3.1405 \(\int \frac{(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=62 \[ -\frac{108 x^6}{25}+\frac{108 x^5}{625}+\frac{7317 x^4}{1250}-\frac{4217 x^3}{3125}-\frac{1816 x^2}{625}+\frac{133659 x}{78125}-\frac{1331}{390625 (5 x+3)}+\frac{15246 \log (5 x+3)}{390625} \]

[Out]

(133659*x)/78125 - (1816*x^2)/625 - (4217*x^3)/3125 + (7317*x^4)/1250 + (108*x^5)/625 - (108*x^6)/25 - 1331/(3
90625*(3 + 5*x)) + (15246*Log[3 + 5*x])/390625

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Rubi [A]  time = 0.0285736, antiderivative size = 62, 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} \[ -\frac{108 x^6}{25}+\frac{108 x^5}{625}+\frac{7317 x^4}{1250}-\frac{4217 x^3}{3125}-\frac{1816 x^2}{625}+\frac{133659 x}{78125}-\frac{1331}{390625 (5 x+3)}+\frac{15246 \log (5 x+3)}{390625} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^2,x]

[Out]

(133659*x)/78125 - (1816*x^2)/625 - (4217*x^3)/3125 + (7317*x^4)/1250 + (108*x^5)/625 - (108*x^6)/25 - 1331/(3
90625*(3 + 5*x)) + (15246*Log[3 + 5*x])/390625

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)^4}{(3+5 x)^2} \, dx &=\int \left (\frac{133659}{78125}-\frac{3632 x}{625}-\frac{12651 x^2}{3125}+\frac{14634 x^3}{625}+\frac{108 x^4}{125}-\frac{648 x^5}{25}+\frac{1331}{78125 (3+5 x)^2}+\frac{15246}{78125 (3+5 x)}\right ) \, dx\\ &=\frac{133659 x}{78125}-\frac{1816 x^2}{625}-\frac{4217 x^3}{3125}+\frac{7317 x^4}{1250}+\frac{108 x^5}{625}-\frac{108 x^6}{25}-\frac{1331}{390625 (3+5 x)}+\frac{15246 \log (3+5 x)}{390625}\\ \end{align*}

Mathematica [A]  time = 0.0321664, size = 61, normalized size = 0.98 \[ \frac{-84375000 x^7-47250000 x^6+116353125 x^5+42240625 x^4-72563750 x^3-635250 x^2+44216865 x+152460 (5 x+3) \log (6 (5 x+3))[phone]}{3906250 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^2,x]

[Out]

(14487499 + 44216865*x - 635250*x^2 - 72563750*x^3 + 42240625*x^4 + 116353125*x^5 - 47250000*x^6 - 84375000*x^
7 + 152460*(3 + 5*x)*Log[6*(3 + 5*x)])/(3906250*(3 + 5*x))

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Maple [A]  time = 0.006, size = 47, normalized size = 0.8 \begin{align*}{\frac{133659\,x}{78125}}-{\frac{1816\,{x}^{2}}{625}}-{\frac{4217\,{x}^{3}}{3125}}+{\frac{7317\,{x}^{4}}{1250}}+{\frac{108\,{x}^{5}}{625}}-{\frac{108\,{x}^{6}}{25}}-{\frac{1331}{1171875+1953125\,x}}+{\frac{15246\,\ln \left ( 3+5\,x \right ) }{390625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x)

[Out]

133659/78125*x-1816/625*x^2-4217/3125*x^3+7317/1250*x^4+108/625*x^5-108/25*x^6-1331/390625/(3+5*x)+15246/39062
5*ln(3+5*x)

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Maxima [A]  time = 1.01946, size = 62, normalized size = 1. \begin{align*} -\frac{108}{25} \, x^{6} + \frac{108}{625} \, x^{5} + \frac{7317}{1250} \, x^{4} - \frac{4217}{3125} \, x^{3} - \frac{1816}{625} \, x^{2} + \frac{133659}{78125} \, x - \frac{1331}{390625 \,{\left (5 \, x + 3\right )}} + \frac{15246}{390625} \, \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)^4/(3+5*x)^2,x, algorithm="maxima")

[Out]

-108/25*x^6 + 108/625*x^5 + 7317/1250*x^4 - 4217/3125*x^3 - 1816/625*x^2 + 133659/78125*x - 1331/390625/(5*x +
3) + 15246/390625*log(5*x + 3)

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Fricas [A]  time = 1.19722, size = 212, normalized size = 3.42 \begin{align*} -\frac{16875000 \, x^{7} + 9450000 \, x^{6} - 23270625 \, x^{5} - 8448125 \, x^{4} + 14512750 \, x^{3} + 127050 \, x^{2} - 30492 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 4009770 \, x + 2662}{781250 \,{\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)^4/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/781250*(16875000*x^7 + 9450000*x^6 - 23270625*x^5 - 8448125*x^4 + 14512750*x^3 + 127050*x^2 - 30492*(5*x +
3)*log(5*x + 3) - 4009770*x + 2662)/(5*x + 3)

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Sympy [A]  time = 0.107096, size = 54, normalized size = 0.87 \begin{align*} - \frac{108 x^{6}}{25} + \frac{108 x^{5}}{625} + \frac{7317 x^{4}}{1250} - \frac{4217 x^{3}}{3125} - \frac{1816 x^{2}}{625} + \frac{133659 x}{78125} + \frac{15246 \log{\left (5 x + 3 \right )}}{390625} - \frac{1331}{1953125 x + 1171875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**4/(3+5*x)**2,x)

[Out]

-108*x**6/25 + 108*x**5/625 + 7317*x**4/1250 - 4217*x**3/3125 - 1816*x**2/625 + 133659*x/78125 + 15246*log(5*x
+ 3)/390625 - 1331/(1953125*x + 1171875)

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Giac [A]  time = 2.80436, size = 113, normalized size = 1.82 \begin{align*} \frac{1}{3906250} \,{\left (5 \, x + 3\right )}^{6}{\left (\frac{19656}{5 \, x + 3} - \frac{112455}{{\left (5 \, x + 3\right )}^{2}} + \frac{121450}{{\left (5 \, x + 3\right )}^{3}} + \frac{530600}{{\left (5 \, x + 3\right )}^{4}} + \frac{632940}{{\left (5 \, x + 3\right )}^{5}} - 1080\right )} - \frac{1331}{390625 \,{\left (5 \, x + 3\right )}} - \frac{15246}{390625} \, \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)^4/(3+5*x)^2,x, algorithm="giac")

[Out]

1/3906250*(5*x + 3)^6*(19656/(5*x + 3) - 112455/(5*x + 3)^2 + 121450/(5*x + 3)^3 + 530600/(5*x + 3)^4 + 632940
/(5*x + 3)^5 - 1080) - 1331/390625/(5*x + 3) - 15246/390625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)