\(\int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx\) [1402]

Optimal result

Integrand size = 22, antiderivative size = 92 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx=\frac {49}{27 (2+3 x)^7}+\frac {1421}{162 (2+3 x)^6}+\frac {7189}{135 (2+3 x)^5}+\frac {1331}{4 (2+3 x)^4}+\frac {6655}{3 (2+3 x)^3}+\frac {33275}{2 (2+3 x)^2}+\frac {166375}{2+3 x}-831875 \log (2+3 x)+831875 \log (3+5 x) \]

[Out]

49/27/(2+3*x)^7+1421/162/(2+3*x)^6+7189/135/(2+3*x)^5+1331/4/(2+3*x)^4+6655/3/(2+3*x)^3+33275/2/(2+3*x)^2+1663
75/(2+3*x)-831875*ln(2+3*x)+831875*ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 92, 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} \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx=\frac {166375}{3 x+2}+\frac {33275}{2 (3 x+2)^2}+\frac {6655}{3 (3 x+2)^3}+\frac {1331}{4 (3 x+2)^4}+\frac {7189}{135 (3 x+2)^5}+\frac {1421}{162 (3 x+2)^6}+\frac {49}{27 (3 x+2)^7}-831875 \log (3 x+2)+831875 \log (5 x+3) \]

[In]

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

[Out]

49/(27*(2 + 3*x)^7) + 1421/(162*(2 + 3*x)^6) + 7189/(135*(2 + 3*x)^5) + 1331/(4*(2 + 3*x)^4) + 6655/(3*(2 + 3*
x)^3) + 33275/(2*(2 + 3*x)^2) + 166375/(2 + 3*x) - 831875*Log[2 + 3*x] + 831875*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}{9 (2+3 x)^8}-\frac {1421}{9 (2+3 x)^7}-\frac {7189}{9 (2+3 x)^6}-\frac {3993}{(2+3 x)^5}-\frac {19965}{(2+3 x)^4}-\frac {99825}{(2+3 x)^3}-\frac {499125}{(2+3 x)^2}-\frac {2495625}{2+3 x}+\frac {4159375}{3+5 x}\right ) \, dx \\ & = \frac {49}{27 (2+3 x)^7}+\frac {1421}{162 (2+3 x)^6}+\frac {7189}{135 (2+3 x)^5}+\frac {1331}{4 (2+3 x)^4}+\frac {6655}{3 (2+3 x)^3}+\frac {33275}{2 (2+3 x)^2}+\frac {166375}{2+3 x}-831875 \log (2+3 x)+831875 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx=\frac {2940+14210 (2+3 x)+86268 (2+3 x)^2+539055 (2+3 x)^3+3593700 (2+3 x)^4+26952750 (2+3 x)^5+269527500 (2+3 x)^6}{1620 (2+3 x)^7}-831875 \log (5 (2+3 x))+831875 \log (3+5 x) \]

[In]

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

[Out]

(2940 + 14210*(2 + 3*x) + 86268*(2 + 3*x)^2 + 539055*(2 + 3*x)^3 + 3593700*(2 + 3*x)^4 + 26952750*(2 + 3*x)^5
+ 269527500*(2 + 3*x)^6)/(1620*(2 + 3*x)^7) - 831875*Log[5*(2 + 3*x)] + 831875*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.61

method	result	size
norman	\(\frac {121287375 x^{6}[phone] x^{4}+\frac {978384825}{2} x^{5}+\frac {2948786577}{4} x^{3}+\frac {11155398233}{30} x^{2}+\frac {27013663171}{270} x +\frac {4543609018}{405}}{\left (2+3 x \right )^{7}}-831875 \ln \left (2+3 x \right )+831875 \ln \left (3+5 x \right )\)	\(56\)
risch	\(\frac {121287375 x^{6}[phone] x^{4}+\frac {978384825}{2} x^{5}+\frac {2948786577}{4} x^{3}+\frac {11155398233}{30} x^{2}+\frac {27013663171}{270} x +\frac {4543609018}{405}}{\left (2+3 x \right )^{7}}-831875 \ln \left (2+3 x \right )+831875 \ln \left (3+5 x \right )\)	\(57\)
default	\(\frac {49}{27 \left (2+3 x \right )^{7}}+\frac {1421}{162 \left (2+3 x \right )^{6}}+\frac {7189}{135 \left (2+3 x \right )^{5}}+\frac {1331}{4 \left (2+3 x \right )^{4}}+\frac {6655}{3 \left (2+3 x \right )^{3}}+\frac {33275}{2 \left (2+3 x \right )^{2}}+\frac {166375}{2+3 x}-831875 \ln \left (2+3 x \right )+831875 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(-\frac {34073599680 x -9659865600000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-2146636800000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-3493076400000 \ln \left (x +\frac {3}{5}\right ) x^{7}-204441600000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-24149664000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-32602046400000 \ln \left (x +\frac {3}{5}\right ) x^{5}-36224496000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-16301023200000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{1920 \left (2+3 x \right )^{7}}\)	\(178\)

