3.1568 \(\int \frac{(3+5 x)^2}{(1-2 x)^2 (2+3 x)^6} \, dx\)

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

[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

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Rubi [A]  time = 0.0418089, antiderivative size = 87, normalized size of antiderivative = 1., 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} \[ \frac{968}{117649 (1-2 x)}-\frac{4180}{117649 (3 x+2)}-\frac{682}{16807 (3 x+2)^2}-\frac{319}{7203 (3 x+2)^3}+\frac{11}{686 (3 x+2)^4}-\frac{1}{735 (3 x+2)^5}-\frac{11264 \log (1-2 x)}{823543}+\frac{11264 \log (3 x+2)}{823543} \]

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*}

Mathematica [A]  time = 0.0553313, size = 64, normalized size = 0.74 \[ \frac{8 \left (-\frac{21 \left (9123840 x^5+25090560 x^4+24288000 x^3+7494080 x^2-1530877 x-913244\right )}{16 (2 x-1) (3 x+2)^5}-21120 \log (1-2 x)+21120 \log (6 x+4)\right )}{12353145} \]

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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Maple [A]  time = 0.01, size = 72, normalized size = 0.8 \begin{align*} -{\frac{968}{235298\,x-117649}}-{\frac{11264\,\ln \left ( 2\,x-1 \right ) }{823543}}-{\frac{1}{735\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{11}{686\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{319}{7203\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{682}{16807\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{4180}{235298+352947\,x}}+{\frac{11264\,\ln \left ( 2+3\,x \right ) }{823543}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-968/117649/(2*x-1)-11264/823543*ln(2*x-1)-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)+11264/823543*ln(2+3*x)

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Maxima [A]  time = 1.07461, size = 103, normalized size = 1.18 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.33964, size = 454, normalized size = 5.22 \begin{align*} -\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{align*}

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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Sympy [A]  time = 0.191898, size = 75, normalized size = 0.86 \begin{align*} - \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{align*}

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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Giac [A]  time = 2.9849, size = 105, normalized size = 1.21 \begin{align*} -\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{align*}

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))