[next] [prev] [prev-tail] [tail] [up]

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

Optimal. Leaf size=97 \[ \frac {64}{3195731 (1-2 x)}+\frac {630342}{2401 (3 x+2)}+\frac {400000}{1331 (5 x+3)}+\frac {8829}{686 (3 x+2)^2}-\frac {3125}{242 (5 x+3)^2}+\frac {27}{49 (3 x+2)^3}-\frac {15168 \log (1-2 x)}{246071287}-\frac {37214802 \log (3 x+2)}{16807}+\frac {32418750 \log (5 x+3)}{14641} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 97, 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 {64}{3195731 (1-2 x)}+\frac {630342}{2401 (3 x+2)}+\frac {400000}{1331 (5 x+3)}+\frac {8829}{686 (3 x+2)^2}-\frac {3125}{242 (5 x+3)^2}+\frac {27}{49 (3 x+2)^3}-\frac {15168 \log (1-2 x)}{246071287}-\frac {37214802 \log (3 x+2)}{16807}+\frac {32418750 \log (5 x+3)}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

64/(3195731*(1 - 2*x)) + 27/(49*(2 + 3*x)^3) + 8829/(686*(2 + 3*x)^2) + 630342/(2401*(2 + 3*x)) - 3125/(242*(3
 + 5*x)^2) + 400000/(1331*(3 + 5*x)) - (15168*Log[1 - 2*x])/246071287 - (37214802*Log[2 + 3*x])/16807 + (32418
750*Log[3 + 5*x])/14641

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)^4 (3+5 x)^3} \, dx &=\int \left (\frac {128}{3195731 (-1+2 x)^2}-\frac {30336}{246071287 (-1+2 x)}-\frac {243}{49 (2+3 x)^4}-\frac {26487}{343 (2+3 x)^3}-\frac {1891026}{2401 (2+3 x)^2}-\frac {111644406}{16807 (2+3 x)}+\frac {15625}{121 (3+5 x)^3}-\frac {2000000}{1331 (3+5 x)^2}+\frac {162093750}{14641 (3+5 x)}\right ) \, dx\\ &=\frac {64}{3195731 (1-2 x)}+\frac {27}{49 (2+3 x)^3}+\frac {8829}{686 (2+3 x)^2}+\frac {630342}{2401 (2+3 x)}-\frac {3125}{242 (3+5 x)^2}+\frac {400000}{1331 (3+5 x)}-\frac {15168 \log (1-2 x)}{246071287}-\frac {37214802 \log (2+3 x)}{16807}+\frac {32418750 \log (3+5 x)}{14641}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 88, normalized size = 0.91 \begin {gather*} \frac {2 \left (\frac {77}{4} \left (\frac {1677970404}{3 x+2}+\frac {1920800000}{5 x+3}+\frac {82259793}{(3 x+2)^2}-\frac {82534375}{(5 x+3)^2}+\frac {3521826}{(3 x+2)^3}+\frac {128}{1-2 x}\right )-7584 \log (1-2 x)-272430958041 \log (6 x+4)+272430965625 \log (10 x+6)\right )}{246071287} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((77*(128/(1 - 2*x) + 3521826/(2 + 3*x)^3 + 82259793/(2 + 3*x)^2 + 1677970404/(2 + 3*x) - 82534375/(3 + 5*x
)^2 + 1920800000/(3 + 5*x)))/4 - 7584*Log[1 - 2*x] - 272430958041*Log[4 + 6*x] + 272430965625*Log[6 + 10*x]))/
246071287

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.26, size = 173, normalized size = 1.78 \begin {gather*} \frac {98075099845800 \, x^{5} + 202688513402760 \, x^{4} + 116200773740898 \, x^{3} - 17675352484482 \, x^{2} + 1089723862500 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} \log \left (5 \, x + 3\right ) - 1089723832164 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} \log \left (3 \, x + 2\right ) - 30336 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} \log \left (2 \, x - 1\right ) - 35145131488467 \, x - 8266584593036}{492142574 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/492142574*(98075099845800*x^5 + 202688513402760*x^4 + 116200773740898*x^3 - 17675352484482*x^2 + 10897238625
00*(1350*x^6 + 3645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)*log(5*x + 3) - 1089723832164*(1350*x^6 +
3645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)*log(3*x + 2) - 30336*(1350*x^6 + 3645*x^5 + 3366*x^4 + 7
69*x^3 - 638*x^2 - 420*x - 72)*log(2*x - 1) - 35145131488467*x - 8266584593036)/(1350*x^6 + 3645*x^5 + 3366*x^
4 + 769*x^3 - 638*x^2 - 420*x - 72)

