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

Optimal. Leaf size=98 \[ \frac {11264}{823543 (1-2 x)}-\frac {24040}{823543 (3 x+2)}+\frac {484}{117649 (1-2 x)^2}-\frac {2875}{117649 (3 x+2)^2}-\frac {829}{50421 (3 x+2)^3}+\frac {16}{2401 (3 x+2)^4}-\frac {1}{1715 (3 x+2)^5}-\frac {11696 \log (1-2 x)}{823543}+\frac {11696 \log (3 x+2)}{823543} \]

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \begin {gather*} \frac {11264}{823543 (1-2 x)}-\frac {24040}{823543 (3 x+2)}+\frac {484}{117649 (1-2 x)^2}-\frac {2875}{117649 (3 x+2)^2}-\frac {829}{50421 (3 x+2)^3}+\frac {16}{2401 (3 x+2)^4}-\frac {1}{1715 (3 x+2)^5}-\frac {11696 \log (1-2 x)}{823543}+\frac {11696 \log (3 x+2)}{823543} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

484/(117649*(1 - 2*x)^2) + 11264/(823543*(1 - 2*x)) - 1/(1715*(2 + 3*x)^5) + 16/(2401*(2 + 3*x)^4) - 829/(5042
1*(2 + 3*x)^3) - 2875/(117649*(2 + 3*x)^2) - 24040/(823543*(2 + 3*x)) - (11696*Log[1 - 2*x])/823543 + (11696*L
og[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)^2}{(1-2 x)^3 (2+3 x)^6} \, dx &=\int \left (-\frac {1936}{117649 (-1+2 x)^3}+\frac {22528}{823543 (-1+2 x)^2}-\frac {23392}{823543 (-1+2 x)}+\frac {3}{343 (2+3 x)^6}-\frac {192}{2401 (2+3 x)^5}+\frac {2487}{16807 (2+3 x)^4}+\frac {17250}{117649 (2+3 x)^3}+\frac {72120}{823543 (2+3 x)^2}+\frac {35088}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {484}{117649 (1-2 x)^2}+\frac {11264}{823543 (1-2 x)}-\frac {1}{1715 (2+3 x)^5}+\frac {16}{2401 (2+3 x)^4}-\frac {829}{50421 (2+3 x)^3}-\frac {2875}{117649 (2+3 x)^2}-\frac {24040}{823543 (2+3 x)}-\frac {11696 \log (1-2 x)}{823543}+\frac {11696 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 69, normalized size = 0.70 \begin {gather*} \frac {4 \left (-\frac {7 \left (28421280 x^6+63947880 x^5+36579240 x^4-14484765 x^3-19495039 x^2-4230956 x+258089\right )}{4 (1-2 x)^2 (3 x+2)^5}-43860 \log (1-2 x)+43860 \log (6 x+4)\right )}{12353145} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-7*(258089 - 4230956*x - 19495039*x^2 - 14484765*x^3 + 36579240*x^4 + 63947880*x^5 + 28421280*x^6))/(4*(1
- 2*x)^2*(2 + 3*x)^5) - 43860*Log[1 - 2*x] + 43860*Log[4 + 6*x]))/12353145

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.61, size = 155, normalized size = 1.58 \begin {gather*} -\frac {198948960 \, x^{6} + 447635160 \, x^{5} + 256054680 \, x^{4} - 101393355 \, x^{3} - 136465273 \, x^{2} - 175440 \, {\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 ) + 175440 \, {\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 ) - 29616692 \, x + 1806623}{12353145 \, {\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)^2/(1-2*x)^3/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/12353145*(198948960*x^6 + 447635160*x^5 + 256054680*x^4 - 101393355*x^3 - 136465273*x^2 - 175440*(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) + 175440*(972*x^7 + 2268*x^6 + 1
323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log(2*x - 1) - 29616692*x + 1806623)/(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 = 1.18, size = 65, normalized size = 0.66 \begin {gather*} -\frac {28421280 \, x^{6} + 63947880 \, x^{5} + 36579240 \, x^{4} - 14484765 \, x^{3} - 19495039 \, x^{2} - 4230956 \, x + 258089}{1764735 \, {\left (3 \, x + 2\right )}^{5} {\left (2 \, x - 1\right )}^{2}} + \frac {11696}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {11696}{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)^2/(1-2*x)^3/(2+3*x)^6,x, algorithm="giac")

