3.1337 \(\int \frac{(1-2 x)^2}{(2+3 x)^8 (3+5 x)^3} \, dx\)

Optimal. Leaf size=110 \[ \frac{19637500}{3 x+2}+\frac{9212500}{5 x+3}+\frac{2958125}{2 (3 x+2)^2}-\frac{378125}{2 (5 x+3)^2}+\frac{424975}{3 (3 x+2)^3}+\frac{28555}{2 (3 x+2)^4}+\frac{6934}{5 (3 x+2)^5}+\frac{707}{6 (3 x+2)^6}+\frac{7}{(3 x+2)^7}-125825000 \log (3 x+2)[phone] \log (5 x+3) \]

[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]

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Rubi [A]  time = 0.059523, 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} \[ \frac{19637500}{3 x+2}+\frac{9212500}{5 x+3}+\frac{2958125}{2 (3 x+2)^2}-\frac{378125}{2 (5 x+3)^2}+\frac{424975}{3 (3 x+2)^3}+\frac{28555}{2 (3 x+2)^4}+\frac{6934}{5 (3 x+2)^5}+\frac{707}{6 (3 x+2)^6}+\frac{7}{(3 x+2)^7}-125825000 \log (3 x+2)[phone] \log (5 x+3) \]

Antiderivative was successfully verified.

[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 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)^2}{(2+3 x)^8 (3+5 x)^3} \, dx &=\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]  time = 0.0764345, size = 112, normalized size = 1.02 \[ \frac{19637500}{3 x+2}+\frac{9212500}{5 x+3}+\frac{2958125}{2 (3 x+2)^2}-\frac{378125}{2 (5 x+3)^2}+\frac{424975}{3 (3 x+2)^3}+\frac{28555}{2 (3 x+2)^4}+\frac{6934}{5 (3 x+2)^5}+\frac{707}{6 (3 x+2)^6}+\frac{7}{(3 x+2)^7}-125825000 \log (5 (3 x+2))[phone] \log (5 x+3) \]

Antiderivative was successfully verified.

[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]

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Maple [A]  time = 0.008, size = 99, normalized size = 0.9 \begin{align*} 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}}}[phone]\, \left ( 2+3\,x \right ) ^{-1}-{\frac{378125}{2\, \left ( 3+5\,x \right ) ^{2}}}+9212500\, \left ( 3+5\,x \right ) ^{-1}-125825000\,\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)^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*ln(2+3*x)[phone]*ln(3+5*x)

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Maxima [A]  time = 1.72165, size = 143, normalized size = 1.3 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.32137, size = 810, normalized size = 7.36 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.244331, size = 102, normalized size = 0.93 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 2.27206, size = 101, normalized size = 0.92 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))