3.1307 \(\int \frac{(1-2 x)^2 (2+3 x)^5}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=62 \[ \frac{162 x^6}{25}+\frac{5508 x^5}{625}-\frac{8721 x^4}{2500}-\frac{25332 x^3}{3125}+\frac{1893 x^2}{6250}+\frac{277174 x}{78125}-\frac{121}{390625 (5 x+3)}+\frac{1771 \log (5 x+3)}{390625} \]

[Out]

(277174*x)/78125 + (1893*x^2)/6250 - (25332*x^3)/3125 - (8721*x^4)/2500 + (5508*x^5)/625 + (162*x^6)/25 - 121/
(390625*(3 + 5*x)) + (1771*Log[3 + 5*x])/390625

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Rubi [A]  time = 0.0298238, 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} \[ \frac{162 x^6}{25}+\frac{5508 x^5}{625}-\frac{8721 x^4}{2500}-\frac{25332 x^3}{3125}+\frac{1893 x^2}{6250}+\frac{277174 x}{78125}-\frac{121}{390625 (5 x+3)}+\frac{1771 \log (5 x+3)}{390625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(277174*x)/78125 + (1893*x^2)/6250 - (25332*x^3)/3125 - (8721*x^4)/2500 + (5508*x^5)/625 + (162*x^6)/25 - 121/
(390625*(3 + 5*x)) + (1771*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)^2 (2+3 x)^5}{(3+5 x)^2} \, dx &=\int \left (\frac{277174}{78125}+\frac{1893 x}{3125}-\frac{75996 x^2}{3125}-\frac{8721 x^3}{625}+\frac{5508 x^4}{125}+\frac{972 x^5}{25}+\frac{121}{78125 (3+5 x)^2}+\frac{1771}{78125 (3+5 x)}\right ) \, dx\\ &=\frac{277174 x}{78125}+\frac{1893 x^2}{6250}-\frac{25332 x^3}{3125}-\frac{8721 x^4}{2500}+\frac{5508 x^5}{625}+\frac{162 x^6}{25}-\frac{121}{390625 (3+5 x)}+\frac{1771 \log (3+5 x)}{390625}\\ \end{align*}

Mathematica [A]  time = 0.0316248, size = 61, normalized size = 0.98 \[ \frac{253125000 x^7+496125000 x^6+70284375 x^5-398409375 x^4-178158750 x^3+145685750 x^2+126267855 x+35420 (5 x+3) \log (6 (5 x+3))[phone]}{7812500 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(25866973 + 126267855*x + 145685750*x^2 - 178158750*x^3 - 398409375*x^4 + 70284375*x^5 + 496125000*x^6 + 25312
5000*x^7 + 35420*(3 + 5*x)*Log[6*(3 + 5*x)])/(7812500*(3 + 5*x))

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Maple [A]  time = 0.005, size = 47, normalized size = 0.8 \begin{align*}{\frac{277174\,x}{78125}}+{\frac{1893\,{x}^{2}}{6250}}-{\frac{25332\,{x}^{3}}{3125}}-{\frac{8721\,{x}^{4}}{2500}}+{\frac{5508\,{x}^{5}}{625}}+{\frac{162\,{x}^{6}}{25}}-{\frac{121}{1171875+1953125\,x}}+{\frac{1771\,\ln \left ( 3+5\,x \right ) }{390625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

277174/78125*x+1893/6250*x^2-25332/3125*x^3-8721/2500*x^4+5508/625*x^5+162/25*x^6-121/390625/(3+5*x)+1771/3906
25*ln(3+5*x)

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Maxima [A]  time = 1.03781, size = 62, normalized size = 1. \begin{align*} \frac{162}{25} \, x^{6} + \frac{5508}{625} \, x^{5} - \frac{8721}{2500} \, x^{4} - \frac{25332}{3125} \, x^{3} + \frac{1893}{6250} \, x^{2} + \frac{277174}{78125} \, x - \frac{121}{390625 \,{\left (5 \, x + 3\right )}} + \frac{1771}{390625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

162/25*x^6 + 5508/625*x^5 - 8721/2500*x^4 - 25332/3125*x^3 + 1893/6250*x^2 + 277174/78125*x - 121/390625/(5*x
+ 3) + 1771/390625*log(5*x + 3)

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Fricas [A]  time = 1.579, size = 216, normalized size = 3.48 \begin{align*} \frac{50625000 \, x^{7} + 99225000 \, x^{6} + 14056875 \, x^{5} - 79681875 \, x^{4} - 35631750 \, x^{3} + 29137150 \, x^{2} + 7084 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 16630440 \, x - 484}{1562500 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1562500*(50625000*x^7 + 99225000*x^6 + 14056875*x^5 - 79681875*x^4 - 35631750*x^3 + 29137150*x^2 + 7084*(5*x
+ 3)*log(5*x + 3) + 16630440*x - 484)/(5*x + 3)

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Sympy [A]  time = 0.108158, size = 54, normalized size = 0.87 \begin{align*} \frac{162 x^{6}}{25} + \frac{5508 x^{5}}{625} - \frac{8721 x^{4}}{2500} - \frac{25332 x^{3}}{3125} + \frac{1893 x^{2}}{6250} + \frac{277174 x}{78125} + \frac{1771 \log{\left (5 x + 3 \right )}}{390625} - \frac{121}{1953125 x + 1171875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

162*x**6/25 + 5508*x**5/625 - 8721*x**4/2500 - 25332*x**3/3125 + 1893*x**2/6250 + 277174*x/78125 + 1771*log(5*
x + 3)/390625 - 121/(1953125*x + 1171875)

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Giac [A]  time = 1.70908, size = 113, normalized size = 1.82 \begin{align*} -\frac{1}{7812500} \,{\left (5 \, x + 3\right )}^{6}{\left (\frac{36288}{5 \, x + 3} - \frac{63315}{{\left (5 \, x + 3\right )}^{2}} - \frac{249900}{{\left (5 \, x + 3\right )}^{3}} - \frac{287700}{{\left (5 \, x + 3\right )}^{4}} - \frac{204680}{{\left (5 \, x + 3\right )}^{5}} - 3240\right )} - \frac{121}{390625 \,{\left (5 \, x + 3\right )}} - \frac{1771}{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)^2*(2+3*x)^5/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/7812500*(5*x + 3)^6*(36288/(5*x + 3) - 63315/(5*x + 3)^2 - 249900/(5*x + 3)^3 - 287700/(5*x + 3)^4 - 204680
/(5*x + 3)^5 - 3240) - 121/390625/(5*x + 3) - 1771/390625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)