3.1403 \(\int \frac{(1-2 x)^3 (2+3 x)^6}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac{729 x^8}{25}-\frac{37908 x^7}{875}+\frac{12231 x^6}{625}+\frac{774981 x^5}{15625}-\frac{5643 x^4}{3125}-\frac{1836723 x^3}{78125}-\frac{461623 x^2}{390625}+\frac{13880997 x}{1953125}-\frac{1331}{9765625 (5 x+3)}+\frac{23232 \log (5 x+3)}{9765625} \]

[Out]

(13880997*x)/1953125 - (461623*x^2)/390625 - (1836723*x^3)/78125 - (5643*x^4)/3125 + (774981*x^5)/15625 + (122
31*x^6)/625 - (37908*x^7)/875 - (729*x^8)/25 - 1331/(9765625*(3 + 5*x)) + (23232*Log[3 + 5*x])/9765625

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Rubi [A]  time = 0.0387137, antiderivative size = 76, 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} \[ -\frac{729 x^8}{25}-\frac{37908 x^7}{875}+\frac{12231 x^6}{625}+\frac{774981 x^5}{15625}-\frac{5643 x^4}{3125}-\frac{1836723 x^3}{78125}-\frac{461623 x^2}{390625}+\frac{13880997 x}{1953125}-\frac{1331}{9765625 (5 x+3)}+\frac{23232 \log (5 x+3)}{9765625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13880997*x)/1953125 - (461623*x^2)/390625 - (1836723*x^3)/78125 - (5643*x^4)/3125 + (774981*x^5)/15625 + (122
31*x^6)/625 - (37908*x^7)/875 - (729*x^8)/25 - 1331/(9765625*(3 + 5*x)) + (23232*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)^3 (2+3 x)^6}{(3+5 x)^2} \, dx &=\int \left (\frac{13880997}{1953125}-\frac{923246 x}{390625}-\frac{5510169 x^2}{78125}-\frac{22572 x^3}{3125}+\frac{774981 x^4}{3125}+\frac{73386 x^5}{625}-\frac{37908 x^6}{125}-\frac{5832 x^7}{25}+\frac{1331}{1953125 (3+5 x)^2}+\frac{23232}{1953125 (3+5 x)}\right ) \, dx\\ &=\frac{13880997 x}{1953125}-\frac{461623 x^2}{390625}-\frac{1836723 x^3}{78125}-\frac{5643 x^4}{3125}+\frac{774981 x^5}{15625}+\frac{12231 x^6}{625}-\frac{37908 x^7}{875}-\frac{729 x^8}{25}-\frac{1331}{9765625 (3+5 x)}+\frac{23232 \log (3+5 x)}{9765625}\\ \end{align*}

Mathematica [A]  time = 0.0349585, size = 71, normalized size = 0.93 \[ \frac{-49833984375 x^9-103939453125 x^8-10979296875 x^7+104830031250 x^6+47772112500 x^5-42029925000 x^4-26126590000 x^3+10934112000 x^2+10818777780 x+813120 (5 x+3) \log (6 (5 x+3))[phone]}{341796875 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2118706028 + 10818777780*x + 10934112000*x^2 - 26126590000*x^3 - 42029925000*x^4 + 47772112500*x^5 + 10483003
1250*x^6 - 10979296875*x^7 - 103939453125*x^8 - 49833984375*x^9 + 813120*(3 + 5*x)*Log[6*(3 + 5*x)])/(34179687
5*(3 + 5*x))

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Maple [A]  time = 0.005, size = 57, normalized size = 0.8 \begin{align*}{\frac{13880997\,x}{1953125}}-{\frac{461623\,{x}^{2}}{390625}}-{\frac{1836723\,{x}^{3}}{78125}}-{\frac{5643\,{x}^{4}}{3125}}+{\frac{774981\,{x}^{5}}{15625}}+{\frac{12231\,{x}^{6}}{625}}-{\frac{37908\,{x}^{7}}{875}}-{\frac{729\,{x}^{8}}{25}}-{\frac{1331}{29296875+48828125\,x}}+{\frac{23232\,\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)^2,x)

[Out]

