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

Optimal. Leaf size=75 \[ \frac {2889}{49 (3 x+2)}+\frac {12125}{121 (5 x+3)}+\frac {27}{14 (3 x+2)^2}-\frac {125}{22 (5 x+3)^2}-\frac {32 \log (1-2 x)}{456533}-\frac {204228}{343} \log (3 x+2)+\frac {792500 \log (5 x+3)}{1331} \]

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Rubi [A]  time = 0.03, 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 {2889}{49 (3 x+2)}+\frac {12125}{121 (5 x+3)}+\frac {27}{14 (3 x+2)^2}-\frac {125}{22 (5 x+3)^2}-\frac {32 \log (1-2 x)}{456533}-\frac {204228}{343} \log (3 x+2)+\frac {792500 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

27/(14*(2 + 3*x)^2) + 2889/(49*(2 + 3*x)) - 125/(22*(3 + 5*x)^2) + 12125/(121*(3 + 5*x)) - (32*Log[1 - 2*x])/4
56533 - (204228*Log[2 + 3*x])/343 + (792500*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+3 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac {64}{456533 (-1+2 x)}-\frac {81}{7 (2+3 x)^3}-\frac {8667}{49 (2+3 x)^2}-\frac {612684}{343 (2+3 x)}+\frac {625}{11 (3+5 x)^3}-\frac {60625}{121 (3+5 x)^2}+\frac {3962500}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {27}{14 (2+3 x)^2}+\frac {2889}{49 (2+3 x)}-\frac {125}{22 (3+5 x)^2}+\frac {12125}{121 (3+5 x)}-\frac {32 \log (1-2 x)}{456533}-\frac {204228}{343} \log (2+3 x)+\frac {792500 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 71, normalized size = 0.95 \begin {gather*} \frac {2889}{147 x+98}+\frac {12125}{605 x+363}+\frac {27}{14 (3 x+2)^2}-\frac {125}{22 (5 x+3)^2}-\frac {32 \log (1-2 x)}{456533}-\frac {204228}{343} \log (6 x+4)+\frac {792500 \log (10 x+6)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

27/(14*(2 + 3*x)^2) - 125/(22*(3 + 5*x)^2) + 2889/(98 + 147*x) + 12125/(363 + 605*x) - (32*Log[1 - 2*x])/45653
3 - (204228*Log[4 + 6*x])/343 + (792500*Log[6 + 10*x])/1331

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.22, size = 123, normalized size = 1.64 \begin {gather*} \frac {8154808200 \, x^{3} + 15494126340 \, x^{2} + 543655000 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (5 \, x + 3\right ) - 543654936 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (3 \, x + 2\right ) - 64 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (2 \, x - 1\right ) + 9797832352 \, x + 2062044985}{913066 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/913066*(8154808200*x^3 + 15494126340*x^2 + 543655000*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(5*x + 3)
- 543654936*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(3*x + 2) - 64*(225*x^4 + 570*x^3 + 541*x^2 + 228*x
+ 36)*log(2*x - 1) + 9797832352*x + 2062044985)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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giac [A]  time = 0.95, size = 59, normalized size = 0.79 \begin {gather*} \frac {105906600 \, x^{3} + 201222420 \, x^{2} + 127244576 \, x + 26779805}{11858 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{2}} + \frac {792500}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {204228}{343} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {32}{456533} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11858*(105906600*x^3 + 201222420*x^2 + 127244576*x + 26779805)/((5*x + 3)^2*(3*x + 2)^2) + 792500/1331*log(a
bs(5*x + 3)) - 204228/343*log(abs(3*x + 2)) - 32/456533*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 62, normalized size = 0.83 \begin {gather*} -\frac {32 \ln \left (2 x -1\right )}{456533}-\frac {204228 \ln \left (3 x +2\right )}{343}+\frac {792500 \ln \left (5 x +3\right )}{1331}-\frac {125}{22 \left (5 x +3\right )^{2}}+\frac {12125}{121 \left (5 x +3\right )}+\frac {27}{14 \left (3 x +2\right )^{2}}+\frac {2889}{49 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/22/(5*x+3)^2+12125/121/(5*x+3)+792500/1331*ln(5*x+3)+27/14/(3*x+2)^2+2889/49/(3*x+2)-204228/343*ln(3*x+2)
-32/456533*ln(2*x-1)

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maxima [A]  time = 0.57, size = 64, normalized size = 0.85 \begin {gather*} \frac {105906600 \, x^{3} + 201222420 \, x^{2} + 127244576 \, x + 26779805}{11858 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} + \frac {792500}{1331} \, \log \left (5 \, x + 3\right ) - \frac {204228}{343} \, \log \left (3 \, x + 2\right ) - \frac {32}{456533} \, \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+3*x)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/11858*(105906600*x^3 + 201222420*x^2 + 127244576*x + 26779805)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36) +
792500/1331*log(5*x + 3) - 204228/343*log(3*x + 2) - 32/456533*log(2*x - 1)

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mupad [B]  time = 0.05, size = 55, normalized size = 0.73 \begin {gather*} \frac {792500\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {204228\,\ln \left (x+\frac {2}{3}\right )}{343}-\frac {32\,\ln \left (x-\frac {1}{2}\right )}{456533}+\frac {\frac {235348\,x^3}{5929}+\frac {136886\,x^2}{1815}+\frac {63622288\,x}{1334025}+\frac {5355961}{533610}}{x^4+\frac {38\,x^3}{15}+\frac {541\,x^2}{225}+\frac {76\,x}{75}+\frac {4}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(792500*log(x + 3/5))/1331 - (204228*log(x + 2/3))/343 - (32*log(x - 1/2))/456533 + ((63622288*x)/1334025 + (1
36886*x^2)/1815 + (235348*x^3)/5929 + 5355961/533610)/((76*x)/75 + (541*x^2)/225 + (38*x^3)/15 + x^4 + 4/25)

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sympy [A]  time = 0.25, size = 66, normalized size = 0.88 \begin {gather*} - \frac {- 105906600 x^{3} - 201222420 x^{2} - 127244576 x - 26779805}{2668050 x^{4} + 6759060 x^{3} + 6415178 x^{2} + 2703624 x + 426888} - \frac {32 \log {\left (x - \frac {1}{2} \right )}}{456533} + \frac {792500 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {204228 \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+3*x)**3/(3+5*x)**3,x)

[Out]

-(-105906600*x**3 - 201222420*x**2 - 127244576*x - 26779805)/(2668050*x**4 + 6759060*x**3 + 6415178*x**2 + 270
3624*x + 426888) - 32*log(x - 1/2)/456533 + 792500*log(x + 3/5)/1331 - 204228*log(x + 2/3)/343

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