3.1317 \(\int \frac {(1-2 x)^2}{(2+3 x)^5 (3+5 x)^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac {7480}{3 x+2}-\frac {3025}{5 x+3}-\frac {1133}{2 (3 x+2)^2}-\frac {154}{3 (3 x+2)^3}-\frac {49}{12 (3 x+2)^4}+46475 \log (3 x+2)-46475 \log (5 x+3) \]

[Out]

-49/12/(2+3*x)^4-154/3/(2+3*x)^3-1133/2/(2+3*x)^2-7480/(2+3*x)-3025/(3+5*x)+46475*ln(2+3*x)-46475*ln(3+5*x)

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Rubi [A]  time = 0.03, antiderivative size = 68, 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} \[ -\frac {7480}{3 x+2}-\frac {3025}{5 x+3}-\frac {1133}{2 (3 x+2)^2}-\frac {154}{3 (3 x+2)^3}-\frac {49}{12 (3 x+2)^4}+46475 \log (3 x+2)-46475 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(12*(2 + 3*x)^4) - 154/(3*(2 + 3*x)^3) - 1133/(2*(2 + 3*x)^2) - 7480/(2 + 3*x) - 3025/(3 + 5*x) + 46475*Lo
g[2 + 3*x] - 46475*Log[3 + 5*x]

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 {(1-2 x)^2}{(2+3 x)^5 (3+5 x)^2} \, dx &=\int \left (\frac {49}{(2+3 x)^5}+\frac {462}{(2+3 x)^4}+\frac {3399}{(2+3 x)^3}+\frac {22440}{(2+3 x)^2}+\frac {139425}{2+3 x}+\frac {15125}{(3+5 x)^2}-\frac {232375}{3+5 x}\right ) \, dx\\ &=-\frac {49}{12 (2+3 x)^4}-\frac {154}{3 (2+3 x)^3}-\frac {1133}{2 (2+3 x)^2}-\frac {7480}{2+3 x}-\frac {3025}{3+5 x}+46475 \log (2+3 x)-46475 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 57, normalized size = 0.84 \[ -\frac {5019300 x^4+13217490 x^3+13046462 x^2+5720639 x+940153}{4 (3 x+2)^4 (5 x+3)}+46475 \log (5 (3 x+2))-46475 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(940153 + 5720639*x + 13046462*x^2 + 13217490*x^3 + 5019300*x^4)/((2 + 3*x)^4*(3 + 5*x)) + 46475*Log[5*(2
+ 3*x)] - 46475*Log[3 + 5*x]

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fricas [A]  time = 0.79, size = 115, normalized size = 1.69 \[ -\frac {5019300 \, x^{4} + 13217490 \, x^{3} + 13046462 \, x^{2} + 185900 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (5 \, x + 3\right ) - 185900 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (3 \, x + 2\right ) + 5720639 \, x + 940153}{4 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(5019300*x^4 + 13217490*x^3 + 13046462*x^2 + 185900*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 4
8)*log(5*x + 3) - 185900*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log(3*x + 2) + 5720639*x + 94
0153)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

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giac [A]  time = 1.04, size = 67, normalized size = 0.99 \[ -\frac {3025}{5 \, x + 3} + \frac {25 \, {\left (\frac {884412}{5 \, x + 3} + \frac {341028}{{\left (5 \, x + 3\right )}^{2}} + \frac {45688}{{\left (5 \, x + 3\right )}^{3}} + 784485\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{4}} + 46475 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3025/(5*x + 3) + 25/4*(884412/(5*x + 3) + 341028/(5*x + 3)^2 + 45688/(5*x + 3)^3 + 784485)/(1/(5*x + 3) + 3)^
4 + 46475*log(abs(-1/(5*x + 3) - 3))

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maple [A]  time = 0.01, size = 63, normalized size = 0.93 \[ 46475 \ln \left (3 x +2\right )-46475 \ln \left (5 x +3\right )-\frac {49}{12 \left (3 x +2\right )^{4}}-\frac {154}{3 \left (3 x +2\right )^{3}}-\frac {1133}{2 \left (3 x +2\right )^{2}}-\frac {7480}{3 x +2}-\frac {3025}{5 x +3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/12/(3*x+2)^4-154/3/(3*x+2)^3-1133/2/(3*x+2)^2-7480/(3*x+2)-3025/(5*x+3)+46475*ln(3*x+2)-46475*ln(5*x+3)

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maxima [A]  time = 0.48, size = 66, normalized size = 0.97 \[ -\frac {5019300 \, x^{4} + 13217490 \, x^{3} + 13046462 \, x^{2} + 5720639 \, x + 940153}{4 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} - 46475 \, \log \left (5 \, x + 3\right ) + 46475 \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(5019300*x^4 + 13217490*x^3 + 13046462*x^2 + 5720639*x + 940153)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^
2 + 368*x + 48) - 46475*log(5*x + 3) + 46475*log(3*x + 2)

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mupad [B]  time = 0.05, size = 56, normalized size = 0.82 \[ 92950\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {9295\,x^4}{3}+\frac {146861\,x^3}{18}+\frac {6523231\,x^2}{810}+\frac {5720639\,x}{1620}+\frac {940153}{1620}}{x^5+\frac {49\,x^4}{15}+\frac {64\,x^3}{15}+\frac {376\,x^2}{135}+\frac {368\,x}{405}+\frac {16}{135}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

92950*atanh(30*x + 19) - ((5720639*x)/1620 + (6523231*x^2)/810 + (146861*x^3)/18 + (9295*x^4)/3 + 940153/1620)
/((368*x)/405 + (376*x^2)/135 + (64*x^3)/15 + (49*x^4)/15 + x^5 + 16/135)

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sympy [A]  time = 0.19, size = 63, normalized size = 0.93 \[ \frac {- 5019300 x^{4} - 13217490 x^{3} - 13046462 x^{2} - 5720639 x - 940153}{1620 x^{5} + 5292 x^{4} + 6912 x^{3} + 4512 x^{2} + 1472 x + 192} - 46475 \log {\left (x + \frac {3}{5} \right )} + 46475 \log {\left (x + \frac {2}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5019300*x**4 - 13217490*x**3 - 13046462*x**2 - 5720639*x - 940153)/(1620*x**5 + 5292*x**4 + 6912*x**3 + 4512
*x**2 + 1472*x + 192) - 46475*log(x + 3/5) + 46475*log(x + 2/3)

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