3.1590 \(\int \frac{(2+3 x)^6}{(1-2 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=58 \[ \frac{729 x^4}{80}+\frac{2673 x^3}{50}+\frac{639819 x^2}{4000}+\frac{3946293 x}{10000}+\frac{117649}{704 (1-2 x)}+\frac{2739541 \log (1-2 x)}{7744}+\frac{\log (5 x+3)}{378125} \]

[Out]

117649/(704*(1 - 2*x)) + (3946293*x)/10000 + (639819*x^2)/4000 + (2673*x^3)/50 + (729*x^4)/80 + (2739541*Log[1
- 2*x])/7744 + Log[3 + 5*x]/378125

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Rubi [A]  time = 0.027014, antiderivative size = 58, 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} \[ \frac{729 x^4}{80}+\frac{2673 x^3}{50}+\frac{639819 x^2}{4000}+\frac{3946293 x}{10000}+\frac{117649}{704 (1-2 x)}+\frac{2739541 \log (1-2 x)}{7744}+\frac{\log (5 x+3)}{378125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

117649/(704*(1 - 2*x)) + (3946293*x)/10000 + (639819*x^2)/4000 + (2673*x^3)/50 + (729*x^4)/80 + (2739541*Log[1
- 2*x])/7744 + Log[3 + 5*x]/378125

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{(2+3 x)^6}{(1-2 x)^2 (3+5 x)} \, dx &=\int \left (\frac{3946293}{10000}+\frac{639819 x}{2000}+\frac{8019 x^2}{50}+\frac{729 x^3}{20}+\frac{117649}{352 (-1+2 x)^2}+\frac{2739541}{3872 (-1+2 x)}+\frac{1}{75625 (3+5 x)}\right ) \, dx\\ &=\frac{117649}{704 (1-2 x)}+\frac{3946293 x}{10000}+\frac{639819 x^2}{4000}+\frac{2673 x^3}{50}+\frac{729 x^4}{80}+\frac{2739541 \log (1-2 x)}{7744}+\frac{\log (3+5 x)}{378125}\\ \end{align*}

Mathematica [A]  time = 0.0234803, size = 55, normalized size = 0.95 \[ \frac{\frac{55 \left (8019000 x^5+43035300 x^4+117237780 x^3+276893694 x^2-6823872 x-156937135\right )}{2 x-1}[phone] \log (5-10 x)+64 \log (5 x+3)}{24200000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((55*(-156937135 - 6823872*x + 276893694*x^2 + 117237780*x^3 + 43035300*x^4 + 8019000*x^5))/(-1 + 2*x) + 85610
65625*Log[5 - 10*x] + 64*Log[3 + 5*x])/24200000

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Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \begin{align*}{\frac{729\,{x}^{4}}{80}}+{\frac{2673\,{x}^{3}}{50}}+{\frac{639819\,{x}^{2}}{4000}}+{\frac{3946293\,x}{10000}}-{\frac{117649}{1408\,x-704}}+{\frac{2739541\,\ln \left ( 2\,x-1 \right ) }{7744}}+{\frac{\ln \left ( 3+5\,x \right ) }{378125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729/80*x^4+2673/50*x^3+639819/4000*x^2+3946293/10000*x-117649/704/(2*x-1)+2739541/7744*ln(2*x-1)+1/378125*ln(3
+5*x)

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Maxima [A]  time = 1.07554, size = 59, normalized size = 1.02 \begin{align*} \frac{729}{80} \, x^{4} + \frac{2673}{50} \, x^{3} + \frac{639819}{4000} \, x^{2} + \frac{3946293}{10000} \, x - \frac{117649}{704 \,{\left (2 \, x - 1\right )}} + \frac{1}{378125} \, \log \left (5 \, x + 3\right ) + \frac{2739541}{7744} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^2/(3+5*x),x, algorithm="maxima")

[Out]

729/80*x^4 + 2673/50*x^3 + 639819/4000*x^2 + 3946293/10000*x - 117649/704/(2*x - 1) + 1/378125*log(5*x + 3) +
2739541/7744*log(2*x - 1)

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Fricas [A]  time = 1.25262, size = 246, normalized size = 4.24 \begin{align*} \frac{441045000 \, x^{5} + 2366941500 \, x^{4} + 6448077900 \, x^{3} + 15229153170 \, x^{2} + 64 \,{\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 8561065625 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 9550029060 \, x - 4044184375}{24200000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^2/(3+5*x),x, algorithm="fricas")

[Out]

1/24200000*(441045000*x^5 + 2366941500*x^4 + 6448077900*x^3 + 15229153170*x^2 + 64*(2*x - 1)*log(5*x + 3) + 85
61065625*(2*x - 1)*log(2*x - 1) - 9550029060*x - 4044184375)/(2*x - 1)

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Sympy [A]  time = 0.13723, size = 49, normalized size = 0.84 \begin{align*} \frac{729 x^{4}}{80} + \frac{2673 x^{3}}{50} + \frac{639819 x^{2}}{4000} + \frac{3946293 x}{10000} + \frac{2739541 \log{\left (x - \frac{1}{2} \right )}}{7744} + \frac{\log{\left (x + \frac{3}{5} \right )}}{378125} - \frac{117649}{1408 x - 704} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729*x**4/80 + 2673*x**3/50 + 639819*x**2/4000 + 3946293*x/10000 + 2739541*log(x - 1/2)/7744 + log(x + 3/5)/378
125 - 117649/(1408*x - 704)

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Giac [A]  time = 1.55711, size = 109, normalized size = 1.88 \begin{align*} \frac{27}{160000} \,{\left (2 \, x - 1\right )}^{4}{\left (\frac{53100}{2 \, x - 1} + \frac{376020}{{\left (2 \, x - 1\right )}^{2}} + \frac{1775512}{{\left (2 \, x - 1\right )}^{3}} + 3375\right )} - \frac{117649}{704 \,{\left (2 \, x - 1\right )}} - \frac{70752609}{200000} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{1}{378125} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^2/(3+5*x),x, algorithm="giac")

[Out]

27/160000*(2*x - 1)^4*(53100/(2*x - 1) + 376020/(2*x - 1)^2 + 1775512/(2*x - 1)^3 + 3375) - 117649/704/(2*x -
1) - 70752609/200000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 1/378125*log(abs(-11/(2*x - 1) - 5))