________________________________________________________________________________________

giac [A]  time = 1.18, size = 106, normalized size = 1.09 \begin {gather*} -\frac {64}{3195731 \, {\left (2 \, x - 1\right )}} - \frac {4 \, {\left (\frac {49415890344165}{2 \, x - 1} + \frac {169212487575969}{{\left (2 \, x - 1\right )}^{2}} + \frac {257446971133345}{{\left (2 \, x - 1\right )}^{3}} + \frac {146840081089779}{{\left (2 \, x - 1\right )}^{4}} + 5410112162850\right )}}{246071287 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2} {\left (\frac {7}{2 \, x - 1} + 3\right )}^{3}} - \frac {37214802}{16807} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) + \frac {32418750}{14641} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-64/3195731/(2*x - 1) - 4/246071287*(49415890344165/(2*x - 1) + 169212487575969/(2*x - 1)^2 + 257446971133345/
(2*x - 1)^3 + 146840081089779/(2*x - 1)^4 + 5410112162850)/((11/(2*x - 1) + 5)^2*(7/(2*x - 1) + 3)^3) - 372148
02/16807*log(abs(-7/(2*x - 1) - 3)) + 32418750/14641*log(abs(-11/(2*x - 1) - 5))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 80, normalized size = 0.82 \begin {gather*} -\frac {15168 \ln \left (2 x -1\right )}{246071287}-\frac {37214802 \ln \left (3 x +2\right )}{16807}+\frac {32418750 \ln \left (5 x +3\right )}{14641}-\frac {3125}{242 \left (5 x +3\right )^{2}}+\frac {400000}{1331 \left (5 x +3\right )}+\frac {27}{49 \left (3 x +2\right )^{3}}+\frac {8829}{686 \left (3 x +2\right )^{2}}+\frac {630342}{2401 \left (3 x +2\right )}-\frac {64}{3195731 \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)^4/(5*x+3)^3,x)

[Out]

-3125/242/(5*x+3)^2+400000/1331/(5*x+3)+32418750/14641*ln(5*x+3)+27/49/(3*x+2)^3+8829/686/(3*x+2)^2+630342/240
1/(3*x+2)-37214802/16807*ln(3*x+2)-64/3195731/(2*x-1)-15168/246071287*ln(2*x-1)

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 84, normalized size = 0.87 \begin {gather*} \frac {1273702595400 \, x^{5} + 2632318355880 \, x^{4} + 1509100957674 \, x^{3} - 229550032266 \, x^{2} - 456430279071 \, x - 107358241468}{6391462 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )}} + \frac {32418750}{14641} \, \log \left (5 \, x + 3\right ) - \frac {37214802}{16807} \, \log \left (3 \, x + 2\right ) - \frac {15168}{246071287} \, \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)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/6391462*(1273702595400*x^5 + 2632318355880*x^4 + 1509100957674*x^3 - 229550032266*x^2 - 456430279071*x - 107
358241468)/(1350*x^6 + 3645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72) + 32418750/14641*log(5*x + 3) - 3
7214802/16807*log(3*x + 2) - 15168/246071287*log(2*x - 1)

________________________________________________________________________________________

mupad [B]  time = 1.07, size = 76, normalized size = 0.78 \begin {gather*} \frac {32418750\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {37214802\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {15168\,\ln \left (x-\frac {1}{2}\right )}{246071287}-\frac {-\frac {471741702\,x^5}{3195731}-\frac {4874663622\,x^4}{15978655}-\frac {27946314031\,x^3}{159786550}+\frac {38258338711\,x^2}{1438078950}+\frac {152143426357\,x}{2876157900}+\frac {26839560367}{2157118425}}{x^6+\frac {27\,x^5}{10}+\frac {187\,x^4}{75}+\frac {769\,x^3}{1350}-\frac {319\,x^2}{675}-\frac {14\,x}{45}-\frac {4}{75}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(32418750*log(x + 3/5))/14641 - (37214802*log(x + 2/3))/16807 - (15168*log(x - 1/2))/246071287 - ((15214342635
7*x)/2876157900 + (38258338711*x^2)/1438078950 - (27946314031*x^3)/159786550 - (4874663622*x^4)/15978655 - (47
1741702*x^5)/3195731 + 26839560367/2157118425)/((769*x^3)/1350 - (319*x^2)/675 - (14*x)/45 + (187*x^4)/75 + (2
7*x^5)/10 + x^6 - 4/75)

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 85, normalized size = 0.88 \begin {gather*} \frac {1273702595400 x^{5} + 2632318355880 x^{4} + 1509100957674 x^{3} - 229550032266 x^{2} - 456430279071 x - 107358241468}{8628473700 x^{6} + 23296878990 x^{5} + 21513661092 x^{4} + 4915034278 x^{3} - 4077752756 x^{2} - 2684414040 x - 460185264} - \frac {15168 \log {\left (x - \frac {1}{2} \right )}}{246071287} + \frac {32418750 \log {\left (x + \frac {3}{5} \right )}}{14641} - \frac {37214802 \log {\left (x + \frac {2}{3} \right )}}{16807} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1273702595400*x**5 + 2632318355880*x**4 + 1509100957674*x**3 - 229550032266*x**2 - 456430279071*x - 107358241
468)/(8628473700*x**6 + 23296878990*x**5 + 21513661092*x**4 + 4915034278*x**3 - 4077752756*x**2 - 2684414040*x
 - 460185264) - 15168*log(x - 1/2)/246071287 + 32418750*log(x + 3/5)/14641 - 37214802*log(x + 2/3)/16807

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]