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

Optimal. Leaf size=87 \[ \frac {14520}{117649 (1-2 x)}-\frac {7755}{117649 (3 x+2)}+\frac {1331}{16807 (1-2 x)^2}+\frac {1023}{33614 (3 x+2)^2}-\frac {11}{2401 (3 x+2)^3}+\frac {1}{4116 (3 x+2)^4}-\frac {59070 \log (1-2 x)}{823543}+\frac {59070 \log (3 x+2)}{823543} \]

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Rubi [A]  time = 0.04, antiderivative size = 87, 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 {14520}{117649 (1-2 x)}-\frac {7755}{117649 (3 x+2)}+\frac {1331}{16807 (1-2 x)^2}+\frac {1023}{33614 (3 x+2)^2}-\frac {11}{2401 (3 x+2)^3}+\frac {1}{4116 (3 x+2)^4}-\frac {59070 \log (1-2 x)}{823543}+\frac {59070 \log (3 x+2)}{823543} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(16807*(1 - 2*x)^2) + 14520/(117649*(1 - 2*x)) + 1/(4116*(2 + 3*x)^4) - 11/(2401*(2 + 3*x)^3) + 1023/(336
14*(2 + 3*x)^2) - 7755/(117649*(2 + 3*x)) - (59070*Log[1 - 2*x])/823543 + (59070*Log[2 + 3*x])/823543

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 {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^5} \, dx &=\int \left (-\frac {5324}{16807 (-1+2 x)^3}+\frac {29040}{117649 (-1+2 x)^2}-\frac {118140}{823543 (-1+2 x)}-\frac {1}{343 (2+3 x)^5}+\frac {99}{2401 (2+3 x)^4}-\frac {3069}{16807 (2+3 x)^3}+\frac {23265}{117649 (2+3 x)^2}+\frac {177210}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {1331}{16807 (1-2 x)^2}+\frac {14520}{117649 (1-2 x)}+\frac {1}{4116 (2+3 x)^4}-\frac {11}{2401 (2+3 x)^3}+\frac {1023}{33614 (2+3 x)^2}-\frac {7755}{117649 (2+3 x)}-\frac {59070 \log (1-2 x)}{823543}+\frac {59070 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 64, normalized size = 0.74 \begin {gather*} \frac {-\frac {7 \left (38277360 x^5+60605820 x^4+8860500 x^3-32767930 x^2-21371408 x-3991495\right )}{4 (1-2 x)^2 (3 x+2)^4}-177210 \log (1-2 x)+177210 \log (6 x+4)}{2470629} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(-3991495 - 21371408*x - 32767930*x^2 + 8860500*x^3 + 60605820*x^4 + 38277360*x^5))/(4*(1 - 2*x)^2*(2 + 3
*x)^4) - 177210*Log[1 - 2*x] + 177210*Log[4 + 6*x])/2470629

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.39, size = 135, normalized size = 1.55 \begin {gather*} -\frac {267941520 \, x^{5} + 424240740 \, x^{4} + 62023500 \, x^{3} - 229375510 \, x^{2} - 708840 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 708840 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (2 \, x - 1\right ) - 149599856 \, x - 27940465}{9882516 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9882516*(267941520*x^5 + 424240740*x^4 + 62023500*x^3 - 229375510*x^2 - 708840*(324*x^6 + 540*x^5 + 81*x^4
- 264*x^3 - 104*x^2 + 32*x + 16)*log(3*x + 2) + 708840*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x
+ 16)*log(2*x - 1) - 149599856*x - 27940465)/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)

