3.1673 \(\int \frac{(2+3 x)^6}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=62 \[ -\frac{243 x^3}{40}-\frac{35721 x^2}{800}-\frac{102303 x}{500}-\frac{2739541}{7744 (1-2 x)}+\frac{117649}{1408 (1-2 x)^2}-\frac{12761315 \log (1-2 x)}{42592}+\frac{\log (5 x+3)}{831875} \]

[Out]

117649/(1408*(1 - 2*x)^2) - 2739541/(7744*(1 - 2*x)) - (102303*x)/500 - (35721*x^2)/800 - (243*x^3)/40 - (1276
1315*Log[1 - 2*x])/42592 + Log[3 + 5*x]/831875

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Rubi [A]  time = 0.0296764, 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} \[ -\frac{243 x^3}{40}-\frac{35721 x^2}{800}-\frac{102303 x}{500}-\frac{2739541}{7744 (1-2 x)}+\frac{117649}{1408 (1-2 x)^2}-\frac{12761315 \log (1-2 x)}{42592}+\frac{\log (5 x+3)}{831875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

117649/(1408*(1 - 2*x)^2) - 2739541/(7744*(1 - 2*x)) - (102303*x)/500 - (35721*x^2)/800 - (243*x^3)/40 - (1276
1315*Log[1 - 2*x])/42592 + Log[3 + 5*x]/831875

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)^3 (3+5 x)} \, dx &=\int \left (-\frac{102303}{500}-\frac{35721 x}{400}-\frac{729 x^2}{40}-\frac{117649}{352 (-1+2 x)^3}-\frac{2739541}{3872 (-1+2 x)^2}-\frac{12761315}{21296 (-1+2 x)}+\frac{1}{166375 (3+5 x)}\right ) \, dx\\ &=\frac{117649}{1408 (1-2 x)^2}-\frac{2739541}{7744 (1-2 x)}-\frac{102303 x}{500}-\frac{35721 x^2}{800}-\frac{243 x^3}{40}-\frac{12761315 \log (1-2 x)}{42592}+\frac{\log (3+5 x)}{831875}\\ \end{align*}

Mathematica [A]  time = 0.0277894, size = 55, normalized size = 0.89 \[ \frac{-\frac{11 \left (235224000 x^5+1493672400 x^4+6252253920 x^3-3308307948 x^2-9050078692 x+3661042443\right )}{(1-2 x)^2}-31903287500 \log (5-10 x)+128 \log (5 x+3)}{106480000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(3661042443 - 9050078692*x - 3308307948*x^2 + 6252253920*x^3 + 1493672400*x^4 + 235224000*x^5))/(1 - 2*x
)^2 - 31903287500*Log[5 - 10*x] + 128*Log[3 + 5*x])/106480000

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Maple [A]  time = 0.007, size = 49, normalized size = 0.8 \begin{align*} -{\frac{243\,{x}^{3}}{40}}-{\frac{35721\,{x}^{2}}{800}}-{\frac{102303\,x}{500}}+{\frac{117649}{1408\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{2739541}{15488\,x-7744}}-{\frac{12761315\,\ln \left ( 2\,x-1 \right ) }{42592}}+{\frac{\ln \left ( 3+5\,x \right ) }{831875}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/40*x^3-35721/800*x^2-102303/500*x+117649/1408/(2*x-1)^2+2739541/7744/(2*x-1)-12761315/42592*ln(2*x-1)+1/8
31875*ln(3+5*x)

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Maxima [A]  time = 1.48885, size = 66, normalized size = 1.06 \begin{align*} -\frac{243}{40} \, x^{3} - \frac{35721}{800} \, x^{2} - \frac{102303}{500} \, x + \frac{16807 \,{\left (652 \, x - 249\right )}}{15488 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1}{831875} \, \log \left (5 \, x + 3\right ) - \frac{12761315}{42592} \, \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)^3/(3+5*x),x, algorithm="maxima")

[Out]

-243/40*x^3 - 35721/800*x^2 - 102303/500*x + 16807/15488*(652*x - 249)/(4*x^2 - 4*x + 1) + 1/831875*log(5*x +
3) - 12761315/42592*log(2*x - 1)

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Fricas [A]  time = 1.57898, size = 290, normalized size = 4.68 \begin{align*} -\frac{2587464000 \, x^{5} + 16430396400 \, x^{4} + 68774793120 \, x^{3} - 82391322420 \, x^{2} - 128 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 31903287500 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 53550930620 \, x + 28771483125}{106480000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/106480000*(2587464000*x^5 + 16430396400*x^4 + 68774793120*x^3 - 82391322420*x^2 - 128*(4*x^2 - 4*x + 1)*log
(5*x + 3) + 31903287500*(4*x^2 - 4*x + 1)*log(2*x - 1) - 53550930620*x + 28771483125)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.166316, size = 51, normalized size = 0.82 \begin{align*} - \frac{243 x^{3}}{40} - \frac{35721 x^{2}}{800} - \frac{102303 x}{500} + \frac{10958164 x - 4184943}{61952 x^{2} - 61952 x + 15488} - \frac{12761315 \log{\left (x - \frac{1}{2} \right )}}{42592} + \frac{\log{\left (x + \frac{3}{5} \right )}}{831875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**3/40 - 35721*x**2/800 - 102303*x/500 + (10958164*x - 4184943)/(61952*x**2 - 61952*x + 15488) - 1276131
5*log(x - 1/2)/42592 + log(x + 3/5)/831875

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Giac [A]  time = 1.91749, size = 62, normalized size = 1. \begin{align*} -\frac{243}{40} \, x^{3} - \frac{35721}{800} \, x^{2} - \frac{102303}{500} \, x + \frac{16807 \,{\left (652 \, x - 249\right )}}{15488 \,{\left (2 \, x - 1\right )}^{2}} + \frac{1}{831875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{12761315}{42592} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/40*x^3 - 35721/800*x^2 - 102303/500*x + 16807/15488*(652*x - 249)/(2*x - 1)^2 + 1/831875*log(abs(5*x + 3)
) - 12761315/42592*log(abs(2*x - 1))