3.1528 \(\int \frac{1}{(1-2 x) (2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{204228}{343 (3 x+2)}+\frac{81250}{121 (5 x+3)}+\frac{2889}{98 (3 x+2)^2}-\frac{625}{22 (5 x+3)^2}+\frac{9}{7 (3 x+2)^3}-\frac{64 \log (1-2 x)}{3195731}-\frac{11984706 \log (3 x+2)}{2401}+\frac{6643750 \log (5 x+3)}{1331} \]

[Out]

9/(7*(2 + 3*x)^3) + 2889/(98*(2 + 3*x)^2) + 204228/(343*(2 + 3*x)) - 625/(22*(3 + 5*x)^2) + 81250/(121*(3 + 5*
x)) - (64*Log[1 - 2*x])/3195731 - (11984706*Log[2 + 3*x])/2401 + (6643750*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0434438, antiderivative size = 86, 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} \[ \frac{204228}{343 (3 x+2)}+\frac{81250}{121 (5 x+3)}+\frac{2889}{98 (3 x+2)^2}-\frac{625}{22 (5 x+3)^2}+\frac{9}{7 (3 x+2)^3}-\frac{64 \log (1-2 x)}{3195731}-\frac{11984706 \log (3 x+2)}{2401}+\frac{6643750 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

9/(7*(2 + 3*x)^3) + 2889/(98*(2 + 3*x)^2) + 204228/(343*(2 + 3*x)) - 625/(22*(3 + 5*x)^2) + 81250/(121*(3 + 5*
x)) - (64*Log[1 - 2*x])/3195731 - (11984706*Log[2 + 3*x])/2401 + (6643750*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)^4 (3+5 x)^3} \, dx &=\int \left (-\frac{128}{3195731 (-1+2 x)}-\frac{81}{7 (2+3 x)^4}-\frac{8667}{49 (2+3 x)^3}-\frac{612684}{343 (2+3 x)^2}-\frac{35954118}{2401 (2+3 x)}+\frac{3125}{11 (3+5 x)^3}-\frac{406250}{121 (3+5 x)^2}+\frac{33218750}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{9}{7 (2+3 x)^3}+\frac{2889}{98 (2+3 x)^2}+\frac{204228}{343 (2+3 x)}-\frac{625}{22 (3+5 x)^2}+\frac{81250}{121 (3+5 x)}-\frac{64 \log (1-2 x)}{3195731}-\frac{11984706 \log (2+3 x)}{2401}+\frac{6643750 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0308134, size = 84, normalized size = 0.98 \[ \frac{204228}{343 (3 x+2)}+\frac{81250}{605 x+363}+\frac{2889}{98 (3 x+2)^2}-\frac{625}{22 (5 x+3)^2}+\frac{9}{7 (3 x+2)^3}-\frac{64 \log (1-2 x)}{3195731}-\frac{11984706 \log (6 x+4)}{2401}+\frac{6643750 \log (10 x+6)}{1331} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

9/(7*(2 + 3*x)^3) + 2889/(98*(2 + 3*x)^2) + 204228/(343*(2 + 3*x)) - 625/(22*(3 + 5*x)^2) + 81250/(363 + 605*x
) - (64*Log[1 - 2*x])/3195731 - (11984706*Log[4 + 6*x])/2401 + (6643750*Log[6 + 10*x])/1331

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Maple [A]  time = 0.01, size = 71, normalized size = 0.8 \begin{align*} -{\frac{64\,\ln \left ( 2\,x-1 \right ) }{3195731}}+{\frac{9}{7\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{2889}{98\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{204228}{686+1029\,x}}-{\frac{11984706\,\ln \left ( 2+3\,x \right ) }{2401}}-{\frac{625}{22\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{81250}{363+605\,x}}+{\frac{6643750\,\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)^4/(3+5*x)^3,x)

[Out]

-64/3195731*ln(2*x-1)+9/7/(2+3*x)^3+2889/98/(2+3*x)^2+204228/343/(2+3*x)-11984706/2401*ln(2+3*x)-625/22/(3+5*x
)^2+81250/121/(3+5*x)+6643750/1331*ln(3+5*x)

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Maxima [A]  time = 1.16461, size = 100, normalized size = 1.16 \begin{align*} \frac{18644777100 \, x^{4} + 47854927170 \, x^{3} + 46018070136 \, x^{2} + 19648830809 \, x + 3143075528}{83006 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + \frac{6643750}{1331} \, \log \left (5 \, x + 3\right ) - \frac{11984706}{2401} \, \log \left (3 \, x + 2\right ) - \frac{64}{3195731} \, \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)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/83006*(18644777100*x^4 + 47854927170*x^3 + 46018070136*x^2 + 19648830809*x + 3143075528)/(675*x^5 + 2160*x^4
+ 2763*x^3 + 1766*x^2 + 564*x + 72) + 6643750/1331*log(5*x + 3) - 11984706/2401*log(3*x + 2) - 64/3195731*log
(2*x - 1)

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Fricas [B]  time = 1.37393, size = 541, normalized size = 6.29 \begin{align*} \frac{1435647836700 \, x^{4} + 3684829392090 \, x^{3} + 3543391400472 \, x^{2} + 31903287500 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 31903287372 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) - 128 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (2 \, x - 1\right ) + 1512959972293 \, x + 242016815656}{6391462 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/6391462*(1435647836700*x^4 + 3684829392090*x^3 + 3543391400472*x^2 + 31903287500*(675*x^5 + 2160*x^4 + 2763*
x^3 + 1766*x^2 + 564*x + 72)*log(5*x + 3) - 31903287372*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72
)*log(3*x + 2) - 128*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(2*x - 1) + 1512959972293*x +
242016815656)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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Sympy [A]  time = 0.241265, size = 75, normalized size = 0.87 \begin{align*} \frac{18644777100 x^{4} + 47854927170 x^{3} + 46018070136 x^{2} + 19648830809 x + 3143075528}{56029050 x^{5} + 179292960 x^{4} + 229345578 x^{3} + 146588596 x^{2} + 46815384 x + 5976432} - \frac{64 \log{\left (x - \frac{1}{2} \right )}}{3195731} + \frac{6643750 \log{\left (x + \frac{3}{5} \right )}}{1331} - \frac{11984706 \log{\left (x + \frac{2}{3} \right )}}{2401} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)**4/(3+5*x)**3,x)

[Out]

(18644777100*x**4 + 47854927170*x**3 + 46018070136*x**2 + 19648830809*x + 3143075528)/(56029050*x**5 + 1792929
60*x**4 + 229345578*x**3 + 146588596*x**2 + 46815384*x + 5976432) - 64*log(x - 1/2)/3195731 + 6643750*log(x +
3/5)/1331 - 11984706*log(x + 2/3)/2401

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Giac [A]  time = 3.43601, size = 86, normalized size = 1. \begin{align*} \frac{18644777100 \, x^{4} + 47854927170 \, x^{3} + 46018070136 \, x^{2} + 19648830809 \, x + 3143075528}{83006 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{3}} + \frac{6643750}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{11984706}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{64}{3195731} \, \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)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

1/83006*(18644777100*x^4 + 47854927170*x^3 + 46018070136*x^2 + 19648830809*x + 3143075528)/((5*x + 3)^2*(3*x +
2)^3) + 6643750/1331*log(abs(5*x + 3)) - 11984706/2401*log(abs(3*x + 2)) - 64/3195731*log(abs(2*x - 1))