3.1629 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^3} \, dx\)

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

[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

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Rubi [A]  time = 0.053388, antiderivative size = 97, 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} \[ \frac{64}{3195731 (1-2 x)}+\frac{630342}{2401 (3 x+2)}+\frac{400000}{1331 (5 x+3)}+\frac{8829}{686 (3 x+2)^2}-\frac{3125}{242 (5 x+3)^2}+\frac{27}{49 (3 x+2)^3}-\frac{15168 \log (1-2 x)}{246071287}-\frac{37214802 \log (3 x+2)}{16807}+\frac{32418750 \log (5 x+3)}{14641} \]

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 = 0.0866555, size = 88, normalized size = 0.91 \[ \frac{2 \left (\frac{77}{4} \left (\frac{1677970404}{3 x+2}+\frac{1920800000}{5 x+3}+\frac{82259793}{(3 x+2)^2}-\frac{82534375}{(5 x+3)^2}+\frac{3521826}{(3 x+2)^3}+\frac{128}{1-2 x}\right )-7584 \log (1-2 x)-272430958041 \log (6 x+4)[phone] \log (10 x+6)\right )}{246071287} \]

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

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Maple [A]  time = 0.011, size = 80, normalized size = 0.8 \begin{align*} -{\frac{64}{6391462\,x-3195731}}-{\frac{15168\,\ln \left ( 2\,x-1 \right ) }{246071287}}+{\frac{27}{49\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{8829}{686\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{630342}{4802+7203\,x}}-{\frac{37214802\,\ln \left ( 2+3\,x \right ) }{16807}}-{\frac{3125}{242\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{400000}{3993+6655\,x}}+{\frac{32418750\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06737, size = 113, normalized size = 1.16 \begin{align*} \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{align*}

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)

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Fricas [B]  time = 1.27368, size = 644, normalized size = 6.64 \begin{align*} \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{align*}

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)

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Sympy [A]  time = 0.254373, size = 85, normalized size = 0.88 \begin{align*} \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{align*}

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

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Giac [A]  time = 2.32259, size = 143, normalized size = 1.47 \begin{align*} -\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{align*}

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))