3.1628 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{32}{456533 (1-2 x)}+\frac{8829}{343 (3 x+2)}+\frac{59375}{1331 (5 x+3)}+\frac{81}{98 (3 x+2)^2}-\frac{625}{242 (5 x+3)^2}-\frac{6528 \log (1-2 x)}{35153041}-\frac{630342 \log (3 x+2)}{2401}+\frac{3843750 \log (5 x+3)}{14641} \]

[Out]

32/(456533*(1 - 2*x)) + 81/(98*(2 + 3*x)^2) + 8829/(343*(2 + 3*x)) - 625/(242*(3 + 5*x)^2) + 59375/(1331*(3 +
5*x)) - (6528*Log[1 - 2*x])/35153041 - (630342*Log[2 + 3*x])/2401 + (3843750*Log[3 + 5*x])/14641

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Rubi [A]  time = 0.0435486, 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} \[ \frac{32}{456533 (1-2 x)}+\frac{8829}{343 (3 x+2)}+\frac{59375}{1331 (5 x+3)}+\frac{81}{98 (3 x+2)^2}-\frac{625}{242 (5 x+3)^2}-\frac{6528 \log (1-2 x)}{35153041}-\frac{630342 \log (3 x+2)}{2401}+\frac{3843750 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

32/(456533*(1 - 2*x)) + 81/(98*(2 + 3*x)^2) + 8829/(343*(2 + 3*x)) - 625/(242*(3 + 5*x)^2) + 59375/(1331*(3 +
5*x)) - (6528*Log[1 - 2*x])/35153041 - (630342*Log[2 + 3*x])/2401 + (3843750*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)^3 (3+5 x)^3} \, dx &=\int \left (\frac{64}{456533 (-1+2 x)^2}-\frac{13056}{35153041 (-1+2 x)}-\frac{243}{49 (2+3 x)^3}-\frac{26487}{343 (2+3 x)^2}-\frac{1891026}{2401 (2+3 x)}+\frac{3125}{121 (3+5 x)^3}-\frac{296875}{1331 (3+5 x)^2}+\frac{19218750}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{32}{456533 (1-2 x)}+\frac{81}{98 (2+3 x)^2}+\frac{8829}{343 (2+3 x)}-\frac{625}{242 (3+5 x)^2}+\frac{59375}{1331 (3+5 x)}-\frac{6528 \log (1-2 x)}{35153041}-\frac{630342 \log (2+3 x)}{2401}+\frac{3843750 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.105074, size = 79, normalized size = 0.92 \[ \frac{2 \left (\frac{77}{4} \left (\frac{23502798}{3 x+2}+\frac{40731250}{5 x+3}+\frac{754677}{(3 x+2)^2}-\frac{2358125}{(5 x+3)^2}+\frac{64}{1-2 x}\right )-3264 \log (1-2 x)-4614418611 \log (6 x+4)[phone] \log (10 x+6)\right )}{35153041} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((77*(64/(1 - 2*x) + 754677/(2 + 3*x)^2 + 23502798/(2 + 3*x) - 2358125/(3 + 5*x)^2 + 40731250/(3 + 5*x)))/4
- 3264*Log[1 - 2*x] - 4614418611*Log[4 + 6*x] + 4614421875*Log[6 + 10*x]))/35153041

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Maple [A]  time = 0.01, size = 71, normalized size = 0.8 \begin{align*} -{\frac{32}{913066\,x-456533}}-{\frac{6528\,\ln \left ( 2\,x-1 \right ) }{35153041}}+{\frac{81}{98\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{8829}{686+1029\,x}}-{\frac{630342\,\ln \left ( 2+3\,x \right ) }{2401}}-{\frac{625}{242\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{59375}{3993+6655\,x}}+{\frac{3843750\,\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)^3/(3+5*x)^3,x)

[Out]

