3.1419 \(\int \frac{(1-2 x)^3 (2+3 x)^6}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=80 \[ -\frac{5832 x^7}{875}-\frac{3402 x^6}{625}+\frac{134622 x^5}{15625}+\frac{74223 x^4}{12500}-\frac{81747 x^3}{15625}-\frac{915777 x^2}{390625}+\frac{4571416 x}{1953125}-\frac{23232}{9765625 (5 x+3)}-\frac{1331}{19531250 (5 x+3)^2}+\frac{166749 \log (5 x+3)}{9765625} \]

[Out]

(4571416*x)/1953125 - (915777*x^2)/390625 - (81747*x^3)/15625 + (74223*x^4)/12500 + (134622*x^5)/15625 - (3402
*x^6)/625 - (5832*x^7)/875 - 1331/(19531250*(3 + 5*x)^2) - 23232/(9765625*(3 + 5*x)) + (166749*Log[3 + 5*x])/9
765625

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Rubi [A]  time = 0.0439333, antiderivative size = 80, 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} \[ -\frac{5832 x^7}{875}-\frac{3402 x^6}{625}+\frac{134622 x^5}{15625}+\frac{74223 x^4}{12500}-\frac{81747 x^3}{15625}-\frac{915777 x^2}{390625}+\frac{4571416 x}{1953125}-\frac{23232}{9765625 (5 x+3)}-\frac{1331}{19531250 (5 x+3)^2}+\frac{166749 \log (5 x+3)}{9765625} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x)^3,x]

[Out]

(4571416*x)/1953125 - (915777*x^2)/390625 - (81747*x^3)/15625 + (74223*x^4)/12500 + (134622*x^5)/15625 - (3402
*x^6)/625 - (5832*x^7)/875 - 1331/(19531250*(3 + 5*x)^2) - 23232/(9765625*(3 + 5*x)) + (166749*Log[3 + 5*x])/9
765625

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)^6}{(3+5 x)^3} \, dx &=\int \left (\frac{4571416}{1953125}-\frac{1831554 x}{390625}-\frac{245241 x^2}{15625}+\frac{74223 x^3}{3125}+\frac{134622 x^4}{3125}-\frac{20412 x^5}{625}-\frac{5832 x^6}{125}+\frac{1331}{1953125 (3+5 x)^3}+\frac{23232}{1953125 (3+5 x)^2}+\frac{166749}{1953125 (3+5 x)}\right ) \, dx\\ &=\frac{4571416 x}{1953125}-\frac{915777 x^2}{390625}-\frac{81747 x^3}{15625}+\frac{74223 x^4}{12500}+\frac{134622 x^5}{15625}-\frac{3402 x^6}{625}-\frac{5832 x^7}{875}-\frac{1331}{19531250 (3+5 x)^2}-\frac{23232}{9765625 (3+5 x)}+\frac{166749 \log (3+5 x)}{9765625}\\ \end{align*}

Mathematica [A]  time = 0.0366185, size = 73, normalized size = 0.91 \[ \frac{-227812500000 x^9-459421875000 x^8-10783125000 x^7+489359390625 x^6+170737481250 x^5-221653096875 x^4-80532567500 x^3+104273484075 x^2+73328526690 x+23344860 (5 x+3)^2 \log (6 (5 x+3))[phone]}{1367187500 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x)^3,x]

[Out]

(13353609877 + 73328526690*x + 104273484075*x^2 - 80532567500*x^3 - 221653096875*x^4 + 170737481250*x^5 + 4893
59390625*x^6 - 10783125000*x^7 - 459421875000*x^8 - 227812500000*x^9 + 23344860*(3 + 5*x)^2*Log[6*(3 + 5*x)])/
(1367187500*(3 + 5*x)^2)

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Maple [A]  time = 0.006, size = 61, normalized size = 0.8 \begin{align*}{\frac{4571416\,x}{1953125}}-{\frac{915777\,{x}^{2}}{390625}}-{\frac{81747\,{x}^{3}}{15625}}+{\frac{74223\,{x}^{4}}{12500}}+{\frac{134622\,{x}^{5}}{15625}}-{\frac{3402\,{x}^{6}}{625}}-{\frac{5832\,{x}^{7}}{875}}-{\frac{1331}{19531250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{23232}{29296875+48828125\,x}}+{\frac{166749\,\ln \left ( 3+5\,x \right ) }{9765625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(2+3*x)^6/(3+5*x)^3,x)

