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

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

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Rubi [A]  time = 0.03, antiderivative size = 58, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 55, normalized size = 0.95 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.34, size = 60, normalized size = 1.03 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.18, size = 81, normalized size = 1.40 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.01, size = 45, normalized size = 0.78 \begin {gather*} \frac {729 x^{4}}{80}+\frac {2673 x^{3}}{50}+\frac {639819 x^{2}}{4000}+\frac {3946293 x}{10000}+\frac {2739541 \ln \left (2 x -1\right )}{7744}+\frac {\ln \left (5 x +3\right )}{378125}-\frac {117649}{704 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 44, normalized size = 0.76 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.04, size = 40, normalized size = 0.69 \begin {gather*} \frac {3946293\,x}{10000}+\frac {2739541\,\ln \left (x-\frac {1}{2}\right )}{7744}+\frac {\ln \left (x+\frac {3}{5}\right )}{378125}-\frac {117649}{1408\,\left (x-\frac {1}{2}\right )}+\frac {639819\,x^2}{4000}+\frac {2673\,x^3}{50}+\frac {729\,x^4}{80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3946293*x)/10000 + (2739541*log(x - 1/2))/7744 + log(x + 3/5)/378125 - 117649/(1408*(x - 1/2)) + (639819*x^2)
/4000 + (2673*x^3)/50 + (729*x^4)/80

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sympy [A]  time = 0.16, size = 49, normalized size = 0.84 \begin {gather*} \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 {gather*}

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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