13880997/1953125*x-461623/390625*x^2-1836723/78125*x^3-5643/3125*x^4+774981/15625*x^5+12231/625*x^6-37908/875*
x^7-729/25*x^8-1331/9765625/(3+5*x)+23232/9765625*ln(3+5*x)

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Maxima [A]  time = 1.01381, size = 76, normalized size = 1. \begin{align*} -\frac{729}{25} \, x^{8} - \frac{37908}{875} \, x^{7} + \frac{12231}{625} \, x^{6} + \frac{774981}{15625} \, x^{5} - \frac{5643}{3125} \, x^{4} - \frac{1836723}{78125} \, x^{3} - \frac{461623}{390625} \, x^{2} + \frac{13880997}{1953125} \, x - \frac{1331}{9765625 \,{\left (5 \, x + 3\right )}} + \frac{23232}{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)^2,x, algorithm="maxima")

[Out]

-729/25*x^8 - 37908/875*x^7 + 12231/625*x^6 + 774981/15625*x^5 - 5643/3125*x^4 - 1836723/78125*x^3 - 461623/39
0625*x^2 + 13880997/1953125*x - 1331/9765625/(5*x + 3) + 23232/9765625*log(5*x + 3)

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Fricas [A]  time = 1.25449, size = 290, normalized size = 3.82 \begin{align*} -\frac{9966796875 \, x^{9} + 20787890625 \, x^{8} + 2195859375 \, x^{7} - 20966006250 \, x^{6} - 9554422500 \, x^{5} + 8405985000 \, x^{4} + 5225318000 \, x^{3} - 2186822400 \, x^{2} - 162624 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 1457504685 \, x + 9317}{68359375 \,{\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)^2,x, algorithm="fricas")

[Out]

-1/68359375*(9966796875*x^9 + 20787890625*x^8 + 2195859375*x^7 - 20966006250*x^6 - 9554422500*x^5 + 8405985000
*x^4 + 5225318000*x^3 - 2186822400*x^2 - 162624*(5*x + 3)*log(5*x + 3) - 1457504685*x + 9317)/(5*x + 3)

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Sympy [A]  time = 0.115776, size = 68, normalized size = 0.89 \begin{align*} - \frac{729 x^{8}}{25} - \frac{37908 x^{7}}{875} + \frac{12231 x^{6}}{625} + \frac{774981 x^{5}}{15625} - \frac{5643 x^{4}}{3125} - \frac{1836723 x^{3}}{78125} - \frac{461623 x^{2}}{390625} + \frac{13880997 x}{1953125} + \frac{23232 \log{\left (5 x + 3 \right )}}{9765625} - \frac{1331}{48828125 x + 29296875} \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)**2,x)

[Out]

-729*x**8/25 - 37908*x**7/875 + 12231*x**6/625 + 774981*x**5/15625 - 5643*x**4/3125 - 1836723*x**3/78125 - 461
623*x**2/390625 + 13880997*x/1953125 + 23232*log(5*x + 3)/9765625 - 1331/(48828125*x + 29296875)

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Giac [A]  time = 3.19128, size = 138, normalized size = 1.82 \begin{align*} \frac{1}{341796875} \,{\left (5 \, x + 3\right )}^{8}{\left (\frac{422820}{5 \, x + 3} - \frac{2021355}{{\left (5 \, x + 3\right )}^{2}} + \frac{474957}{{\left (5 \, x + 3\right )}^{3}} + \frac{9876195}{{\left (5 \, x + 3\right )}^{4}} + \frac{14499345}{{\left (5 \, x + 3\right )}^{5}} + \frac{10904215}{{\left (5 \, x + 3\right )}^{6}} + \frac{5836215}{{\left (5 \, x + 3\right )}^{7}} - 25515\right )} - \frac{1331}{9765625 \,{\left (5 \, x + 3\right )}} - \frac{23232}{9765625} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\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)^2,x, algorithm="giac")

[Out]

1/341796875*(5*x + 3)^8*(422820/(5*x + 3) - 2021355/(5*x + 3)^2 + 474957/(5*x + 3)^3 + 9876195/(5*x + 3)^4 + 1
4499345/(5*x + 3)^5 + 10904215/(5*x + 3)^6 + 5836215/(5*x + 3)^7 - 25515) - 1331/9765625/(5*x + 3) - 23232/976
5625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)