3.1321 \(\int \frac {(1-2 x)^2 (2+3 x)^8}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=87 \[ \frac {6561 x^8}{250}+\frac {332424 x^7}{4375}+\frac {376407 x^6}{6250}-\frac {74601 x^5}{3125}-\frac {1700919 x^4}{31250}-\frac {5350194 x^3}{390625}+\frac {55559043 x^2}{3906250}+\frac {92582457 x}{9765625}-\frac {572}{9765625 (5 x+3)}-\frac {121}{97656250 (5 x+3)^2}+\frac {5888 \log (5 x+3)}{9765625} \]

[Out]

92582457/9765625*x+55559043/3906250*x^2-5350194/390625*x^3-1700919/31250*x^4-74601/3125*x^5+376407/6250*x^6+33
2424/4375*x^7+6561/250*x^8-121/97656250/(3+5*x)^2-572/9765625/(3+5*x)+5888/9765625*ln(3+5*x)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, 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} \[ \frac {6561 x^8}{250}+\frac {332424 x^7}{4375}+\frac {376407 x^6}{6250}-\frac {74601 x^5}{3125}-\frac {1700919 x^4}{31250}-\frac {5350194 x^3}{390625}+\frac {55559043 x^2}{3906250}+\frac {92582457 x}{9765625}-\frac {572}{9765625 (5 x+3)}-\frac {121}{97656250 (5 x+3)^2}+\frac {5888 \log (5 x+3)}{9765625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(92582457*x)/9765625 + (55559043*x^2)/3906250 - (5350194*x^3)/390625 - (1700919*x^4)/31250 - (74601*x^5)/3125
+ (376407*x^6)/6250 + (332424*x^7)/4375 + (6561*x^8)/250 - 121/(97656250*(3 + 5*x)^2) - 572/(9765625*(3 + 5*x)
) + (5888*Log[3 + 5*x])/9765625

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 {92582457}{9765625}+\frac {55559043 x}{1953125}-\frac {16050582 x^2}{390625}-\frac {3401838 x^3}{15625}-\frac {74601 x^4}{625}+\frac {1129221 x^5}{3125}+\frac {332424 x^6}{625}+\frac {26244 x^7}{125}+\frac {121}{9765625 (3+5 x)^3}+\frac {572}{1953125 (3+5 x)^2}+\frac {5888}{1953125 (3+5 x)}\right ) \, dx\\ &=\frac {92582457 x}{9765625}+\frac {55559043 x^2}{3906250}-\frac {5350194 x^3}{390625}-\frac {1700919 x^4}{31250}-\frac {74601 x^5}{3125}+\frac {376407 x^6}{6250}+\frac {332424 x^7}{4375}+\frac {6561 x^8}{250}-\frac {121}{97656250 (3+5 x)^2}-\frac {572}{9765625 (3+5 x)}+\frac {5888 \log (3+5 x)}{9765625}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 76, normalized size = 0.87 \[ \frac {448505859375 x^{10}[phone] x^9+2748937500000 x^8+1294582500000 x^7-1049233500000 x^6-1497169800000 x^5-372682800000 x^4+369438720000 x^3+310701230325 x^2+92853841190 x+412160 (5 x+3)^2 \log (5 x+3)[phone]}{683593750 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10358007077 + 92853841190*x + 310701230325*x^2 + 369438720000*x^3 - 372682800000*x^4 - 1497169800000*x^5 - 10
49233500000*x^6 + 1294582500000*x^7 + 2748937500000*x^8 + 1836738281250*x^9 + 448505859375*x^10 + 412160*(3 +
5*x)^2*Log[3 + 5*x])/(683593750*(3 + 5*x)^2)

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 82, normalized size = 0.94 \[ \frac {448505859375 \, x^{10} + 1836738281250 \, x^{9} + 2748937500000 \, x^{8} + 1294582500000 \, x^{7} - 1049233500000 \, x^{6} - 1497169800000 \, x^{5} - 372682800000 \, x^{4} + 369438720000 \, x^{3} + 281928652425 \, x^{2} + 412160 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 58326747710 \, x - 120967}{683593750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

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]

1/683593750*(448505859375*x^10 + 1836738281250*x^9 + 2748937500000*x^8 + 1294582500000*x^7 - 1049233500000*x^6
- 1497169800000*x^5 - 372682800000*x^4 + 369438720000*x^3 + 281928652425*x^2 + 412160*(25*x^2 + 30*x + 9)*log
(5*x + 3) + 58326747710*x - 120967)/(25*x^2 + 30*x + 9)

