3.1422 \(\int \frac{(1-2 x)^3 (2+3 x)^3}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=59 \[ -\frac{54 x^4}{125}+\frac{468 x^3}{625}-\frac{927 x^2}{3125}-\frac{1303 x}{3125}-\frac{11253}{78125 (5 x+3)}-\frac{1331}{156250 (5 x+3)^2}+\frac{5907 \log (5 x+3)}{15625} \]

[Out]

(-1303*x)/3125 - (927*x^2)/3125 + (468*x^3)/625 - (54*x^4)/125 - 1331/(156250*(3 + 5*x)^2) - 11253/(78125*(3 +
5*x)) + (5907*Log[3 + 5*x])/15625

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Rubi [A]  time = 0.0290739, antiderivative size = 59, 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} \[ -\frac{54 x^4}{125}+\frac{468 x^3}{625}-\frac{927 x^2}{3125}-\frac{1303 x}{3125}-\frac{11253}{78125 (5 x+3)}-\frac{1331}{156250 (5 x+3)^2}+\frac{5907 \log (5 x+3)}{15625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1303*x)/3125 - (927*x^2)/3125 + (468*x^3)/625 - (54*x^4)/125 - 1331/(156250*(3 + 5*x)^2) - 11253/(78125*(3 +
5*x)) + (5907*Log[3 + 5*x])/15625

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)^3}{(3+5 x)^3} \, dx &=\int \left (-\frac{1303}{3125}-\frac{1854 x}{3125}+\frac{1404 x^2}{625}-\frac{216 x^3}{125}+\frac{1331}{15625 (3+5 x)^3}+\frac{11253}{15625 (3+5 x)^2}+\frac{5907}{3125 (3+5 x)}\right ) \, dx\\ &=-\frac{1303 x}{3125}-\frac{927 x^2}{3125}+\frac{468 x^3}{625}-\frac{54 x^4}{125}-\frac{1331}{156250 (3+5 x)^2}-\frac{11253}{78125 (3+5 x)}+\frac{5907 \log (3+5 x)}{15625}\\ \end{align*}

Mathematica [A]  time = 0.0152593, size = 56, normalized size = 0.95 \[ -\frac{337500 x^6-180000 x^5-348750 x^4+393250 x^3+416250 x^2+70080 x-11814 (5 x+3)^2 \log (5 x+3)-7139}{31250 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-7139 + 70080*x + 416250*x^2 + 393250*x^3 - 348750*x^4 - 180000*x^5 + 337500*x^6 - 11814*(3 + 5*x)^2*Log[3 +
5*x])/(31250*(3 + 5*x)^2)

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Maple [A]  time = 0.004, size = 46, normalized size = 0.8 \begin{align*} -{\frac{1303\,x}{3125}}-{\frac{927\,{x}^{2}}{3125}}+{\frac{468\,{x}^{3}}{625}}-{\frac{54\,{x}^{4}}{125}}-{\frac{1331}{156250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{11253}{234375+390625\,x}}+{\frac{5907\,\ln \left ( 3+5\,x \right ) }{15625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1303/3125*x-927/3125*x^2+468/625*x^3-54/125*x^4-1331/156250/(3+5*x)^2-11253/78125/(3+5*x)+5907/15625*ln(3+5*x
)

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Maxima [A]  time = 1.05507, size = 62, normalized size = 1.05 \begin{align*} -\frac{54}{125} \, x^{4} + \frac{468}{625} \, x^{3} - \frac{927}{3125} \, x^{2} - \frac{1303}{3125} \, x - \frac{121 \,{\left (930 \, x + 569\right )}}{156250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{5907}{15625} \, \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)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

-54/125*x^4 + 468/625*x^3 - 927/3125*x^2 - 1303/3125*x - 121/156250*(930*x + 569)/(25*x^2 + 30*x + 9) + 5907/1
5625*log(5*x + 3)

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Fricas [A]  time = 1.2471, size = 216, normalized size = 3.66 \begin{align*} -\frac{1687500 \, x^{6} - 900000 \, x^{5} - 1743750 \, x^{4} + 1966250 \, x^{3} + 2371650 \, x^{2} - 59070 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 698880 \, x + 68849}{156250 \,{\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)^3/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/156250*(1687500*x^6 - 900000*x^5 - 1743750*x^4 + 1966250*x^3 + 2371650*x^2 - 59070*(25*x^2 + 30*x + 9)*log(
5*x + 3) + 698880*x + 68849)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.121139, size = 49, normalized size = 0.83 \begin{align*} - \frac{54 x^{4}}{125} + \frac{468 x^{3}}{625} - \frac{927 x^{2}}{3125} - \frac{1303 x}{3125} - \frac{112530 x + 68849}{3906250 x^{2} + 4687500 x + 1406250} + \frac{5907 \log{\left (5 x + 3 \right )}}{15625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-54*x**4/125 + 468*x**3/625 - 927*x**2/3125 - 1303*x/3125 - (112530*x + 68849)/(3906250*x**2 + 4687500*x + 140
6250) + 5907*log(5*x + 3)/15625

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Giac [A]  time = 2.96413, size = 57, normalized size = 0.97 \begin{align*} -\frac{54}{125} \, x^{4} + \frac{468}{625} \, x^{3} - \frac{927}{3125} \, x^{2} - \frac{1303}{3125} \, x - \frac{121 \,{\left (930 \, x + 569\right )}}{156250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{5907}{15625} \, \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)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

-54/125*x^4 + 468/625*x^3 - 927/3125*x^2 - 1303/3125*x - 121/156250*(930*x + 569)/(5*x + 3)^2 + 5907/15625*log
(abs(5*x + 3))