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

Optimal. Leaf size=75 \[ \frac {16}{41503 (1-2 x)}-\frac {1998}{343 (3 x+2)}-\frac {625}{121 (5 x+3)}-\frac {27}{98 (3 x+2)^2}-\frac {2704 \log (1-2 x)}{3195731}+\frac {107109 \log (3 x+2)}{2401}-\frac {59375 \log (5 x+3)}{1331} \]

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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}{41503 (1-2 x)}-\frac {1998}{343 (3 x+2)}-\frac {625}{121 (5 x+3)}-\frac {27}{98 (3 x+2)^2}-\frac {2704 \log (1-2 x)}{3195731}+\frac {107109 \log (3 x+2)}{2401}-\frac {59375 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

16/(41503*(1 - 2*x)) - 27/(98*(2 + 3*x)^2) - 1998/(343*(2 + 3*x)) - 625/(121*(3 + 5*x)) - (2704*Log[1 - 2*x])/
3195731 + (107109*Log[2 + 3*x])/2401 - (59375*Log[3 + 5*x])/1331

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)^3 (3+5 x)^2} \, dx &=\int \left (\frac {32}{41503 (-1+2 x)^2}-\frac {5408}{3195731 (-1+2 x)}+\frac {81}{49 (2+3 x)^3}+\frac {5994}{343 (2+3 x)^2}+\frac {321327}{2401 (2+3 x)}+\frac {3125}{121 (3+5 x)^2}-\frac {296875}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {16}{41503 (1-2 x)}-\frac {27}{98 (2+3 x)^2}-\frac {1998}{343 (2+3 x)}-\frac {625}{121 (3+5 x)}-\frac {2704 \log (1-2 x)}{3195731}+\frac {107109 \log (2+3 x)}{2401}-\frac {59375 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 65, normalized size = 0.87 \begin {gather*} \frac {-\frac {77 \left (22224420 x^3+17783592 x^2-5074951 x-4684319\right )}{(3 x+2)^2 \left (10 x^2+x-3\right )}-5408 \log (3-6 x)[phone] \log (3 x+2)-285118750 \log (-3 (5 x+3))}{6391462} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-77*(-4684319 - 5074951*x + 17783592*x^2 + 22224420*x^3))/((2 + 3*x)^2*(-3 + x + 10*x^2)) - 5408*Log[3 - 6*x
] + 285124158*Log[2 + 3*x] - 285118750*Log[-3*(3 + 5*x)])/6391462

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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)^3 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.62, size = 123, normalized size = 1.64 \begin {gather*} -\frac {1711280340 \, x^{3} + 1369336584 \, x^{2} + 285118750 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (5 \, x + 3\right ) - 285124158 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (3 \, x + 2\right ) + 5408 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (2 \, x - 1\right ) - 390771227 \, x - 360692563}{6391462 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6391462*(1711280340*x^3 + 1369336584*x^2 + 285118750*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*log(5*x + 3) -
285124158*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*log(3*x + 2) + 5408*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)
*log(2*x - 1) - 390771227*x - 360692563)/(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)

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giac [A]  time = 1.14, size = 86, normalized size = 1.15 \begin {gather*} -\frac {625}{121 \, {\left (5 \, x + 3\right )}} + \frac {5 \, {\left (\frac {604065417}{5 \, x + 3} + \frac {258530842}{{\left (5 \, x + 3\right )}^{2}} - 118375902\right )}}{913066 \, {\left (\frac {11}{5 \, x + 3} - 2\right )} {\left (\frac {1}{5 \, x + 3} + 3\right )}^{2}} + \frac {107109}{2401} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {2704}{3195731} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 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)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

-625/121/(5*x + 3) + 5/913066*(604065417/(5*x + 3) + 258530842/(5*x + 3)^2 - 118375902)/((11/(5*x + 3) - 2)*(1
/(5*x + 3) + 3)^2) + 107109/2401*log(abs(-1/(5*x + 3) - 3)) - 2704/3195731*log(abs(-11/(5*x + 3) + 2))

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maple [A]  time = 0.01, size = 62, normalized size = 0.83 \begin {gather*} -\frac {2704 \ln \left (2 x -1\right )}{3195731}+\frac {107109 \ln \left (3 x +2\right )}{2401}-\frac {59375 \ln \left (5 x +3\right )}{1331}-\frac {625}{121 \left (5 x +3\right )}-\frac {27}{98 \left (3 x +2\right )^{2}}-\frac {1998}{343 \left (3 x +2\right )}-\frac {16}{41503 \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)^3/(5*x+3)^2,x)

[Out]

-625/121/(5*x+3)-59375/1331*ln(5*x+3)-27/98/(3*x+2)^2-1998/343/(3*x+2)+107109/2401*ln(3*x+2)-16/41503/(2*x-1)-
2704/3195731*ln(2*x-1)

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maxima [A]  time = 0.59, size = 64, normalized size = 0.85 \begin {gather*} -\frac {22224420 \, x^{3} + 17783592 \, x^{2} - 5074951 \, x - 4684319}{83006 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} - \frac {59375}{1331} \, \log \left (5 \, x + 3\right ) + \frac {107109}{2401} \, \log \left (3 \, x + 2\right ) - \frac {2704}{3195731} \, \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)^3/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/83006*(22224420*x^3 + 17783592*x^2 - 5074951*x - 4684319)/(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12) - 59375/1
331*log(5*x + 3) + 107109/2401*log(3*x + 2) - 2704/3195731*log(2*x - 1)

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mupad [B]  time = 0.04, size = 55, normalized size = 0.73 \begin {gather*} \frac {107109\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {2704\,\ln \left (x-\frac {1}{2}\right )}{3195731}-\frac {59375\,\ln \left (x+\frac {3}{5}\right )}{1331}+\frac {-\frac {123469\,x^3}{41503}-\frac {1481966\,x^2}{622545}+\frac {724993\,x}{1067220}+\frac {4684319}{7470540}}{x^4+\frac {43\,x^3}{30}+\frac {5\,x^2}{18}-\frac {16\,x}{45}-\frac {2}{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(107109*log(x + 2/3))/2401 - (2704*log(x - 1/2))/3195731 - (59375*log(x + 3/5))/1331 + ((724993*x)/1067220 - (
1481966*x^2)/622545 - (123469*x^3)/41503 + 4684319/7470540)/((5*x^2)/18 - (16*x)/45 + (43*x^3)/30 + x^4 - 2/15
)

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sympy [A]  time = 0.24, size = 65, normalized size = 0.87 \begin {gather*} \frac {- 22224420 x^{3} - 17783592 x^{2} + 5074951 x + 4684319}{7470540 x^{4} + 10707774 x^{3} + 2075150 x^{2} - 2656192 x - 996072} - \frac {2704 \log {\left (x - \frac {1}{2} \right )}}{3195731} - \frac {59375 \log {\left (x + \frac {3}{5} \right )}}{1331} + \frac {107109 \log {\left (x + \frac {2}{3} \right )}}{2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-22224420*x**3 - 17783592*x**2 + 5074951*x + 4684319)/(7470540*x**4 + 10707774*x**3 + 2075150*x**2 - 2656192*
x - 996072) - 2704*log(x - 1/2)/3195731 - 59375*log(x + 3/5)/1331 + 107109*log(x + 2/3)/2401

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