3.1306 \(\int \frac{(1-2 x)^2 (2+3 x)^6}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=69 \[ \frac{2916 x^7}{175}+\frac{4374 x^6}{125}+\frac{28917 x^5}{3125}-\frac{157599 x^4}{6250}-\frac{48771 x^3}{3125}+\frac{463086 x^2}{78125}+\frac{2777053 x}{390625}-\frac{121}{1953125 (5 x+3)}+\frac{2134 \log (5 x+3)}{1953125} \]

[Out]

(2777053*x)/390625 + (463086*x^2)/78125 - (48771*x^3)/3125 - (157599*x^4)/6250 + (28917*x^5)/3125 + (4374*x^6)
/125 + (2916*x^7)/175 - 121/(1953125*(3 + 5*x)) + (2134*Log[3 + 5*x])/1953125

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Rubi [A]  time = 0.035115, antiderivative size = 69, 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} \[ \frac{2916 x^7}{175}+\frac{4374 x^6}{125}+\frac{28917 x^5}{3125}-\frac{157599 x^4}{6250}-\frac{48771 x^3}{3125}+\frac{463086 x^2}{78125}+\frac{2777053 x}{390625}-\frac{121}{1953125 (5 x+3)}+\frac{2134 \log (5 x+3)}{1953125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2777053*x)/390625 + (463086*x^2)/78125 - (48771*x^3)/3125 - (157599*x^4)/6250 + (28917*x^5)/3125 + (4374*x^6)
/125 + (2916*x^7)/175 - 121/(1953125*(3 + 5*x)) + (2134*Log[3 + 5*x])/1953125

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)^6}{(3+5 x)^2} \, dx &=\int \left (\frac{2777053}{390625}+\frac{926172 x}{78125}-\frac{146313 x^2}{3125}-\frac{315198 x^3}{3125}+\frac{28917 x^4}{625}+\frac{26244 x^5}{125}+\frac{2916 x^6}{25}+\frac{121}{390625 (3+5 x)^2}+\frac{2134}{390625 (3+5 x)}\right ) \, dx\\ &=\frac{2777053 x}{390625}+\frac{463086 x^2}{78125}-\frac{48771 x^3}{3125}-\frac{157599 x^4}{6250}+\frac{28917 x^5}{3125}+\frac{4374 x^6}{125}+\frac{2916 x^7}{175}-\frac{121}{1953125 (3+5 x)}+\frac{2134 \log (3+5 x)}{1953125}\\ \end{align*}

Mathematica [A]  time = 0.0339339, size = 66, normalized size = 0.96 \[ \frac{11390625000 x^8+30754687500 x^7+20677781250 x^6-13442034375 x^5-21011090625 x^4-2349191250 x^3+7291044250 x^2+3997343145 x+149380 (5 x+3) \log (6 (5 x+3))[phone]}{136718750 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(648854027 + 3997343145*x + 7291044250*x^2 - 2349191250*x^3 - 21011090625*x^4 - 13442034375*x^5 + 20677781250*
x^6 + 30754687500*x^7 + 11390625000*x^8 + 149380*(3 + 5*x)*Log[6*(3 + 5*x)])/(136718750*(3 + 5*x))

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Maple [A]  time = 0.005, size = 52, normalized size = 0.8 \begin{align*}{\frac{2777053\,x}{390625}}+{\frac{463086\,{x}^{2}}{78125}}-{\frac{48771\,{x}^{3}}{3125}}-{\frac{157599\,{x}^{4}}{6250}}+{\frac{28917\,{x}^{5}}{3125}}+{\frac{4374\,{x}^{6}}{125}}+{\frac{2916\,{x}^{7}}{175}}-{\frac{121}{5859375+9765625\,x}}+{\frac{2134\,\ln \left ( 3+5\,x \right ) }{1953125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(2+3*x)^6/(3+5*x)^2,x)

[Out]

