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

Optimal. Leaf size=98 \[ \frac {2608}{823543 (1-2 x)}-\frac {7680}{823543 (3 x+2)}+\frac {88}{117649 (1-2 x)^2}-\frac {1140}{117649 (3 x+2)^2}-\frac {186}{16807 (3 x+2)^3}-\frac {87}{9604 (3 x+2)^4}+\frac {3}{1715 (3 x+2)^5}-\frac {3312 \log (1-2 x)}{823543}+\frac {3312 \log (3 x+2)}{823543} \]

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Rubi [A]  time = 0.05, antiderivative size = 98, 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} \begin {gather*} \frac {2608}{823543 (1-2 x)}-\frac {7680}{823543 (3 x+2)}+\frac {88}{117649 (1-2 x)^2}-\frac {1140}{117649 (3 x+2)^2}-\frac {186}{16807 (3 x+2)^3}-\frac {87}{9604 (3 x+2)^4}+\frac {3}{1715 (3 x+2)^5}-\frac {3312 \log (1-2 x)}{823543}+\frac {3312 \log (3 x+2)}{823543} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

88/(117649*(1 - 2*x)^2) + 2608/(823543*(1 - 2*x)) + 3/(1715*(2 + 3*x)^5) - 87/(9604*(2 + 3*x)^4) - 186/(16807*
(2 + 3*x)^3) - 1140/(117649*(2 + 3*x)^2) - 7680/(823543*(2 + 3*x)) - (3312*Log[1 - 2*x])/823543 + (3312*Log[2
+ 3*x])/823543

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)^3 (2+3 x)^6} \, dx &=\int \left (-\frac {352}{117649 (-1+2 x)^3}+\frac {5216}{823543 (-1+2 x)^2}-\frac {6624}{823543 (-1+2 x)}-\frac {9}{343 (2+3 x)^6}+\frac {261}{2401 (2+3 x)^5}+\frac {1674}{16807 (2+3 x)^4}+\frac {6840}{117649 (2+3 x)^3}+\frac {23040}{823543 (2+3 x)^2}+\frac {9936}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {88}{117649 (1-2 x)^2}+\frac {2608}{823543 (1-2 x)}+\frac {3}{1715 (2+3 x)^5}-\frac {87}{9604 (2+3 x)^4}-\frac {186}{16807 (2+3 x)^3}-\frac {1140}{117649 (2+3 x)^2}-\frac {7680}{823543 (2+3 x)}-\frac {3312 \log (1-2 x)}{823543}+\frac {3312 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 69, normalized size = 0.70 \begin {gather*} \frac {3 \left (-\frac {7 \left (10730880 x^6+24144480 x^5+13811040 x^4-5468940 x^3-7360644 x^2-1134751 x+381394\right )}{3 (1-2 x)^2 (3 x+2)^5}-22080 \log (3-6 x)+22080 \log (3 x+2)\right )}{16470860} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((-7*(381394 - 1134751*x - 7360644*x^2 - 5468940*x^3 + 13811040*x^4 + 24144480*x^5 + 10730880*x^6))/(3*(1 -
2*x)^2*(2 + 3*x)^5) - 22080*Log[3 - 6*x] + 22080*Log[2 + 3*x]))/16470860

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.19, size = 155, normalized size = 1.58 \begin {gather*} -\frac {75116160 \, x^{6} + 169011360 \, x^{5} + 96677280 \, x^{4} - 38282580 \, x^{3} - 51524508 \, x^{2} - 66240 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 66240 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (2 \, x - 1\right ) - 7943257 \, x + 2669758}{16470860 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16470860*(75116160*x^6 + 169011360*x^5 + 96677280*x^4 - 38282580*x^3 - 51524508*x^2 - 66240*(972*x^7 + 2268
*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log(3*x + 2) + 66240*(972*x^7 + 2268*x^6 + 1323*x^
5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log(2*x - 1) - 7943257*x + 2669758)/(972*x^7 + 2268*x^6 + 1323*x
^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)