[Out]

-1/1764735*(28421280*x^6 + 63947880*x^5 + 36579240*x^4 - 14484765*x^3 - 19495039*x^2 - 4230956*x + 258089)/((3
*x + 2)^5*(2*x - 1)^2) + 11696/823543*log(abs(3*x + 2)) - 11696/823543*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 81, normalized size = 0.83 \begin {gather*} -\frac {11696 \ln \left (2 x -1\right )}{823543}+\frac {11696 \ln \left (3 x +2\right )}{823543}-\frac {1}{1715 \left (3 x +2\right )^{5}}+\frac {16}{2401 \left (3 x +2\right )^{4}}-\frac {829}{50421 \left (3 x +2\right )^{3}}-\frac {2875}{117649 \left (3 x +2\right )^{2}}-\frac {24040}{823543 \left (3 x +2\right )}+\frac {484}{117649 \left (2 x -1\right )^{2}}-\frac {11264}{823543 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1715/(3*x+2)^5+16/2401/(3*x+2)^4-829/50421/(3*x+2)^3-2875/117649/(3*x+2)^2-24040/823543/(3*x+2)+11696/82354
3*ln(3*x+2)+484/117649/(2*x-1)^2-11264/823543/(2*x-1)-11696/823543*ln(2*x-1)

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maxima [A]  time = 0.58, size = 86, normalized size = 0.88 \begin {gather*} -\frac {28421280 \, x^{6} + 63947880 \, x^{5} + 36579240 \, x^{4} - 14484765 \, x^{3} - 19495039 \, x^{2} - 4230956 \, x + 258089}{1764735 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} + \frac {11696}{823543} \, \log \left (3 \, x + 2\right ) - \frac {11696}{823543} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1764735*(28421280*x^6 + 63947880*x^5 + 36579240*x^4 - 14484765*x^3 - 19495039*x^2 - 4230956*x + 258089)/(97
2*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32) + 11696/823543*log(3*x + 2) - 11696/82
3543*log(2*x - 1)

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mupad [B]  time = 1.06, size = 76, normalized size = 0.78 \begin {gather*} \frac {23392\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {5848\,x^6}{352947}+\frac {4386\,x^5}{117649}+\frac {203218\,x^4}{9529569}-\frac {965651\,x^3}{114354828}-\frac {19495039\,x^2}{1715322420}-\frac {1057739\,x}{428830605}+\frac {258089}{1715322420}}{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/((2*x - 1)^3*(3*x + 2)^6),x)

[Out]

(23392*atanh((12*x)/7 + 1/7))/823543 - ((203218*x^4)/9529569 - (19495039*x^2)/1715322420 - (965651*x^3)/114354
828 - (1057739*x)/428830605 + (4386*x^5)/117649 + (5848*x^6)/352947 + 258089/1715322420)/((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 = 0.23, size = 85, normalized size = 0.87 \begin {gather*} - \frac {28421280 x^{6} + 63947880 x^{5} + 36579240 x^{4} - 14484765 x^{3} - 19495039 x^{2} - 4230956 x + 258089}{1715322420 x^{7} + 4002418980 x^{6} + 2334744405 x^{5} - 1111783050 x^{4} - 1482377400 x^{3} - 197650320 x^{2} + 197650320 x + 56471520} - \frac {11696 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {11696 \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)**2/(1-2*x)**3/(2+3*x)**6,x)

[Out]

-(28421280*x**6 + 63947880*x**5 + 36579240*x**4 - 14484765*x**3 - 19495039*x**2 - 4230956*x + 258089)/(1715322
420*x**7 + 4002418980*x**6 + 2334744405*x**5 - 1111783050*x**4 - 1482377400*x**3 - 197650320*x**2 + 197650320*
x + 56471520) - 11696*log(x - 1/2)/823543 + 11696*log(x + 2/3)/823543

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