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)-64/3195731*ln(1-2*x)-119
84706/2401*ln(2+3*x)+6643750/1331*ln(3+5*x)

________________________________________________________________________________________

Rubi [A]  time = 0.04, 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} \[ \frac {204228}{343 (3 x+2)}+\frac {81250}{121 (5 x+3)}+\frac {2889}{98 (3 x+2)^2}-\frac {625}{22 (5 x+3)^2}+\frac {9}{7 (3 x+2)^3}-\frac {64 \log (1-2 x)}{3195731}-\frac {11984706 \log (3 x+2)}{2401}+\frac {6643750 \log (5 x+3)}{1331} \]

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 = 0.06, 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

________________________________________________________________________________________

fricas [B]  time = 0.87, size = 148, normalized size = 1.72 \[ \frac {1435647836700 \, x^{4} + 3684829392090 \, x^{3} + 3543391400472 \, x^{2} + 31903287500 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 31903287372 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) - 128 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (2 \, x - 1\right ) + 1512959972293 \, x + 242016815656}{6391462 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 1.13, size = 64, normalized size = 0.74 \[ \frac {18644777100 \, x^{4} + 47854927170 \, x^{3} + 46018070136 \, x^{2} + 19648830809 \, x + 3143075528}{83006 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{3}} + \frac {6643750}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {11984706}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {64}{3195731} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 71, normalized size = 0.83 \[ -\frac {64 \ln \left (2 x -1\right )}{3195731}-\frac {11984706 \ln \left (3 x +2\right )}{2401}+\frac {6643750 \ln \left (5 x +3\right )}{1331}-\frac {625}{22 \left (5 x +3\right )^{2}}+\frac {81250}{121 \left (5 x +3\right )}+\frac {9}{7 \left (3 x +2\right )^{3}}+\frac {2889}{98 \left (3 x +2\right )^{2}}+\frac {204228}{343 \left (3 x +2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 74, normalized size = 0.86 \[ \frac {18644777100 \, x^{4} + 47854927170 \, x^{3} + 46018070136 \, x^{2} + 19648830809 \, x + 3143075528}{83006 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + \frac {6643750}{1331} \, \log \left (5 \, x + 3\right ) - \frac {11984706}{2401} \, \log \left (3 \, x + 2\right ) - \frac {64}{3195731} \, \log \left (2 \, x - 1\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.12, size = 65, normalized size = 0.76 \[ \frac {6643750\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {11984706\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {64\,\ln \left (x-\frac {1}{2}\right )}{3195731}+\frac {\frac {13810946\,x^4}{41503}+\frac {177240471\,x^3}{207515}+\frac {852186484\,x^2}{1037575}+\frac {19648830809\,x}{56029050}+\frac {1571537764}{28014525}}{x^5+\frac {16\,x^4}{5}+\frac {307\,x^3}{75}+\frac {1766\,x^2}{675}+\frac {188\,x}{225}+\frac {8}{75}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6643750*log(x + 3/5))/1331 - (11984706*log(x + 2/3))/2401 - (64*log(x - 1/2))/3195731 + ((19648830809*x)/5602
9050 + (852186484*x^2)/1037575 + (177240471*x^3)/207515 + (13810946*x^4)/41503 + 1571537764/28014525)/((188*x)
/225 + (1766*x^2)/675 + (307*x^3)/75 + (16*x^4)/5 + x^5 + 8/75)

________________________________________________________________________________________

sympy [A]  time = 0.26, size = 76, normalized size = 0.88 \[ - \frac {- 18644777100 x^{4} - 47854927170 x^{3} - 46018070136 x^{2} - 19648830809 x - 3143075528}{56029050 x^{5} + 179292960 x^{4} + 229345578 x^{3} + 146588596 x^{2} + 46815384 x + 5976432} - \frac {64 \log {\left (x - \frac {1}{2} \right )}}{3195731} + \frac {6643750 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {11984706 \log {\left (x + \frac {2}{3} \right )}}{2401} \]

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 + 17929
2960*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

________________________________________________________________________________________