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

Optimal. Leaf size=86 \[ \frac {32}{184877 (1-2 x)}+\frac {434043}{16807 (3 x+2)}+\frac {12393}{4802 (3 x+2)^2}+\frac {117}{343 (3 x+2)^3}+\frac {9}{196 (3 x+2)^4}-\frac {6400 \log (1-2 x)}{14235529}-\frac {15192225 \log (3 x+2)}{117649}+\frac {15625}{121} \log (5 x+3) \]

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Rubi [A]  time = 0.05, antiderivative size = 86, 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 {32}{184877 (1-2 x)}+\frac {434043}{16807 (3 x+2)}+\frac {12393}{4802 (3 x+2)^2}+\frac {117}{343 (3 x+2)^3}+\frac {9}{196 (3 x+2)^4}-\frac {6400 \log (1-2 x)}{14235529}-\frac {15192225 \log (3 x+2)}{117649}+\frac {15625}{121} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

32/(184877*(1 - 2*x)) + 9/(196*(2 + 3*x)^4) + 117/(343*(2 + 3*x)^3) + 12393/(4802*(2 + 3*x)^2) + 434043/(16807
*(2 + 3*x)) - (6400*Log[1 - 2*x])/14235529 - (15192225*Log[2 + 3*x])/117649 + (15625*Log[3 + 5*x])/121

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)^5 (3+5 x)} \, dx &=\int \left (\frac {64}{184877 (-1+2 x)^2}-\frac {12800}{14235529 (-1+2 x)}-\frac {27}{49 (2+3 x)^5}-\frac {1053}{343 (2+3 x)^4}-\frac {37179}{2401 (2+3 x)^3}-\frac {1302129}{16807 (2+3 x)^2}-\frac {45576675}{117649 (2+3 x)}+\frac {78125}{121 (3+5 x)}\right ) \, dx\\ &=\frac {32}{184877 (1-2 x)}+\frac {9}{196 (2+3 x)^4}+\frac {117}{343 (2+3 x)^3}+\frac {12393}{4802 (2+3 x)^2}+\frac {434043}{16807 (2+3 x)}-\frac {6400 \log (1-2 x)}{14235529}-\frac {15192225 \log (2+3 x)}{117649}+\frac {15625}{121} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 81, normalized size = 0.94 \begin {gather*} \frac {5 \left (\frac {77}{5} \left (\frac {19097892}{3 x+2}+\frac {1908522}{(3 x+2)^2}+\frac {252252}{(3 x+2)^3}+\frac {33957}{(3 x+2)^4}+\frac {128}{1-2 x}\right )-5120 \log (5-10 x)-1470607380 \log (5 (3 x+2))[phone] \log (5 x+3)\right )}{56942116} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*((77*(128/(1 - 2*x) + 33957/(2 + 3*x)^4 + 252252/(2 + 3*x)^3 + 1908522/(2 + 3*x)^2 + 19097892/(2 + 3*x)))/5
- 5120*Log[5 - 10*x] - 1470607380*Log[5*(2 + 3*x)] + 1470612500*Log[3 + 5*x]))/56942116

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.78, size = 148, normalized size = 1.72 \begin {gather*} \frac {79408236600 \, x^{4} + 121756635000 \, x^{3} + 28788432750 \, x^{2} + 7353062500 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (5 \, x + 3\right ) - 7353036900 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) - 25600 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) - 29974871850 \, x - 12393745441}{56942116 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/56942116*(79408236600*x^4 + 121756635000*x^3 + 28788432750*x^2 + 7353062500*(162*x^5 + 351*x^4 + 216*x^3 - 2
4*x^2 - 64*x - 16)*log(5*x + 3) - 7353036900*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(3*x + 2) -
25600*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) - 29974871850*x - 12393745441)/(162*x^5
+ 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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giac [A]  time = 1.19, size = 82, normalized size = 0.95 \begin {gather*} \frac {434043}{16807 \, {\left (3 \, x + 2\right )}} + \frac {192}{1294139 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}} + \frac {12393}{4802 \, {\left (3 \, x + 2\right )}^{2}} + \frac {117}{343 \, {\left (3 \, x + 2\right )}^{3}} + \frac {9}{196 \, {\left (3 \, x + 2\right )}^{4}} + \frac {15625}{121} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {6400}{14235529} \, \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)^5/(3+5*x),x, algorithm="giac")

