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

Optimal. Leaf size=87 \[ \frac {968}{117649 (1-2 x)}-\frac {4180}{117649 (3 x+2)}-\frac {682}{16807 (3 x+2)^2}-\frac {319}{7203 (3 x+2)^3}+\frac {11}{686 (3 x+2)^4}-\frac {1}{735 (3 x+2)^5}-\frac {11264 \log (1-2 x)}{823543}+\frac {11264 \log (3 x+2)}{823543} \]

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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 {968}{117649 (1-2 x)}-\frac {4180}{117649 (3 x+2)}-\frac {682}{16807 (3 x+2)^2}-\frac {319}{7203 (3 x+2)^3}+\frac {11}{686 (3 x+2)^4}-\frac {1}{735 (3 x+2)^5}-\frac {11264 \log (1-2 x)}{823543}+\frac {11264 \log (3 x+2)}{823543} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

968/(117649*(1 - 2*x)) - 1/(735*(2 + 3*x)^5) + 11/(686*(2 + 3*x)^4) - 319/(7203*(2 + 3*x)^3) - 682/(16807*(2 +
3*x)^2) - 4180/(117649*(2 + 3*x)) - (11264*Log[1 - 2*x])/823543 + (11264*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)^2}{(1-2 x)^2 (2+3 x)^6} \, dx &=\int \left (\frac {1936}{117649 (-1+2 x)^2}-\frac {22528}{823543 (-1+2 x)}+\frac {1}{49 (2+3 x)^6}-\frac {66}{343 (2+3 x)^5}+\frac {957}{2401 (2+3 x)^4}+\frac {4092}{16807 (2+3 x)^3}+\frac {12540}{117649 (2+3 x)^2}+\frac {33792}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {968}{117649 (1-2 x)}-\frac {1}{735 (2+3 x)^5}+\frac {11}{686 (2+3 x)^4}-\frac {319}{7203 (2+3 x)^3}-\frac {682}{16807 (2+3 x)^2}-\frac {4180}{117649 (2+3 x)}-\frac {11264 \log (1-2 x)}{823543}+\frac {11264 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 64, normalized size = 0.74 \begin {gather*} \frac {8 \left (-\frac {21 \left (9123840 x^5+25090560 x^4+24288000 x^3+7494080 x^2-1530877 x-913244\right )}{16 (2 x-1) (3 x+2)^5}-21120 \log (1-2 x)+21120 \log (6 x+4)\right )}{12353145} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*((-21*(-913244 - 1530877*x + 7494080*x^2 + 24288000*x^3 + 25090560*x^4 + 9123840*x^5))/(16*(-1 + 2*x)*(2 +
3*x)^5) - 21120*Log[1 - 2*x] + 21120*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)^2 (2+3 x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.12, size = 135, normalized size = 1.55 \begin {gather*} -\frac {63866880 \, x^{5} + 175633920 \, x^{4} + 170016000 \, x^{3} + 52458560 \, x^{2} - 112640 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (3 \, x + 2\right ) + 112640 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (2 \, x - 1\right ) - 10716139 \, x - 6392708}{8235430 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8235430*(63866880*x^5 + 175633920*x^4 + 170016000*x^3 + 52458560*x^2 - 112640*(486*x^6 + 1377*x^5 + 1350*x^
4 + 360*x^3 - 240*x^2 - 176*x - 32)*log(3*x + 2) + 112640*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 -
176*x - 32)*log(2*x - 1) - 10716139*x - 6392708)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x -
32)

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giac [A]  time = 0.86, size = 78, normalized size = 0.90 \begin {gather*} -\frac {968}{117649 \, {\left (2 \, x - 1\right )}} + \frac {8 \, {\left (\frac {18039105}{2 \, x - 1} + \frac {68101425}{{\left (2 \, x - 1\right )}^{2}} + \frac {114476250}{{\left (2 \, x - 1\right )}^{3}} + \frac {72150050}{{\left (2 \, x - 1\right )}^{4}} + 1800144\right )}}{4117715 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{5}} + \frac {11264}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-968/117649/(2*x - 1) + 8/4117715*(18039105/(2*x - 1) + 68101425/(2*x - 1)^2 + 114476250/(2*x - 1)^3 + 7215005
0/(2*x - 1)^4 + 1800144)/(7/(2*x - 1) + 3)^5 + 11264/823543*log(abs(-7/(2*x - 1) - 3))

