized size = 0.93 \[ \frac{30672675000000 x^{8} + 160520332500000 x^{7} + 367435926000000 x^{6} + 480493891350000 x^{5} + 392612784696000 x^{4} + 205262100529200 x^{3} + 67053019228048 x^{2} + 12513316868859 x + 1021373267628}{1640250 x^{9} + 9622800 x^{8} + 25084890 x^{7} + 38136420 x^{6} + 37263240 x^{5} + 24267600 x^{4} + 10533600 x^{3} + 2938560 x^{2} + 478080 x + 34560} + 280500000 \log{\left (x + \frac{3}{5} \right )} - 280500000 \log{\left (x + \frac{2}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 + 48049389135
0000*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)

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GIAC/XCAS [A]  time = 0.210887, size = 101, normalized size = 0.92 \[ \frac{30672675000000 \, x^{8} + 160520332500000 \, x^{7} + 367435926000000 \, x^{6} + 480493891350000 \, x^{5} + 392612784696000 \, x^{4} + 205262100529200 \, x^{3} + 67053019228048 \, x^{2} + 12513316868859 \, x + 1021373267628}{30 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{7}} + 280500000 \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ) - 280500000 \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(30672675000000*x^8 + 160520332500000*x^7 + 367435926000000*x^6 + 480493891
350000*x^5 + 392612784696000*x^4 + 205262100529200*x^3 + 67053019228048*x^2 + 12
513316868859*x + 1021373267628)/((5*x + 3)^2*(3*x + 2)^7) + 280500000*ln(abs(5*x
+ 3)) - 280500000*ln(abs(3*x + 2))