3.1433 \(\int \frac{(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx\)

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) - 831875/(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 = 0.0612491, 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, Rules used = {88} \[ \frac{43848750}{3 x+2}+\frac{20418750}{5 x+3}+\frac{6618975}{2 (3 x+2)^2}-\frac{831875}{2 (5 x+3)^2}+\frac{317845}{(3 x+2)^3}+\frac{64317}{2 (3 x+2)^4}+\frac{15708}{5 (3 x+2)^5}+\frac{539}{2 (3 x+2)^6}+\frac{49}{3 (3 x+2)^7}-280500000 \log (3 x+2)[phone] \log (5 x+3) \]

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) - 831875/(2*(3 + 5*x)^2) + 20418750/(3 + 5*x) - 280500000*Log[2
+ 3*x] + 280500000*Log[3 + 5*x]

Rule 88

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*} \int \frac{(1-2 x)^3}{(2+3 x)^8 (3+5 x)^3} \, dx &=\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]  time = 0.0971567, 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) - 831875/(2*(3 + 5*x)^2) + 20418750/(3 + 5*x) - 280500000*Log[5
*(2 + 3*x)] + 280500000*Log[3 + 5*x]

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Maple [A]  time = 0.01, size = 99, normalized size = 0.9 \begin{align*}{\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 ) \end{align*}

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+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)

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Maxima [A]  time = 1.48417, size = 143, normalized size = 1.3 \begin{align*} \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 ) \end{align*}

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, 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)

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Fricas [A]  time = 1.39402, size = 821, normalized size = 7.46 \begin{align*} \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 )}} \end{align*}

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, 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)

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Sympy [A]  time = 0.260878, size = 102, normalized size = 0.93 \begin{align*} \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 )} \end{align*}

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 + 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)

________________________________________________________________________________________

Giac [A]  time = 2.58471, size = 101, normalized size = 0.92 \begin{align*} \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 ) \end{align*}

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, 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))