3.1619 \(\int \frac{(2+3 x)^6}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=66 \[ \frac{729 x^2}{1000}+\frac{2916 x}{625}+\frac{117649}{21296 (1-2 x)}-\frac{202}{4159375 (5 x+3)}-\frac{1}{756250 (5 x+3)^2}+\frac{1563051 \log (1-2 x)}{234256}+\frac{17139 \log (5 x+3)}{45753125} \]

[Out]

117649/(21296*(1 - 2*x)) + (2916*x)/625 + (729*x^2)/1000 - 1/(756250*(3 + 5*x)^2) - 202/(4159375*(3 + 5*x)) +
(1563051*Log[1 - 2*x])/234256 + (17139*Log[3 + 5*x])/45753125

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Rubi [A]  time = 0.0317153, antiderivative size = 66, 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} \[ \frac{729 x^2}{1000}+\frac{2916 x}{625}+\frac{117649}{21296 (1-2 x)}-\frac{202}{4159375 (5 x+3)}-\frac{1}{756250 (5 x+3)^2}+\frac{1563051 \log (1-2 x)}{234256}+\frac{17139 \log (5 x+3)}{45753125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

117649/(21296*(1 - 2*x)) + (2916*x)/625 + (729*x^2)/1000 - 1/(756250*(3 + 5*x)^2) - 202/(4159375*(3 + 5*x)) +
(1563051*Log[1 - 2*x])/234256 + (17139*Log[3 + 5*x])/45753125

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)^3} \, dx &=\int \left (\frac{2916}{625}+\frac{729 x}{500}+\frac{117649}{10648 (-1+2 x)^2}+\frac{1563051}{117128 (-1+2 x)}+\frac{1}{75625 (3+5 x)^3}+\frac{202}{831875 (3+5 x)^2}+\frac{17139}{9150625 (3+5 x)}\right ) \, dx\\ &=\frac{117649}{21296 (1-2 x)}+\frac{2916 x}{625}+\frac{729 x^2}{1000}-\frac{1}{756250 (3+5 x)^2}-\frac{202}{4159375 (3+5 x)}+\frac{1563051 \log (1-2 x)}{234256}+\frac{17139 \log (3+5 x)}{45753125}\\ \end{align*}

Mathematica [A]  time = 0.0467237, size = 60, normalized size = 0.91 \[ \frac{11 \left (97029900 x^2+620991360 x+\frac{735306250}{1-2 x}-\frac{6464}{5 x+3}-\frac{176}{(5 x+3)^2}-334753155\right )[phone] \log (1-2 x)+548448 \log (10 x+6)}{1464100000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*(-334753155 + 735306250/(1 - 2*x) + 620991360*x + 97029900*x^2 - 176/(3 + 5*x)^2 - 6464/(3 + 5*x)) + 97690
68750*Log[1 - 2*x] + 548448*Log[6 + 10*x])/1464100000

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Maple [A]  time = 0.008, size = 53, normalized size = 0.8 \begin{align*}{\frac{729\,{x}^{2}}{1000}}+{\frac{2916\,x}{625}}-{\frac{117649}{42592\,x-21296}}+{\frac{1563051\,\ln \left ( 2\,x-1 \right ) }{234256}}-{\frac{1}{756250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{202}{12478125+20796875\,x}}+{\frac{17139\,\ln \left ( 3+5\,x \right ) }{45753125}} \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)^3,x)

[Out]

729/1000*x^2+2916/625*x-117649/21296/(2*x-1)+1563051/234256*ln(2*x-1)-1/756250/(3+5*x)^2-202/4159375/(3+5*x)+1
7139/45753125*ln(3+5*x)

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Maxima [A]  time = 1.12374, size = 73, normalized size = 1.11 \begin{align*} \frac{729}{1000} \, x^{2} + \frac{2916}{625} \, x - \frac{9191360445 \, x^{2} + 11029597158 \, x + 3308868341}{66550000 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{17139}{45753125} \, \log \left (5 \, x + 3\right ) + \frac{1563051}{234256} \, \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)^3,x, algorithm="maxima")

[Out]

729/1000*x^2 + 2916/625*x - 1/66550000*(9191360445*x^2 + 11029597158*x + 3308868341)/(50*x^3 + 35*x^2 - 12*x -
9) + 17139/45753125*log(5*x + 3) + 1563051/234256*log(2*x - 1)

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Fricas [A]  time = 1.43315, size = 343, normalized size = 5.2 \begin{align*} \frac{26683222500 \, x^{5} + 189450879750 \, x^{4} + 113136863400 \, x^{3} - 146893374705 \, x^{2} + 274224 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 4884534375 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 152064641058 \, x - 36397551751}{732050000 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\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)^3,x, algorithm="fricas")

[Out]

1/732050000*(26683222500*x^5 + 189450879750*x^4 + 113136863400*x^3 - 146893374705*x^2 + 274224*(50*x^3 + 35*x^
2 - 12*x - 9)*log(5*x + 3) + 4884534375*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1) - 152064641058*x - 363975517
51)/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [A]  time = 0.174834, size = 56, normalized size = 0.85 \begin{align*} \frac{729 x^{2}}{1000} + \frac{2916 x}{625} - \frac{9191360445 x^{2} + 11029597158 x + 3308868341}{3327500000 x^{3} + 2329250000 x^{2} - 798600000 x - 598950000} + \frac{1563051 \log{\left (x - \frac{1}{2} \right )}}{234256} + \frac{17139 \log{\left (x + \frac{3}{5} \right )}}{45753125} \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)**3,x)

[Out]

729*x**2/1000 + 2916*x/625 - (9191360445*x**2 + 11029597158*x + 3308868341)/(3327500000*x**3 + 2329250000*x**2
- 798600000*x - 598950000) + 1563051*log(x - 1/2)/234256 + 17139*log(x + 3/5)/45753125

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Giac [A]  time = 1.68707, size = 127, normalized size = 1.92 \begin{align*} \frac{{\left (2 \, x - 1\right )}^{2}{\left (\frac{25615893600}{2 \, x - 1} + \frac{93337977265}{{\left (2 \, x - 1\right )}^{2}} + \frac{95568773322}{{\left (2 \, x - 1\right )}^{3}} + 1334161125\right )}}{292820000 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} - \frac{117649}{21296 \,{\left (2 \, x - 1\right )}} - \frac{333639}{50000} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{17139}{45753125} \, \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)^3,x, algorithm="giac")

[Out]

1/292820000*(2*x - 1)^2*(25615893600/(2*x - 1) + 93337977265/(2*x - 1)^2 + 95568773322/(2*x - 1)^3 + 133416112
5)/(11/(2*x - 1) + 5)^2 - 117649/21296/(2*x - 1) - 333639/50000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 17139/4575
3125*log(abs(-11/(2*x - 1) - 5))