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)-102303/500*x-35721/800*x^2-243/40*x^3-12761315/42592*ln(1-2*x)+1/83
1875*ln(3+5*x)

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Rubi [A]  time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.79, size = 75, normalized size = 1.21 \[ -\frac {2587464000 \, x^{5} + 16430396400 \, x^{4} + 68774793120 \, x^{3} - 82391322420 \, x^{2} - 128 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 31903287500 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 53550930620 \, x + 28771483125}{106480000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

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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giac [A]  time = 1.17, size = 46, normalized size = 0.74 \[ -\frac {243}{40} \, x^{3} - \frac {35721}{800} \, x^{2} - \frac {102303}{500} \, x + \frac {16807 \, {\left (652 \, x - 249\right )}}{15488 \, {\left (2 \, x - 1\right )}^{2}} + \frac {1}{831875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {12761315}{42592} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 49, normalized size = 0.79 \[ -\frac {243 x^{3}}{40}-\frac {35721 x^{2}}{800}-\frac {102303 x}{500}-\frac {12761315 \ln \left (2 x -1\right )}{42592}+\frac {\ln \left (5 x +3\right )}{831875}+\frac {117649}{1408 \left (2 x -1\right )^{2}}+\frac {2739541}{7744 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/40*x^3-35721/800*x^2-102303/500*x+1/831875*ln(5*x+3)+117649/1408/(2*x-1)^2+2739541/7744/(2*x-1)-12761315/
42592*ln(2*x-1)

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maxima [A]  time = 0.59, size = 49, normalized size = 0.79 \[ -\frac {243}{40} \, x^{3} - \frac {35721}{800} \, x^{2} - \frac {102303}{500} \, x + \frac {16807 \, {\left (652 \, x - 249\right )}}{15488 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1}{831875} \, \log \left (5 \, x + 3\right ) - \frac {12761315}{42592} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 0.04, size = 42, normalized size = 0.68 \[ \frac {\ln \left (x+\frac {3}{5}\right )}{831875}-\frac {12761315\,\ln \left (x-\frac {1}{2}\right )}{42592}-\frac {102303\,x}{500}+\frac {\frac {2739541\,x}{15488}-\frac {4184943}{61952}}{x^2-x+\frac {1}{4}}-\frac {35721\,x^2}{800}-\frac {243\,x^3}{40} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3/5)/831875 - (12761315*log(x - 1/2))/42592 - (102303*x)/500 + ((2739541*x)/15488 - 4184943/61952)/(x^
2 - x + 1/4) - (35721*x^2)/800 - (243*x^3)/40

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sympy [A]  time = 0.18, size = 51, normalized size = 0.82 \[ - \frac {243 x^{3}}{40} - \frac {35721 x^{2}}{800} - \frac {102303 x}{500} - \frac {4184943 - 10958164 x}{61952 x^{2} - 61952 x + 15488} - \frac {12761315 \log {\left (x - \frac {1}{2} \right )}}{42592} + \frac {\log {\left (x + \frac {3}{5} \right )}}{831875} \]

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 - (4184943 - 10958164*x)/(61952*x**2 - 61952*x + 15488) - 1276131
5*log(x - 1/2)/42592 + log(x + 3/5)/831875

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