3.1406 \(\int \frac{(1-2 x)^3 (2+3 x)^3}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=55 \[ -\frac{216 x^5}{125}+\frac{189 x^4}{125}+\frac{786 x^3}{625}-\frac{12077 x^2}{6250}+\frac{1998 x}{3125}-\frac{1331}{78125 (5 x+3)}+\frac{11253 \log (5 x+3)}{78125} \]

[Out]

(1998*x)/3125 - (12077*x^2)/6250 + (786*x^3)/625 + (189*x^4)/125 - (216*x^5)/125 - 1331/(78125*(3 + 5*x)) + (1
1253*Log[3 + 5*x])/78125

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Rubi [A]  time = 0.0267051, antiderivative size = 55, 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} \[ -\frac{216 x^5}{125}+\frac{189 x^4}{125}+\frac{786 x^3}{625}-\frac{12077 x^2}{6250}+\frac{1998 x}{3125}-\frac{1331}{78125 (5 x+3)}+\frac{11253 \log (5 x+3)}{78125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1998*x)/3125 - (12077*x^2)/6250 + (786*x^3)/625 + (189*x^4)/125 - (216*x^5)/125 - 1331/(78125*(3 + 5*x)) + (1
1253*Log[3 + 5*x])/78125

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)^2} \, dx &=\int \left (\frac{1998}{3125}-\frac{12077 x}{3125}+\frac{2358 x^2}{625}+\frac{756 x^3}{125}-\frac{216 x^4}{25}+\frac{1331}{15625 (3+5 x)^2}+\frac{11253}{15625 (3+5 x)}\right ) \, dx\\ &=\frac{1998 x}{3125}-\frac{12077 x^2}{6250}+\frac{786 x^3}{625}+\frac{189 x^4}{125}-\frac{216 x^5}{125}-\frac{1331}{78125 (3+5 x)}+\frac{11253 \log (3+5 x)}{78125}\\ \end{align*}

Mathematica [A]  time = 0.0155753, size = 54, normalized size = 0.98 \[ \frac{-6750000 x^6+1856250 x^5+8456250 x^4-4600625 x^3-2031375 x^2+5485095 x+112530 (5 x+3) \log (5 x+3)+2378647}{781250 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2378647 + 5485095*x - 2031375*x^2 - 4600625*x^3 + 8456250*x^4 + 1856250*x^5 - 6750000*x^6 + 112530*(3 + 5*x)*
Log[3 + 5*x])/(781250*(3 + 5*x))

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Maple [A]  time = 0.006, size = 42, normalized size = 0.8 \begin{align*}{\frac{1998\,x}{3125}}-{\frac{12077\,{x}^{2}}{6250}}+{\frac{786\,{x}^{3}}{625}}+{\frac{189\,{x}^{4}}{125}}-{\frac{216\,{x}^{5}}{125}}-{\frac{1331}{234375+390625\,x}}+{\frac{11253\,\ln \left ( 3+5\,x \right ) }{78125}} \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)^2,x)

[Out]

1998/3125*x-12077/6250*x^2+786/625*x^3+189/125*x^4-216/125*x^5-1331/78125/(3+5*x)+11253/78125*ln(3+5*x)

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Maxima [A]  time = 1.41195, size = 55, normalized size = 1. \begin{align*} -\frac{216}{125} \, x^{5} + \frac{189}{125} \, x^{4} + \frac{786}{625} \, x^{3} - \frac{12077}{6250} \, x^{2} + \frac{1998}{3125} \, x - \frac{1331}{78125 \,{\left (5 \, x + 3\right )}} + \frac{11253}{78125} \, \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)^2,x, algorithm="maxima")

[Out]

-216/125*x^5 + 189/125*x^4 + 786/625*x^3 - 12077/6250*x^2 + 1998/3125*x - 1331/78125/(5*x + 3) + 11253/78125*l
og(5*x + 3)

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Fricas [A]  time = 1.26284, size = 185, normalized size = 3.36 \begin{align*} -\frac{1350000 \, x^{6} - 371250 \, x^{5} - 1691250 \, x^{4} + 920125 \, x^{3} + 406275 \, x^{2} - 22506 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 299700 \, x + 2662}{156250 \,{\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)^2,x, algorithm="fricas")

[Out]

-1/156250*(1350000*x^6 - 371250*x^5 - 1691250*x^4 + 920125*x^3 + 406275*x^2 - 22506*(5*x + 3)*log(5*x + 3) - 2
99700*x + 2662)/(5*x + 3)

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Sympy [A]  time = 0.10523, size = 48, normalized size = 0.87 \begin{align*} - \frac{216 x^{5}}{125} + \frac{189 x^{4}}{125} + \frac{786 x^{3}}{625} - \frac{12077 x^{2}}{6250} + \frac{1998 x}{3125} + \frac{11253 \log{\left (5 x + 3 \right )}}{78125} - \frac{1331}{390625 x + 234375} \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)**2,x)

[Out]

-216*x**5/125 + 189*x**4/125 + 786*x**3/625 - 12077*x**2/6250 + 1998*x/3125 + 11253*log(5*x + 3)/78125 - 1331/
(390625*x + 234375)

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Giac [A]  time = 2.76644, size = 101, normalized size = 1.84 \begin{align*} \frac{1}{781250} \,{\left (5 \, x + 3\right )}^{5}{\left (\frac{8370}{5 \, x + 3} - \frac{53700}{{\left (5 \, x + 3\right )}^{2}} + \frac{87575}{{\left (5 \, x + 3\right )}^{3}} + \frac{295350}{{\left (5 \, x + 3\right )}^{4}} - 432\right )} - \frac{1331}{78125 \,{\left (5 \, x + 3\right )}} - \frac{11253}{78125} \, \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)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

1/781250*(5*x + 3)^5*(8370/(5*x + 3) - 53700/(5*x + 3)^2 + 87575/(5*x + 3)^3 + 295350/(5*x + 3)^4 - 432) - 133
1/78125/(5*x + 3) - 11253/78125*log(1/5*abs(5*x + 3)/(5*x + 3)^2)