[Out]

4571416/1953125*x-915777/390625*x^2-81747/15625*x^3+74223/12500*x^4+134622/15625*x^5-3402/625*x^6-5832/875*x^7
-1331/19531250/(3+5*x)^2-23232/9765625/(3+5*x)+166749/9765625*ln(3+5*x)

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Maxima [A]  time = 1.01757, size = 82, normalized size = 1.02 \begin{align*} -\frac{5832}{875} \, x^{7} - \frac{3402}{625} \, x^{6} + \frac{134622}{15625} \, x^{5} + \frac{74223}{12500} \, x^{4} - \frac{81747}{15625} \, x^{3} - \frac{915777}{390625} \, x^{2} + \frac{4571416}{1953125} \, x - \frac{121 \,{\left (1920 \, x + 1163\right )}}{19531250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{166749}{9765625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^6/(3+5*x)^3,x, algorithm="maxima")

[Out]

-5832/875*x^7 - 3402/625*x^6 + 134622/15625*x^5 + 74223/12500*x^4 - 81747/15625*x^3 - 915777/390625*x^2 + 4571
416/1953125*x - 121/19531250*(1920*x + 1163)/(25*x^2 + 30*x + 9) + 166749/9765625*log(5*x + 3)

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Fricas [A]  time = 1.23322, size = 331, normalized size = 4.14 \begin{align*} -\frac{45562500000 \, x^{9} + 91884375000 \, x^{8} + 2156625000 \, x^{7} - 97871878125 \, x^{6} - 34147496250 \, x^{5} + 44330619375 \, x^{4} + 16106513500 \, x^{3} - 13430552100 \, x^{2} - 4668972 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 5756731680 \, x + 1970122}{273437500 \,{\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)^3*(2+3*x)^6/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/273437500*(45562500000*x^9 + 91884375000*x^8 + 2156625000*x^7 - 97871878125*x^6 - 34147496250*x^5 + 4433061
9375*x^4 + 16106513500*x^3 - 13430552100*x^2 - 4668972*(25*x^2 + 30*x + 9)*log(5*x + 3) - 5756731680*x + 19701
22)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.130646, size = 70, normalized size = 0.88 \begin{align*} - \frac{5832 x^{7}}{875} - \frac{3402 x^{6}}{625} + \frac{134622 x^{5}}{15625} + \frac{74223 x^{4}}{12500} - \frac{81747 x^{3}}{15625} - \frac{915777 x^{2}}{390625} + \frac{4571416 x}{1953125} - \frac{232320 x + 140723}{488281250 x^{2} + 585937500 x + 175781250} + \frac{166749 \log{\left (5 x + 3 \right )}}{9765625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**6/(3+5*x)**3,x)

[Out]

-5832*x**7/875 - 3402*x**6/625 + 134622*x**5/15625 + 74223*x**4/12500 - 81747*x**3/15625 - 915777*x**2/390625
+ 4571416*x/1953125 - (232320*x + 140723)/(488281250*x**2 + 585937500*x + 175781250) + 166749*log(5*x + 3)/976
5625

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Giac [A]  time = 3.31787, size = 77, normalized size = 0.96 \begin{align*} -\frac{5832}{875} \, x^{7} - \frac{3402}{625} \, x^{6} + \frac{134622}{15625} \, x^{5} + \frac{74223}{12500} \, x^{4} - \frac{81747}{15625} \, x^{3} - \frac{915777}{390625} \, x^{2} + \frac{4571416}{1953125} \, x - \frac{121 \,{\left (1920 \, x + 1163\right )}}{19531250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{166749}{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)^3*(2+3*x)^6/(3+5*x)^3,x, algorithm="giac")

[Out]

-5832/875*x^7 - 3402/625*x^6 + 134622/15625*x^5 + 74223/12500*x^4 - 81747/15625*x^3 - 915777/390625*x^2 + 4571
416/1953125*x - 121/19531250*(1920*x + 1163)/(5*x + 3)^2 + 166749/9765625*log(abs(5*x + 3))