[In]

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

[Out]

(121287375*x^6+822238560*x^4+978384825/2*x^5+2948786577/4*x^3+11155398233/30*x^2+27013663171/270*x+4543609018/
405)/(2+3*x)^7-831875*ln(2+3*x)+831875*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.68 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx=\frac {196485547500 \, x^{6} + 792491708250 \, x^{5} + 1332026467200 \, x^{4} + 1194258563685 \, x^{3} + 602391504582 \, x^{2} + 1347637500 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (5 \, x + 3\right ) - 1347637500 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 162081979026 \, x + 18174436072}{1620 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[In]

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

[Out]

1/1620*(196485547500*x^6 + 792491708250*x^5 + 1332026467200*x^4 + 1194258563685*x^3 + 602391504582*x^2 + 13476
37500*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(5*x + 3) - 1347
637500*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(3*x + 2) + 162
081979026*x + 18174436072)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128
)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx=- \frac {- 196485547500 x^{6} - 792491708250 x^{5} - 1332026467200 x^{4} - 1194258563685 x^{3} - 602391504582 x^{2} - 162081979026 x - 18174436072}{3542940 x^{7} + 16533720 x^{6} + 33067440 x^{5} + 36741600 x^{4} + 24494400 x^{3} + 9797760 x^{2} + 2177280 x + 207360} + 831875 \log {\left (x + \frac {3}{5} \right )} - 831875 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

-(-196485547500*x**6 - 792491708250*x**5 - 1332026467200*x**4 - 1194258563685*x**3 - 602391504582*x**2 - 16208
1979026*x - 18174436072)/(3542940*x**7 + 16533720*x**6 + 33067440*x**5 + 36741600*x**4 + 24494400*x**3 + 97977
60*x**2 + 2177280*x + 207360) + 831875*log(x + 3/5) - 831875*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx=\frac {196485547500 \, x^{6} + 792491708250 \, x^{5} + 1332026467200 \, x^{4} + 1194258563685 \, x^{3} + 602391504582 \, x^{2} + 162081979026 \, x + 18174436072}{1620 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + 831875 \, \log \left (5 \, x + 3\right ) - 831875 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/1620*(196485547500*x^6 + 792491708250*x^5 + 1332026467200*x^4 + 1194258563685*x^3 + 602391504582*x^2 + 16208
1979026*x + 18174436072)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
+ 831875*log(5*x + 3) - 831875*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx=\frac {196485547500 \, x^{6} + 792491708250 \, x^{5} + 1332026467200 \, x^{4} + 1194258563685 \, x^{3} + 602391504582 \, x^{2} + 162081979026 \, x + 18174436072}{1620 \, {\left (3 \, x + 2\right )}^{7}} + 831875 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 831875 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

1/1620*(196485547500*x^6 + 792491708250*x^5 + 1332026467200*x^4 + 1194258563685*x^3 + 602391504582*x^2 + 16208
1979026*x + 18174436072)/(3*x + 2)^7 + 831875*log(abs(5*x + 3)) - 831875*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)} \, dx=\frac {\frac {166375\,x^6}{3}+\frac {4026275\,x^5}{18}+\frac {30453280\,x^4}{81}+\frac {327642953\,x^3}{972}+\frac {11155398233\,x^2}{65610}+\frac {27013663171\,x}{590490}+\frac {4543609018}{885735}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}}-1663750\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

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

[Out]

((27013663171*x)/590490 + (11155398233*x^2)/65610 + (327642953*x^3)/972 + (30453280*x^4)/81 + (4026275*x^5)/18
+ (166375*x^6)/3 + 4543609018/885735)/((448*x)/729 + (224*x^2)/81 + (560*x^3)/81 + (280*x^4)/27 + (28*x^5)/3
+ (14*x^6)/3 + x^7 + 128/2187) - 1663750*atanh(30*x + 19)