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*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

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Rubi [A]  time = 0.0474229, antiderivative size = 87, 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} \[ \frac{6561 x^8}{250}+\frac{332424 x^7}{4375}+\frac{376407 x^6}{6250}-\frac{74601 x^5}{3125}-\frac{1700919 x^4}{31250}-\frac{5350194 x^3}{390625}+\frac{55559043 x^2}{3906250}+\frac{92582457 x}{9765625}-\frac{572}{9765625 (5 x+3)}-\frac{121}{97656250 (5 x+3)^2}+\frac{5888 \log (5 x+3)}{9765625} \]

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 = 0.0302863, 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)

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Maple [A]  time = 0.007, size = 66, normalized size = 0.8 \begin{align*}{\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\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{572}{29296875+48828125\,x}}+{\frac{5888\,\ln \left ( 3+5\,x \right ) }{9765625}} \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]

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)

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Maxima [A]  time = 1.49222, size = 89, normalized size = 1.02 \begin{align*} \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 ) \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]

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)

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Fricas [A]  time = 1.33491, size = 374, normalized size = 4.3 \begin{align*} \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 )}} \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]

1/683593750*(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)

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Sympy [A]  time = 0.132468, size = 76, normalized size = 0.87 \begin{align*} \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} \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]

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 + 87890625
0) + 5888*log(5*x + 3)/9765625

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Giac [A]  time = 2.90283, size = 84, normalized size = 0.97 \begin{align*} \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 ) \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]

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))