-32/456533/(2*x-1)-6528/35153041*ln(2*x-1)+81/98/(2+3*x)^2+8829/343/(2+3*x)-630342/2401*ln(2+3*x)-625/242/(3+5
*x)^2+59375/1331/(3+5*x)+3843750/14641*ln(3+5*x)

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Maxima [A]  time = 1.02986, size = 100, normalized size = 1.16 \begin{align*} \frac{7191217800 \, x^{4} + 10067655960 \, x^{3} + 1808383578 \, x^{2} - 2501680914 \, x - 909187261}{913066 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} + \frac{3843750}{14641} \, \log \left (5 \, x + 3\right ) - \frac{630342}{2401} \, \log \left (3 \, x + 2\right ) - \frac{6528}{35153041} \, \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)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/913066*(7191217800*x^4 + 10067655960*x^3 + 1808383578*x^2 - 2501680914*x - 909187261)/(450*x^5 + 915*x^4 + 5
12*x^3 - 85*x^2 - 156*x - 36) + 3843750/14641*log(5*x + 3) - 630342/2401*log(3*x + 2) - 6528/35153041*log(2*x
- 1)

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Fricas [B]  time = 1.49463, size = 517, normalized size = 6.01 \begin{align*} \frac{553723770600 \, x^{4} + 775209508920 \, x^{3} + 139245535506 \, x^{2} + 18457687500 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (5 \, x + 3\right ) - 18457674444 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (3 \, x + 2\right ) - 13056 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (2 \, x - 1\right ) - 192629430378 \, x - 70007419097}{70306082 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/70306082*(553723770600*x^4 + 775209508920*x^3 + 139245535506*x^2 + 18457687500*(450*x^5 + 915*x^4 + 512*x^3
- 85*x^2 - 156*x - 36)*log(5*x + 3) - 18457674444*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*log(3*x
+ 2) - 13056*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*log(2*x - 1) - 192629430378*x - 70007419097)/
(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)

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Sympy [A]  time = 0.234236, size = 75, normalized size = 0.87 \begin{align*} \frac{7191217800 x^{4} + 10067655960 x^{3} + 1808383578 x^{2} - 2501680914 x - 909187261}{410879700 x^{5} + 835455390 x^{4} + 467489792 x^{3} - 77610610 x^{2} - 142438296 x - 32870376} - \frac{6528 \log{\left (x - \frac{1}{2} \right )}}{35153041} + \frac{3843750 \log{\left (x + \frac{3}{5} \right )}}{14641} - \frac{630342 \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/(2+3*x)**3/(3+5*x)**3,x)

[Out]

(7191217800*x**4 + 10067655960*x**3 + 1808383578*x**2 - 2501680914*x - 909187261)/(410879700*x**5 + 835455390*
x**4 + 467489792*x**3 - 77610610*x**2 - 142438296*x - 32870376) - 6528*log(x - 1/2)/35153041 + 3843750*log(x +
3/5)/14641 - 630342*log(x + 2/3)/2401

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Giac [A]  time = 1.40539, size = 131, normalized size = 1.52 \begin{align*} -\frac{32}{456533 \,{\left (2 \, x - 1\right )}} - \frac{4 \,{\left (\frac{207724651275}{2 \, x - 1} + \frac{470659858850}{{\left (2 \, x - 1\right )}^{2}} + \frac{355299675423}{{\left (2 \, x - 1\right )}^{3}} + 30544881750\right )}}{35153041 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}{\left (\frac{7}{2 \, x - 1} + 3\right )}^{2}} - \frac{630342}{2401} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) + \frac{3843750}{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)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

-32/456533/(2*x - 1) - 4/35153041*(207724651275/(2*x - 1) + 470659858850/(2*x - 1)^2 + 355299675423/(2*x - 1)^
3 + 30544881750)/((11/(2*x - 1) + 5)^2*(7/(2*x - 1) + 3)^2) - 630342/2401*log(abs(-7/(2*x - 1) - 3)) + 3843750
/14641*log(abs(-11/(2*x - 1) - 5))