3.1527 \(\int \frac{1}{(1-2 x) (2+3 x)^3 (3+5 x)^3} \, dx\)

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

[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

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Rubi [A]  time = 0.0348291, antiderivative size = 75, normalized size of antiderivative = 1., 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} \[ \frac{2889}{49 (3 x+2)}+\frac{12125}{121 (5 x+3)}+\frac{27}{14 (3 x+2)^2}-\frac{125}{22 (5 x+3)^2}-\frac{32 \log (1-2 x)}{456533}-\frac{204228}{343} \log (3 x+2)+\frac{792500 \log (5 x+3)}{1331} \]

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*}

Mathematica [A]  time = 0.0278655, size = 71, normalized size = 0.95 \[ \frac{2889}{147 x+98}+\frac{12125}{605 x+363}+\frac{27}{14 (3 x+2)^2}-\frac{125}{22 (5 x+3)^2}-\frac{32 \log (1-2 x)}{456533}-\frac{204228}{343} \log (6 x+4)+\frac{792500 \log (10 x+6)}{1331} \]

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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Maple [A]  time = 0.01, size = 62, normalized size = 0.8 \begin{align*} -{\frac{32\,\ln \left ( 2\,x-1 \right ) }{456533}}+{\frac{27}{14\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{2889}{98+147\,x}}-{\frac{204228\,\ln \left ( 2+3\,x \right ) }{343}}-{\frac{125}{22\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{12125}{363+605\,x}}+{\frac{792500\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/456533*ln(2*x-1)+27/14/(2+3*x)^2+2889/49/(2+3*x)-204228/343*ln(2+3*x)-125/22/(3+5*x)^2+12125/121/(3+5*x)+7
92500/1331*ln(3+5*x)

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Maxima [A]  time = 1.18479, size = 86, normalized size = 1.15 \begin{align*} \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{align*}

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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Fricas [B]  time = 1.38534, size = 423, normalized size = 5.64 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.224259, size = 65, normalized size = 0.87 \begin{align*} \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{align*}

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 + 27036
24*x + 426888) - 32*log(x - 1/2)/456533 + 792500*log(x + 3/5)/1331 - 204228*log(x + 2/3)/343

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Giac [A]  time = 2.5801, size = 80, normalized size = 1.07 \begin{align*} \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{align*}

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))