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giac [A]  time = 1.20, size = 65, normalized size = 0.66 \begin {gather*} -\frac {10730880 \, x^{6} + 24144480 \, x^{5} + 13811040 \, x^{4} - 5468940 \, x^{3} - 7360644 \, x^{2} - 1134751 \, x + 381394}{2352980 \, {\left (3 \, x + 2\right )}^{5} {\left (2 \, x - 1\right )}^{2}} + \frac {3312}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {3312}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2352980*(10730880*x^6 + 24144480*x^5 + 13811040*x^4 - 5468940*x^3 - 7360644*x^2 - 1134751*x + 381394)/((3*x
+ 2)^5*(2*x - 1)^2) + 3312/823543*log(abs(3*x + 2)) - 3312/823543*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 81, normalized size = 0.83 \begin {gather*} -\frac {3312 \ln \left (2 x -1\right )}{823543}+\frac {3312 \ln \left (3 x +2\right )}{823543}+\frac {3}{1715 \left (3 x +2\right )^{5}}-\frac {87}{9604 \left (3 x +2\right )^{4}}-\frac {186}{16807 \left (3 x +2\right )^{3}}-\frac {1140}{117649 \left (3 x +2\right )^{2}}-\frac {7680}{823543 \left (3 x +2\right )}+\frac {88}{117649 \left (2 x -1\right )^{2}}-\frac {2608}{823543 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1715/(3*x+2)^5-87/9604/(3*x+2)^4-186/16807/(3*x+2)^3-1140/117649/(3*x+2)^2-7680/823543/(3*x+2)+3312/823543*l
n(3*x+2)+88/117649/(2*x-1)^2-2608/823543/(2*x-1)-3312/823543*ln(2*x-1)

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maxima [A]  time = 0.55, size = 86, normalized size = 0.88 \begin {gather*} -\frac {10730880 \, x^{6} + 24144480 \, x^{5} + 13811040 \, x^{4} - 5468940 \, x^{3} - 7360644 \, x^{2} - 1134751 \, x + 381394}{2352980 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} + \frac {3312}{823543} \, \log \left (3 \, x + 2\right ) - \frac {3312}{823543} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2352980*(10730880*x^6 + 24144480*x^5 + 13811040*x^4 - 5468940*x^3 - 7360644*x^2 - 1134751*x + 381394)/(972*
x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32) + 3312/823543*log(3*x + 2) - 3312/823543
*log(2*x - 1)

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mupad [B]  time = 1.07, size = 76, normalized size = 0.78 \begin {gather*} \frac {6624\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {552\,x^6}{117649}+\frac {1242\,x^5}{117649}+\frac {6394\,x^4}{1058841}-\frac {30383\,x^3}{12706092}-\frac {613387\,x^2}{190591380}-\frac {1134751\,x}{2287096560}+\frac {190697}{1143548280}}{x^7+\frac {7\,x^6}{3}+\frac {49\,x^5}{36}-\frac {35\,x^4}{54}-\frac {70\,x^3}{81}-\frac {28\,x^2}{243}+\frac {28\,x}{243}+\frac {8}{243}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6624*atanh((12*x)/7 + 1/7))/823543 - ((6394*x^4)/1058841 - (613387*x^2)/190591380 - (30383*x^3)/12706092 - (1
134751*x)/2287096560 + (1242*x^5)/117649 + (552*x^6)/117649 + 190697/1143548280)/((28*x)/243 - (28*x^2)/243 -
(70*x^3)/81 - (35*x^4)/54 + (49*x^5)/36 + (7*x^6)/3 + x^7 + 8/243)

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sympy [A]  time = 0.22, size = 85, normalized size = 0.87 \begin {gather*} - \frac {10730880 x^{6} + 24144480 x^{5} + 13811040 x^{4} - 5468940 x^{3} - 7360644 x^{2} - 1134751 x + 381394}{2287096560 x^{7} + 5336558640 x^{6} + 3112992540 x^{5} - 1482377400 x^{4} - 1976503200 x^{3} - 263533760 x^{2} + 263533760 x + 75295360} - \frac {3312 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {3312 \log {\left (x + \frac {2}{3} \right )}}{823543} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(10730880*x**6 + 24144480*x**5 + 13811040*x**4 - 5468940*x**3 - 7360644*x**2 - 1134751*x + 381394)/(228709656
0*x**7 + 5336558640*x**6 + 3112992540*x**5 - 1482377400*x**4 - 1976503200*x**3 - 263533760*x**2 + 263533760*x
+ 75295360) - 3312*log(x - 1/2)/823543 + 3312*log(x + 2/3)/823543

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