2777053/390625*x+463086/78125*x^2-48771/3125*x^3-157599/6250*x^4+28917/3125*x^5+4374/125*x^6+2916/175*x^7-121/
1953125/(3+5*x)+2134/1953125*ln(3+5*x)

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Maxima [A]  time = 1.0572, size = 69, normalized size = 1. \begin{align*} \frac{2916}{175} \, x^{7} + \frac{4374}{125} \, x^{6} + \frac{28917}{3125} \, x^{5} - \frac{157599}{6250} \, x^{4} - \frac{48771}{3125} \, x^{3} + \frac{463086}{78125} \, x^{2} + \frac{2777053}{390625} \, x - \frac{121}{1953125 \,{\left (5 \, x + 3\right )}} + \frac{2134}{1953125} \, \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)^6/(3+5*x)^2,x, algorithm="maxima")

[Out]

2916/175*x^7 + 4374/125*x^6 + 28917/3125*x^5 - 157599/6250*x^4 - 48771/3125*x^3 + 463086/78125*x^2 + 2777053/3
90625*x - 121/1953125/(5*x + 3) + 2134/1953125*log(5*x + 3)

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Fricas [A]  time = 1.44883, size = 259, normalized size = 3.75 \begin{align*} \frac{2278125000 \, x^{8} + 6150937500 \, x^{7} + 4135556250 \, x^{6} - 2688406875 \, x^{5} - 4202218125 \, x^{4} - 469838250 \, x^{3} + 1458208850 \, x^{2} + 29876 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 583181130 \, x - 1694}{27343750 \,{\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)^6/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/27343750*(2278125000*x^8 + 6150937500*x^7 + 4135556250*x^6 - 2688406875*x^5 - 4202218125*x^4 - 469838250*x^3
+ 1458208850*x^2 + 29876*(5*x + 3)*log(5*x + 3) + 583181130*x - 1694)/(5*x + 3)

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Sympy [A]  time = 0.11183, size = 61, normalized size = 0.88 \begin{align*} \frac{2916 x^{7}}{175} + \frac{4374 x^{6}}{125} + \frac{28917 x^{5}}{3125} - \frac{157599 x^{4}}{6250} - \frac{48771 x^{3}}{3125} + \frac{463086 x^{2}}{78125} + \frac{2777053 x}{390625} + \frac{2134 \log{\left (5 x + 3 \right )}}{1953125} - \frac{121}{9765625 x + 5859375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(2+3*x)**6/(3+5*x)**2,x)

[Out]

2916*x**7/175 + 4374*x**6/125 + 28917*x**5/3125 - 157599*x**4/6250 - 48771*x**3/3125 + 463086*x**2/78125 + 277
7053*x/390625 + 2134*log(5*x + 3)/1953125 - 121/(9765625*x + 5859375)

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Giac [A]  time = 1.6539, size = 126, normalized size = 1.83 \begin{align*} -\frac{1}{136718750} \,{\left (5 \, x + 3\right )}^{7}{\left (\frac{306180}{5 \, x + 3} - \frac{404838}{{\left (5 \, x + 3\right )}^{2}} - \frac{2189565}{{\left (5 \, x + 3\right )}^{3}} - \frac{2888550}{{\left (5 \, x + 3\right )}^{4}} - \frac{2081520}{{\left (5 \, x + 3\right )}^{5}} - \frac{1088290}{{\left (5 \, x + 3\right )}^{6}} - 29160\right )} - \frac{121}{1953125 \,{\left (5 \, x + 3\right )}} - \frac{2134}{1953125} \, \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)^6/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/136718750*(5*x + 3)^7*(306180/(5*x + 3) - 404838/(5*x + 3)^2 - 2189565/(5*x + 3)^3 - 2888550/(5*x + 3)^4 -
2081520/(5*x + 3)^5 - 1088290/(5*x + 3)^6 - 29160) - 121/1953125/(5*x + 3) - 2134/1953125*log(1/5*abs(5*x + 3)
/(5*x + 3)^2)