3.1602 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=97 \[ \frac{64}{1294139 (1-2 x)}+\frac{15192225}{117649 (3 x+2)}+\frac{434043}{33614 (3 x+2)^2}+\frac{4131}{2401 (3 x+2)^3}+\frac{351}{1372 (3 x+2)^4}+\frac{9}{245 (3 x+2)^5}-\frac{14912 \log (1-2 x)}{99648703}-\frac{531729603 \log (3 x+2)}{823543}+\frac{78125}{121} \log (5 x+3) \]

[Out]

64/(1294139*(1 - 2*x)) + 9/(245*(2 + 3*x)^5) + 351/(1372*(2 + 3*x)^4) + 4131/(2401*(2 + 3*x)^3) + 434043/(3361
4*(2 + 3*x)^2) + 15192225/(117649*(2 + 3*x)) - (14912*Log[1 - 2*x])/99648703 - (531729603*Log[2 + 3*x])/823543
+ (78125*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0544353, 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} \[ \frac{64}{1294139 (1-2 x)}+\frac{15192225}{117649 (3 x+2)}+\frac{434043}{33614 (3 x+2)^2}+\frac{4131}{2401 (3 x+2)^3}+\frac{351}{1372 (3 x+2)^4}+\frac{9}{245 (3 x+2)^5}-\frac{14912 \log (1-2 x)}{99648703}-\frac{531729603 \log (3 x+2)}{823543}+\frac{78125}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^2*(2 + 3*x)^6*(3 + 5*x)),x]

[Out]

64/(1294139*(1 - 2*x)) + 9/(245*(2 + 3*x)^5) + 351/(1372*(2 + 3*x)^4) + 4131/(2401*(2 + 3*x)^3) + 434043/(3361
4*(2 + 3*x)^2) + 15192225/(117649*(2 + 3*x)) - (14912*Log[1 - 2*x])/99648703 - (531729603*Log[2 + 3*x])/823543
+ (78125*Log[3 + 5*x])/121

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)^6 (3+5 x)} \, dx &=\int \left (\frac{128}{1294139 (-1+2 x)^2}-\frac{29824}{99648703 (-1+2 x)}-\frac{27}{49 (2+3 x)^6}-\frac{1053}{343 (2+3 x)^5}-\frac{37179}{2401 (2+3 x)^4}-\frac{1302129}{16807 (2+3 x)^3}-\frac{45576675}{117649 (2+3 x)^2}-\frac{1595188809}{823543 (2+3 x)}+\frac{390625}{121 (3+5 x)}\right ) \, dx\\ &=\frac{64}{1294139 (1-2 x)}+\frac{9}{245 (2+3 x)^5}+\frac{351}{1372 (2+3 x)^4}+\frac{4131}{2401 (2+3 x)^3}+\frac{434043}{33614 (2+3 x)^2}+\frac{15192225}{117649 (2+3 x)}-\frac{14912 \log (1-2 x)}{99648703}-\frac{531729603 \log (2+3 x)}{823543}+\frac{78125}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.047487, size = 87, normalized size = 0.9 \[ \frac{\frac{19712}{1-2 x}+\frac{51471258300}{3 x+2}+\frac{5146881894}{(3 x+2)^2}+\frac{685795572}{(3 x+2)^3}+\frac{101972871}{(3 x+2)^4}+\frac{73211292}{5 (3 x+2)^5}-59648 \log (5-10 x)-257357127852 \log (5 (3 x+2))[phone] \log (5 x+3)}{398594812} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^2*(2 + 3*x)^6*(3 + 5*x)),x]

[Out]

(19712/(1 - 2*x) + 73211292/(5*(2 + 3*x)^5) + 101972871/(2 + 3*x)^4 + 685795572/(2 + 3*x)^3 + 5146881894/(2 +
3*x)^2 + 51471258300/(2 + 3*x) - 59648*Log[5 - 10*x] - 257357127852*Log[5*(2 + 3*x)] + 257357187500*Log[3 + 5*
x])/398594812

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Maple [A]  time = 0.011, size = 80, normalized size = 0.8 \begin{align*} -{\frac{64}{2588278\,x-1294139}}-{\frac{14912\,\ln \left ( 2\,x-1 \right ) }{99648703}}+{\frac{9}{245\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{351}{1372\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{4131}{2401\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{434043}{33614\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{15192225}{235298+352947\,x}}-{\frac{531729603\,\ln \left ( 2+3\,x \right ) }{823543}}+{\frac{78125\,\ln \left ( 3+5\,x \right ) }{121}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^2/(2+3*x)^6/(3+5*x),x)

[Out]