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giac [A]  time = 1.21, size = 78, normalized size = 0.90 \begin {gather*} -\frac {7755}{117649 \, {\left (3 \, x + 2\right )}} + \frac {4356 \, {\left (\frac {217}{3 \, x + 2} - 51\right )}}{823543 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2}} + \frac {1023}{33614 \, {\left (3 \, x + 2\right )}^{2}} - \frac {11}{2401 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1}{4116 \, {\left (3 \, x + 2\right )}^{4}} - \frac {59070}{823543} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7755/117649/(3*x + 2) + 4356/823543*(217/(3*x + 2) - 51)/(7/(3*x + 2) - 2)^2 + 1023/33614/(3*x + 2)^2 - 11/24
01/(3*x + 2)^3 + 1/4116/(3*x + 2)^4 - 59070/823543*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 72, normalized size = 0.83 \begin {gather*} -\frac {59070 \ln \left (2 x -1\right )}{823543}+\frac {59070 \ln \left (3 x +2\right )}{823543}+\frac {1}{4116 \left (3 x +2\right )^{4}}-\frac {11}{2401 \left (3 x +2\right )^{3}}+\frac {1023}{33614 \left (3 x +2\right )^{2}}-\frac {7755}{117649 \left (3 x +2\right )}+\frac {1331}{16807 \left (2 x -1\right )^{2}}-\frac {14520}{117649 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4116/(3*x+2)^4-11/2401/(3*x+2)^3+1023/33614/(3*x+2)^2-7755/117649/(3*x+2)+59070/823543*ln(3*x+2)+1331/16807/
(2*x-1)^2-14520/117649/(2*x-1)-59070/823543*ln(2*x-1)

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maxima [A]  time = 0.53, size = 76, normalized size = 0.87 \begin {gather*} -\frac {38277360 \, x^{5} + 60605820 \, x^{4} + 8860500 \, x^{3} - 32767930 \, x^{2} - 21371408 \, x - 3991495}{1411788 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} + \frac {59070}{823543} \, \log \left (3 \, x + 2\right ) - \frac {59070}{823543} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1411788*(38277360*x^5 + 60605820*x^4 + 8860500*x^3 - 32767930*x^2 - 21371408*x - 3991495)/(324*x^6 + 540*x^
5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16) + 59070/823543*log(3*x + 2) - 59070/823543*log(2*x - 1)

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mupad [B]  time = 1.09, size = 65, normalized size = 0.75 \begin {gather*} \frac {118140\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}+\frac {-\frac {9845\,x^5}{117649}-\frac {187055\,x^4}{1411788}-\frac {246125\,x^3}{12706092}+\frac {16383965\,x^2}{228709656}+\frac {1335713\,x}{28588707}+\frac {3991495}{457419312}}{x^6+\frac {5\,x^5}{3}+\frac {x^4}{4}-\frac {22\,x^3}{27}-\frac {26\,x^2}{81}+\frac {8\,x}{81}+\frac {4}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(118140*atanh((12*x)/7 + 1/7))/823543 + ((1335713*x)/28588707 + (16383965*x^2)/228709656 - (246125*x^3)/127060
92 - (187055*x^4)/1411788 - (9845*x^5)/117649 + 3991495/457419312)/((8*x)/81 - (26*x^2)/81 - (22*x^3)/27 + x^4
/4 + (5*x^5)/3 + x^6 + 4/81)

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sympy [A]  time = 0.22, size = 75, normalized size = 0.86 \begin {gather*} - \frac {38277360 x^{5} + 60605820 x^{4} + 8860500 x^{3} - 32767930 x^{2} - 21371408 x - 3991495}{457419312 x^{6} + 762365520 x^{5} + 114354828 x^{4} - 372712032 x^{3} - 146825952 x^{2} + 45177216 x + 22588608} - \frac {59070 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {59070 \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)**3/(1-2*x)**3/(2+3*x)**5,x)

[Out]

-(38277360*x**5 + 60605820*x**4 + 8860500*x**3 - 32767930*x**2 - 21371408*x - 3991495)/(457419312*x**6 + 76236
5520*x**5 + 114354828*x**4 - 372712032*x**3 - 146825952*x**2 + 45177216*x + 22588608) - 59070*log(x - 1/2)/823
543 + 59070*log(x + 2/3)/823543

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