[Out]

434043/16807/(3*x + 2) + 192/1294139/(7/(3*x + 2) - 2) + 12393/4802/(3*x + 2)^2 + 117/343/(3*x + 2)^3 + 9/196/
(3*x + 2)^4 + 15625/121*log(abs(-1/(3*x + 2) + 5)) - 6400/14235529*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 71, normalized size = 0.83 \begin {gather*} -\frac {6400 \ln \left (2 x -1\right )}{14235529}-\frac {15192225 \ln \left (3 x +2\right )}{117649}+\frac {15625 \ln \left (5 x +3\right )}{121}+\frac {9}{196 \left (3 x +2\right )^{4}}+\frac {117}{343 \left (3 x +2\right )^{3}}+\frac {12393}{4802 \left (3 x +2\right )^{2}}+\frac {434043}{16807 \left (3 x +2\right )}-\frac {32}{184877 \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)^5/(5*x+3),x)

[Out]

15625/121*ln(5*x+3)+9/196/(3*x+2)^4+117/343/(3*x+2)^3+12393/4802/(3*x+2)^2+434043/16807/(3*x+2)-15192225/11764
9*ln(3*x+2)-32/184877/(2*x-1)-6400/14235529*ln(2*x-1)

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maxima [A]  time = 0.57, size = 74, normalized size = 0.86 \begin {gather*} \frac {1031275800 \, x^{4} + 1581255000 \, x^{3} + 373875750 \, x^{2} - 389284050 \, x - 160957733}{739508 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac {15625}{121} \, \log \left (5 \, x + 3\right ) - \frac {15192225}{117649} \, \log \left (3 \, x + 2\right ) - \frac {6400}{14235529} \, \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)^5/(3+5*x),x, algorithm="maxima")

[Out]

1/739508*(1031275800*x^4 + 1581255000*x^3 + 373875750*x^2 - 389284050*x - 160957733)/(162*x^5 + 351*x^4 + 216*
x^3 - 24*x^2 - 64*x - 16) + 15625/121*log(5*x + 3) - 15192225/117649*log(3*x + 2) - 6400/14235529*log(2*x - 1)

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mupad [B]  time = 1.09, size = 68, normalized size = 0.79 \begin {gather*} \frac {15625\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {15192225\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {6400\,\ln \left (x-\frac {1}{2}\right )}{14235529}-\frac {\frac {1591475\,x^4}{184877}+\frac {7320625\,x^3}{554631}+\frac {2307875\,x^2}{739508}-\frac {64880675\,x}{19966716}-\frac {160957733}{119800296}}{-x^5-\frac {13\,x^4}{6}-\frac {4\,x^3}{3}+\frac {4\,x^2}{27}+\frac {32\,x}{81}+\frac {8}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(15625*log(x + 3/5))/121 - (15192225*log(x + 2/3))/117649 - (6400*log(x - 1/2))/14235529 - ((2307875*x^2)/7395
08 - (64880675*x)/19966716 + (7320625*x^3)/554631 + (1591475*x^4)/184877 - 160957733/119800296)/((32*x)/81 + (
4*x^2)/27 - (4*x^3)/3 - (13*x^4)/6 - x^5 + 8/81)

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sympy [A]  time = 0.26, size = 75, normalized size = 0.87 \begin {gather*} \frac {1031275800 x^{4} + 1581255000 x^{3} + 373875750 x^{2} - 389284050 x - 160957733}{119800296 x^{5} + 259567308 x^{4} + 159733728 x^{3} - 17748192 x^{2} - 47328512 x - 11832128} - \frac {6400 \log {\left (x - \frac {1}{2} \right )}}{14235529} + \frac {15625 \log {\left (x + \frac {3}{5} \right )}}{121} - \frac {15192225 \log {\left (x + \frac {2}{3} \right )}}{117649} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1031275800*x**4 + 1581255000*x**3 + 373875750*x**2 - 389284050*x - 160957733)/(119800296*x**5 + 259567308*x**
4 + 159733728*x**3 - 17748192*x**2 - 47328512*x - 11832128) - 6400*log(x - 1/2)/14235529 + 15625*log(x + 3/5)/
121 - 15192225*log(x + 2/3)/117649

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