-64/1294139/(2*x-1)-14912/99648703*ln(2*x-1)+9/245/(2+3*x)^5+351/1372/(2+3*x)^4+4131/2401/(2+3*x)^3+434043/336
14/(2+3*x)^2+15192225/117649/(2+3*x)-531729603/823543*ln(2+3*x)+78125/121*ln(3+5*x)

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Maxima [A]  time = 1.03486, size = 113, normalized size = 1.16 \begin{align*} \frac{541450587960 \, x^{5} + 1191190085640 \, x^{4} + 749805990750 \, x^{3} - 73492321230 \, x^{2} - 220760702913 \, x - 56342700586}{25882780 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} + \frac{78125}{121} \, \log \left (5 \, x + 3\right ) - \frac{531729603}{823543} \, \log \left (3 \, x + 2\right ) - \frac{14912}{99648703} \, \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)^6/(3+5*x),x, algorithm="maxima")

[Out]

1/25882780*(541450587960*x^5 + 1191190085640*x^4 + 749805990750*x^3 - 73492321230*x^2 - 220760702913*x - 56342
700586)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32) + 78125/121*log(5*x + 3) - 531729603/
823543*log(3*x + 2) - 14912/99648703*log(2*x - 1)

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Fricas [B]  time = 1.27316, size = 637, normalized size = 6.57 \begin{align*} \frac{41691695272920 \, x^{5} + 91721636594280 \, x^{4} + 57735061287750 \, x^{3} - 5658908734710 \, x^{2} + 1286785937500 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (5 \, x + 3\right ) - 1286785639260 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (3 \, x + 2\right ) - 298240 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (2 \, x - 1\right ) - 16998574124301 \, x - 4338387945122}{1992974060 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^2/(2+3*x)^6/(3+5*x),x, algorithm="fricas")

[Out]

1/1992974060*(41691695272920*x^5 + 91721636594280*x^4 + 57735061287750*x^3 - 5658908734710*x^2 + 1286785937500
*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log(5*x + 3) - 1286785639260*(486*x^6 + 1377
*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log(3*x + 2) - 298240*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x
^3 - 240*x^2 - 176*x - 32)*log(2*x - 1) - 16998574124301*x - 4338387945122)/(486*x^6 + 1377*x^5 + 1350*x^4 + 3
60*x^3 - 240*x^2 - 176*x - 32)

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Sympy [A]  time = 0.255427, size = 85, normalized size = 0.88 \begin{align*} \frac{541450587960 x^{5} + 1191190085640 x^{4} + 749805990750 x^{3} - 73492321230 x^{2} - 220760702913 x - 56342700586}{12579031080 x^{6} + 35640588060 x^{5} + 34941753000 x^{4} + 9317800800 x^{3} - 6211867200 x^{2} - 4555369280 x - 828248960} - \frac{14912 \log{\left (x - \frac{1}{2} \right )}}{99648703} + \frac{78125 \log{\left (x + \frac{3}{5} \right )}}{121} - \frac{531729603 \log{\left (x + \frac{2}{3} \right )}}{823543} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)**2/(2+3*x)**6/(3+5*x),x)

[Out]

(541450587960*x**5 + 1191190085640*x**4 + 749805990750*x**3 - 73492321230*x**2 - 220760702913*x - 56342700586)
/(12579031080*x**6 + 35640588060*x**5 + 34941753000*x**4 + 9317800800*x**3 - 6211867200*x**2 - 4555369280*x -
828248960) - 14912*log(x - 1/2)/99648703 + 78125*log(x + 3/5)/121 - 531729603*log(x + 2/3)/823543

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Giac [A]  time = 2.28072, size = 126, normalized size = 1.3 \begin{align*} -\frac{64}{1294139 \,{\left (2 \, x - 1\right )}} - \frac{54 \,{\left (\frac{6617665845}{2 \, x - 1} + \frac{23331909825}{{\left (2 \, x - 1\right )}^{2}} + \frac{36565643625}{{\left (2 \, x - 1\right )}^{3}} + \frac{21492731575}{{\left (2 \, x - 1\right )}^{4}} + 703958526\right )}}{4117715 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{5}} - \frac{531729603}{823543} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) + \frac{78125}{121} \, \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)^6/(3+5*x),x, algorithm="giac")

[Out]

-64/1294139/(2*x - 1) - 54/4117715*(6617665845/(2*x - 1) + 23331909825/(2*x - 1)^2 + 36565643625/(2*x - 1)^3 +
21492731575/(2*x - 1)^4 + 703958526)/(7/(2*x - 1) + 3)^5 - 531729603/823543*log(abs(-7/(2*x - 1) - 3)) + 7812
5/121*log(abs(-11/(2*x - 1) - 5))