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

Optimal. Leaf size=75 \[ \frac {16}{65219 (1-2 x)}+\frac {81}{49 (3 x+2)}+\frac {7750}{1331 (5 x+3)}-\frac {125}{242 (5 x+3)^2}-\frac {2736 \log (1-2 x)}{5021863}-\frac {8829}{343} \log (3 x+2)+\frac {376875 \log (5 x+3)}{14641} \]

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Rubi [A]  time = 0.04, antiderivative size = 75, 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 {16}{65219 (1-2 x)}+\frac {81}{49 (3 x+2)}+\frac {7750}{1331 (5 x+3)}-\frac {125}{242 (5 x+3)^2}-\frac {2736 \log (1-2 x)}{5021863}-\frac {8829}{343} \log (3 x+2)+\frac {376875 \log (5 x+3)}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

16/(65219*(1 - 2*x)) + 81/(49*(2 + 3*x)) - 125/(242*(3 + 5*x)^2) + 7750/(1331*(3 + 5*x)) - (2736*Log[1 - 2*x])
/5021863 - (8829*Log[2 + 3*x])/343 + (376875*Log[3 + 5*x])/14641

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}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx &=\int \left (\frac {32}{65219 (-1+2 x)^2}-\frac {5472}{5021863 (-1+2 x)}-\frac {243}{49 (2+3 x)^2}-\frac {26487}{343 (2+3 x)}+\frac {625}{121 (3+5 x)^3}-\frac {38750}{1331 (3+5 x)^2}+\frac {1884375}{14641 (3+5 x)}\right ) \, dx\\ &=\frac {16}{65219 (1-2 x)}+\frac {81}{49 (2+3 x)}-\frac {125}{242 (3+5 x)^2}+\frac {7750}{1331 (3+5 x)}-\frac {2736 \log (1-2 x)}{5021863}-\frac {8829}{343} \log (2+3 x)+\frac {376875 \log (3+5 x)}{14641}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 65, normalized size = 0.87 \begin {gather*} \frac {\frac {77 \left (33563700 x^3+24606540 x^2-7974123 x-6363424\right )}{(5 x+3)^2 \left (6 x^2+x-2\right )}-5472 \log (5-10 x)-258530778 \log (5 (3 x+2))[phone] \log (5 x+3)}{10043726} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((77*(-6363424 - 7974123*x + 24606540*x^2 + 33563700*x^3))/((3 + 5*x)^2*(-2 + x + 6*x^2)) - 5472*Log[5 - 10*x]
- 258530778*Log[5*(2 + 3*x)] + 258536250*Log[3 + 5*x])/10043726

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.55, size = 123, normalized size = 1.64 \begin {gather*} \frac {2584404900 \, x^{3} + 1894703580 \, x^{2} + 258536250 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (5 \, x + 3\right ) - 258530778 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (3 \, x + 2\right ) - 5472 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (2 \, x - 1\right ) - 614007471 \, x - 489983648}{10043726 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10043726*(2584404900*x^3 + 1894703580*x^2 + 258536250*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log(5*x + 3)
- 258530778*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log(3*x + 2) - 5472*(150*x^4 + 205*x^3 + 34*x^2 - 51*x -
18)*log(2*x - 1) - 614007471*x - 489983648)/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)

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giac [A]  time = 1.20, size = 86, normalized size = 1.15 \begin {gather*} \frac {81}{49 \, {\left (3 \, x + 2\right )}} + \frac {27 \, {\left (\frac {139939165}{3 \, x + 2} - \frac {31679854}{{\left (3 \, x + 2\right )}^{2}} - 37396350\right )}}{913066 \, {\left (\frac {7}{3 \, x + 2} - 2\right )} {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {376875}{14641} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {2736}{5021863} \, \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(1/(1-2*x)^2/(2+3*x)^2/(3+5*x)^3,x, algorithm="giac")

[Out]

81/49/(3*x + 2) + 27/913066*(139939165/(3*x + 2) - 31679854/(3*x + 2)^2 - 37396350)/((7/(3*x + 2) - 2)*(1/(3*x
+ 2) - 5)^2) + 376875/14641*log(abs(-1/(3*x + 2) + 5)) - 2736/5021863*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.02, size = 62, normalized size = 0.83 \begin {gather*} -\frac {2736 \ln \left (2 x -1\right )}{5021863}-\frac {8829 \ln \left (3 x +2\right )}{343}+\frac {376875 \ln \left (5 x +3\right )}{14641}-\frac {125}{242 \left (5 x +3\right )^{2}}+\frac {7750}{1331 \left (5 x +3\right )}+\frac {81}{49 \left (3 x +2\right )}-\frac {16}{65219 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/242/(5*x+3)^2+7750/1331/(5*x+3)+376875/14641*ln(5*x+3)+81/49/(3*x+2)-8829/343*ln(3*x+2)-16/65219/(2*x-1)-
2736/5021863*ln(2*x-1)

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maxima [A]  time = 0.57, size = 64, normalized size = 0.85 \begin {gather*} \frac {33563700 \, x^{3} + 24606540 \, x^{2} - 7974123 \, x - 6363424}{130438 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} + \frac {376875}{14641} \, \log \left (5 \, x + 3\right ) - \frac {8829}{343} \, \log \left (3 \, x + 2\right ) - \frac {2736}{5021863} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/130438*(33563700*x^3 + 24606540*x^2 - 7974123*x - 6363424)/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18) + 376875
/14641*log(5*x + 3) - 8829/343*log(3*x + 2) - 2736/5021863*log(2*x - 1)

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mupad [B]  time = 0.05, size = 56, normalized size = 0.75 \begin {gather*} \frac {376875\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {8829\,\ln \left (x+\frac {2}{3}\right )}{343}-\frac {2736\,\ln \left (x-\frac {1}{2}\right )}{5021863}-\frac {-\frac {111879\,x^3}{65219}-\frac {58587\,x^2}{46585}+\frac {2658041\,x}{6521900}+\frac {1590856}{4891425}}{x^4+\frac {41\,x^3}{30}+\frac {17\,x^2}{75}-\frac {17\,x}{50}-\frac {3}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(376875*log(x + 3/5))/14641 - (8829*log(x + 2/3))/343 - (2736*log(x - 1/2))/5021863 - ((2658041*x)/6521900 - (
58587*x^2)/46585 - (111879*x^3)/65219 + 1590856/4891425)/((17*x^2)/75 - (17*x)/50 + (41*x^3)/30 + x^4 - 3/25)

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sympy [A]  time = 0.24, size = 65, normalized size = 0.87 \begin {gather*} \frac {33563700 x^{3} + 24606540 x^{2} - 7974123 x - 6363424}{19565700 x^{4} + 26739790 x^{3} + 4434892 x^{2} - 6652338 x - 2347884} - \frac {2736 \log {\left (x - \frac {1}{2} \right )}}{5021863} + \frac {376875 \log {\left (x + \frac {3}{5} \right )}}{14641} - \frac {8829 \log {\left (x + \frac {2}{3} \right )}}{343} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(33563700*x**3 + 24606540*x**2 - 7974123*x - 6363424)/(19565700*x**4 + 26739790*x**3 + 4434892*x**2 - 6652338*
x - 2347884) - 2736*log(x - 1/2)/5021863 + 376875*log(x + 3/5)/14641 - 8829*log(x + 2/3)/343

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