________________________________________________________________________________________

giac [A]  time = 0.95, size = 62, normalized size = 0.71 \[ \frac {6561}{250} \, x^{8} + \frac {332424}{4375} \, x^{7} + \frac {376407}{6250} \, x^{6} - \frac {74601}{3125} \, x^{5} - \frac {1700919}{31250} \, x^{4} - \frac {5350194}{390625} \, x^{3} + \frac {55559043}{3906250} \, x^{2} + \frac {92582457}{9765625} \, x - \frac {11 \, {\left (2600 \, x + 1571\right )}}{97656250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {5888}{9765625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

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]

6561/250*x^8 + 332424/4375*x^7 + 376407/6250*x^6 - 74601/3125*x^5 - 1700919/31250*x^4 - 5350194/390625*x^3 + 5
5559043/3906250*x^2 + 92582457/9765625*x - 11/97656250*(2600*x + 1571)/(5*x + 3)^2 + 5888/9765625*log(abs(5*x
+ 3))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 66, normalized size = 0.76 \[ \frac {6561 x^{8}}{250}+\frac {332424 x^{7}}{4375}+\frac {376407 x^{6}}{6250}-\frac {74601 x^{5}}{3125}-\frac {1700919 x^{4}}{31250}-\frac {5350194 x^{3}}{390625}+\frac {55559043 x^{2}}{3906250}+\frac {92582457 x}{9765625}+\frac {5888 \ln \left (5 x +3\right )}{9765625}-\frac {121}{97656250 \left (5 x +3\right )^{2}}-\frac {572}{9765625 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

92582457/9765625*x+55559043/3906250*x^2-5350194/390625*x^3-1700919/31250*x^4-74601/3125*x^5+376407/6250*x^6+33
2424/4375*x^7+6561/250*x^8-121/97656250/(5*x+3)^2-572/9765625/(5*x+3)+5888/9765625*ln(5*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.58, size = 66, normalized size = 0.76 \[ \frac {6561}{250} \, x^{8} + \frac {332424}{4375} \, x^{7} + \frac {376407}{6250} \, x^{6} - \frac {74601}{3125} \, x^{5} - \frac {1700919}{31250} \, x^{4} - \frac {5350194}{390625} \, x^{3} + \frac {55559043}{3906250} \, x^{2} + \frac {92582457}{9765625} \, x - \frac {11 \, {\left (2600 \, x + 1571\right )}}{97656250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {5888}{9765625} \, \log \left (5 \, x + 3\right ) \]

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]

6561/250*x^8 + 332424/4375*x^7 + 376407/6250*x^6 - 74601/3125*x^5 - 1700919/31250*x^4 - 5350194/390625*x^3 + 5
5559043/3906250*x^2 + 92582457/9765625*x - 11/97656250*(2600*x + 1571)/(25*x^2 + 30*x + 9) + 5888/9765625*log(
5*x + 3)

________________________________________________________________________________________

mupad [B]  time = 1.11, size = 62, normalized size = 0.71 \[ \frac {92582457\,x}{9765625}+\frac {5888\,\ln \left (x+\frac {3}{5}\right )}{9765625}-\frac {\frac {572\,x}{48828125}+\frac {17281}{2441406250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\frac {55559043\,x^2}{3906250}-\frac {5350194\,x^3}{390625}-\frac {1700919\,x^4}{31250}-\frac {74601\,x^5}{3125}+\frac {376407\,x^6}{6250}+\frac {332424\,x^7}{4375}+\frac {6561\,x^8}{250} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(92582457*x)/9765625 + (5888*log(x + 3/5))/9765625 - ((572*x)/48828125 + 17281/2441406250)/((6*x)/5 + x^2 + 9/
25) + (55559043*x^2)/3906250 - (5350194*x^3)/390625 - (1700919*x^4)/31250 - (74601*x^5)/3125 + (376407*x^6)/62
50 + (332424*x^7)/4375 + (6561*x^8)/250

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 78, normalized size = 0.90 \[ \frac {6561 x^{8}}{250} + \frac {332424 x^{7}}{4375} + \frac {376407 x^{6}}{6250} - \frac {74601 x^{5}}{3125} - \frac {1700919 x^{4}}{31250} - \frac {5350194 x^{3}}{390625} + \frac {55559043 x^{2}}{3906250} + \frac {92582457 x}{9765625} + \frac {- 28600 x - 17281}{2441406250 x^{2} + 2929687500 x + 878906250} + \frac {5888 \log {\left (5 x + 3 \right )}}{9765625} \]

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]

6561*x**8/250 + 332424*x**7/4375 + 376407*x**6/6250 - 74601*x**5/3125 - 1700919*x**4/31250 - 5350194*x**3/3906
25 + 55559043*x**2/3906250 + 92582457*x/9765625 + (-28600*x - 17281)/(2441406250*x**2 + 2929687500*x + 8789062
50) + 5888*log(5*x + 3)/9765625

________________________________________________________________________________________