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maple [A]  time = 0.01, size = 72, normalized size = 0.83 \begin {gather*} -\frac {11264 \ln \left (2 x -1\right )}{823543}+\frac {11264 \ln \left (3 x +2\right )}{823543}-\frac {1}{735 \left (3 x +2\right )^{5}}+\frac {11}{686 \left (3 x +2\right )^{4}}-\frac {319}{7203 \left (3 x +2\right )^{3}}-\frac {682}{16807 \left (3 x +2\right )^{2}}-\frac {4180}{117649 \left (3 x +2\right )}-\frac {968}{117649 \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)^2/(3*x+2)^6,x)

[Out]

-1/735/(3*x+2)^5+11/686/(3*x+2)^4-319/7203/(3*x+2)^3-682/16807/(3*x+2)^2-4180/117649/(3*x+2)+11264/823543*ln(3
*x+2)-968/117649/(2*x-1)-11264/823543*ln(2*x-1)

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maxima [A]  time = 0.55, size = 76, normalized size = 0.87 \begin {gather*} -\frac {9123840 \, x^{5} + 25090560 \, x^{4} + 24288000 \, x^{3} + 7494080 \, x^{2} - 1530877 \, x - 913244}{1176490 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} + \frac {11264}{823543} \, \log \left (3 \, x + 2\right ) - \frac {11264}{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)^2/(2+3*x)^6,x, algorithm="maxima")

[Out]

-1/1176490*(9123840*x^5 + 25090560*x^4 + 24288000*x^3 + 7494080*x^2 - 1530877*x - 913244)/(486*x^6 + 1377*x^5
+ 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32) + 11264/823543*log(3*x + 2) - 11264/823543*log(2*x - 1)

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mupad [B]  time = 1.11, size = 66, normalized size = 0.76 \begin {gather*} \frac {22528\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {5632\,x^5}{352947}+\frac {15488\,x^4}{352947}+\frac {404800\,x^3}{9529569}+\frac {374704\,x^2}{28588707}-\frac {1530877\,x}{571774140}-\frac {228311}{142943535}}{x^6+\frac {17\,x^5}{6}+\frac {25\,x^4}{9}+\frac {20\,x^3}{27}-\frac {40\,x^2}{81}-\frac {88\,x}{243}-\frac {16}{243}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(22528*atanh((12*x)/7 + 1/7))/823543 - ((374704*x^2)/28588707 - (1530877*x)/571774140 + (404800*x^3)/9529569 +
(15488*x^4)/352947 + (5632*x^5)/352947 - 228311/142943535)/((20*x^3)/27 - (40*x^2)/81 - (88*x)/243 + (25*x^4)
/9 + (17*x^5)/6 + x^6 - 16/243)

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sympy [A]  time = 0.20, size = 75, normalized size = 0.86 \begin {gather*} \frac {- 9123840 x^{5} - 25090560 x^{4} - 24288000 x^{3} - 7494080 x^{2} + 1530877 x + 913244}{571774140 x^{6} + 1620026730 x^{5} + 1588261500 x^{4} + 423536400 x^{3} - 282357600 x^{2} - 207062240 x - 37647680} - \frac {11264 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {11264 \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)**2/(2+3*x)**6,x)

[Out]

(-9123840*x**5 - 25090560*x**4 - 24288000*x**3 - 7494080*x**2 + 1530877*x + 913244)/(571774140*x**6 + 16200267
30*x**5 + 1588261500*x**4 + 423536400*x**3 - 282357600*x**2 - 207062240*x - 37647680) - 11264*log(x - 1/2)/823
543 + 11264*log(x + 2/3)/823543

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