3.1448 \(\int \frac {3+5 x}{(1-2 x) (2+3 x)^6} \, dx\)

Optimal. Leaf size=76 \[ -\frac {88}{16807 (3 x+2)}-\frac {22}{2401 (3 x+2)^2}-\frac {22}{1029 (3 x+2)^3}-\frac {11}{196 (3 x+2)^4}+\frac {1}{105 (3 x+2)^5}-\frac {176 \log (1-2 x)}{117649}+\frac {176 \log (3 x+2)}{117649} \]

[Out]

1/105/(2+3*x)^5-11/196/(2+3*x)^4-22/1029/(2+3*x)^3-22/2401/(2+3*x)^2-88/16807/(2+3*x)-176/117649*ln(1-2*x)+176
/117649*ln(2+3*x)

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Rubi [A]  time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {88}{16807 (3 x+2)}-\frac {22}{2401 (3 x+2)^2}-\frac {22}{1029 (3 x+2)^3}-\frac {11}{196 (3 x+2)^4}+\frac {1}{105 (3 x+2)^5}-\frac {176 \log (1-2 x)}{117649}+\frac {176 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(105*(2 + 3*x)^5) - 11/(196*(2 + 3*x)^4) - 22/(1029*(2 + 3*x)^3) - 22/(2401*(2 + 3*x)^2) - 88/(16807*(2 + 3*
x)) - (176*Log[1 - 2*x])/117649 + (176*Log[2 + 3*x])/117649

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {3+5 x}{(1-2 x) (2+3 x)^6} \, dx &=\int \left (-\frac {352}{117649 (-1+2 x)}-\frac {1}{7 (2+3 x)^6}+\frac {33}{49 (2+3 x)^5}+\frac {66}{343 (2+3 x)^4}+\frac {132}{2401 (2+3 x)^3}+\frac {264}{16807 (2+3 x)^2}+\frac {528}{117649 (2+3 x)}\right ) \, dx\\ &=\frac {1}{105 (2+3 x)^5}-\frac {11}{196 (2+3 x)^4}-\frac {22}{1029 (2+3 x)^3}-\frac {22}{2401 (2+3 x)^2}-\frac {88}{16807 (2+3 x)}-\frac {176 \log (1-2 x)}{117649}+\frac {176 \log (2+3 x)}{117649}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 50, normalized size = 0.66 \[ \frac {-\frac {7 \left (427680 x^4+1389960 x^3+1833480 x^2+1268025 x+348226\right )}{(3 x+2)^5}-10560 \log (3-6 x)+10560 \log (3 x+2)}{7058940} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(348226 + 1268025*x + 1833480*x^2 + 1389960*x^3 + 427680*x^4))/(2 + 3*x)^5 - 10560*Log[3 - 6*x] + 10560*L
og[2 + 3*x])/7058940

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fricas [A]  time = 0.75, size = 115, normalized size = 1.51 \[ -\frac {2993760 \, x^{4} + 9729720 \, x^{3} + 12834360 \, x^{2} - 10560 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 10560 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (2 \, x - 1\right ) + 8876175 \, x + 2437582}{7058940 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7058940*(2993760*x^4 + 9729720*x^3 + 12834360*x^2 - 10560*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x +
32)*log(3*x + 2) + 10560*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(2*x - 1) + 8876175*x + 243
7582)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 0.88, size = 48, normalized size = 0.63 \[ -\frac {427680 \, x^{4} + 1389960 \, x^{3} + 1833480 \, x^{2} + 1268025 \, x + 348226}{1008420 \, {\left (3 \, x + 2\right )}^{5}} + \frac {176}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {176}{117649} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1008420*(427680*x^4 + 1389960*x^3 + 1833480*x^2 + 1268025*x + 348226)/(3*x + 2)^5 + 176/117649*log(abs(3*x
+ 2)) - 176/117649*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 63, normalized size = 0.83 \[ -\frac {176 \ln \left (2 x -1\right )}{117649}+\frac {176 \ln \left (3 x +2\right )}{117649}+\frac {1}{105 \left (3 x +2\right )^{5}}-\frac {11}{196 \left (3 x +2\right )^{4}}-\frac {22}{1029 \left (3 x +2\right )^{3}}-\frac {22}{2401 \left (3 x +2\right )^{2}}-\frac {88}{16807 \left (3 x +2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105/(3*x+2)^5-11/196/(3*x+2)^4-22/1029/(3*x+2)^3-22/2401/(3*x+2)^2-88/16807/(3*x+2)+176/117649*ln(3*x+2)-176
/117649*ln(2*x-1)

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maxima [A]  time = 0.51, size = 66, normalized size = 0.87 \[ -\frac {427680 \, x^{4} + 1389960 \, x^{3} + 1833480 \, x^{2} + 1268025 \, x + 348226}{1008420 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {176}{117649} \, \log \left (3 \, x + 2\right ) - \frac {176}{117649} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1008420*(427680*x^4 + 1389960*x^3 + 1833480*x^2 + 1268025*x + 348226)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x
^2 + 240*x + 32) + 176/117649*log(3*x + 2) - 176/117649*log(2*x - 1)

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mupad [B]  time = 1.12, size = 56, normalized size = 0.74 \[ \frac {352\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}-\frac {\frac {88\,x^4}{50421}+\frac {286\,x^3}{50421}+\frac {10186\,x^2}{1361367}+\frac {84535\,x}{16336404}+\frac {174113}{122523030}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(352*atanh((12*x)/7 + 1/7))/117649 - ((84535*x)/16336404 + (10186*x^2)/1361367 + (286*x^3)/50421 + (88*x^4)/50
421 + 174113/122523030)/((80*x)/81 + (80*x^2)/27 + (40*x^3)/9 + (10*x^4)/3 + x^5 + 32/243)

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sympy [A]  time = 0.19, size = 65, normalized size = 0.86 \[ - \frac {427680 x^{4} + 1389960 x^{3} + 1833480 x^{2} + 1268025 x + 348226}{245046060 x^{5} + 816820200 x^{4} + 1089093600 x^{3} + 726062400 x^{2} + 242020800 x + 32269440} - \frac {176 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {176 \log {\left (x + \frac {2}{3} \right )}}{117649} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(427680*x**4 + 1389960*x**3 + 1833480*x**2 + 1268025*x + 348226)/(245046060*x**5 + 816820200*x**4 + 108909360
0*x**3 + 726062400*x**2 + 242020800*x + 32269440) - 176*log(x - 1/2)/117649 + 176*log(x + 2/3)/117649

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