3.17.1 \(\int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx\) [1601]

3.17.1.1 Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx=\frac {32}{184877 (1-2 x)}+\frac {9}{196 (2+3 x)^4}+\frac {117}{343 (2+3 x)^3}+\frac {12393}{4802 (2+3 x)^2}+\frac {434043}{16807 (2+3 x)}-\frac {6400 \log (1-2 x)}{14235529}-\frac {15192225 \log (2+3 x)}{117649}+\frac {15625}{121} \log (3+5 x) \]

output

32/184877/(1-2*x)+9/196/(2+3*x)^4+117/343/(2+3*x)^3+12393/4802/(2+3*x)^2+4 
34043/16807/(2+3*x)-6400/14235529*ln(1-2*x)-15192225/117649*ln(2+3*x)+1562 
5/121*ln(3+5*x)
 

3.17.1.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx=\frac {5 \left (\frac {77}{5} \left (\frac {128}{1-2 x}+\frac {33957}{(2+3 x)^4}+\frac {252252}{(2+3 x)^3}+\frac {1908522}{(2+3 x)^2}+\frac {19097892}{2+3 x}\right )-5120 \log (5-10 x)-1470607380 \log (5 (2+3 x))[phone] \log (3+5 x)\right )}{56942116} \]

input

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

output

(5*((77*(128/(1 - 2*x) + 33957/(2 + 3*x)^4 + 252252/(2 + 3*x)^3 + 1908522/ 
(2 + 3*x)^2 + 19097892/(2 + 3*x)))/5 - 5120*Log[5 - 10*x] - 1470607380*Log 
[5*(2 + 3*x)] + 1470612500*Log[3 + 5*x]))/56942116
 

3.17.1.3 Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {45576675}{117649 (3 x+2)}+\frac {78125}{121 (5 x+3)}-\frac {1302129}{16807 (3 x+2)^2}-\frac {37179}{2401 (3 x+2)^3}-\frac {1053}{343 (3 x+2)^4}-\frac {27}{49 (3 x+2)^5}-\frac {12800}{14235529 (2 x-1)}+\frac {64}{184877 (2 x-1)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {32}{184877 (1-2 x)}+\frac {434043}{16807 (3 x+2)}+\frac {12393}{4802 (3 x+2)^2}+\frac {117}{343 (3 x+2)^3}+\frac {9}{196 (3 x+2)^4}-\frac {6400 \log (1-2 x)}{14235529}-\frac {15192225 \log (3 x+2)}{117649}+\frac {15625}{121} \log (5 x+3)\)

input

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

output

32/(184877*(1 - 2*x)) + 9/(196*(2 + 3*x)^4) + 117/(343*(2 + 3*x)^3) + 1239 
3/(4802*(2 + 3*x)^2) + 434043/(16807*(2 + 3*x)) - (6400*Log[1 - 2*x])/1423 
5529 - (15192225*Log[2 + 3*x])/117649 + (15625*Log[3 + 5*x])/121
 

3.17.1.3.1 Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(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]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.17.1.4 Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.71

method	result	size
norman	\(\frac {-\frac {194642025}{369754} x +\frac {186937875}{369754} x^{2}+\frac {257818950}{184877} x^{4}+\frac {395313750}{184877} x^{3}-\frac {160957733}{739508}}{\left (-1+2 x \right ) \left (2+3 x \right )^{4}}-\frac {6400 \ln \left (-1+2 x \right )}{14235529}-\frac {15192225 \ln \left (2+3 x \right )}{117649}+\frac {15625 \ln \left (3+5 x \right )}{121}\)	\(61\)
risch	\(\frac {-\frac {194642025}{369754} x +\frac {186937875}{369754} x^{2}+\frac {257818950}{184877} x^{4}+\frac {395313750}{184877} x^{3}-\frac {160957733}{739508}}{\left (-1+2 x \right ) \left (2+3 x \right )^{4}}-\frac {6400 \ln \left (-1+2 x \right )}{14235529}-\frac {15192225 \ln \left (2+3 x \right )}{117649}+\frac {15625 \ln \left (3+5 x \right )}{121}\)	\(62\)
default	\(\frac {15625 \ln \left (3+5 x \right )}{121}-\frac {32}{184877 \left (-1+2 x \right )}-\frac {6400 \ln \left (-1+2 x \right )}{14235529}+\frac {9}{196 \left (2+3 x \right )^{4}}+\frac {117}{343 \left (2+3 x \right )^{3}}+\frac {12393}{4802 \left (2+3 x \right )^{2}}+\frac {434043}{16807 \left (2+3 x \right )}-\frac {15192225 \ln \left (2+3 x \right )}{117649}\)	\(71\)
parallelrisch	\(-\frac {-313601758624 x [phone] \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}[phone] \ln \left (x +\frac {3}{5}\right ) x -2823566169600 \ln \left (\frac {2}{3}+x \right ) x^{2}-7529509785600 \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{3}[phone] x^{4}-758064814584 x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-1882377446400 \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (x -\frac {1}{2}\right ) x^{3}-9830400 \ln \left (x -\frac {1}{2}\right ) x^{2}-26214400 \ln \left (x -\frac {1}{2}\right ) x [phone] \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-25412184000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-19059138000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-41294799000000 \ln \left (x +\frac {3}{5}\right ) x^{4}-6553600 \ln \left (x -\frac {1}{2}\right )[phone] \ln \left (x -\frac {1}{2}\right ) x^{5}}{911073856 \left (-1+2 x \right ) \left (2+3 x \right )^{4}}\)	\(188\)

input

int(1/(1-2*x)^2/(2+3*x)^5/(3+5*x),x,method=_RETURNVERBOSE)
 

output

(-194642025/369754*x[phone]/369754*x^2+257818950/184877*x^4+395313750/1 
84877*x^3-160957733/739508)/(-1+2*x)/(2+3*x)^4-6400/14235529*ln(-1+2*x)-15 
192225/117649*ln(2+3*x)+15625/121*ln(3+5*x)
 

3.17.1.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).

Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx=\frac {79408236600 \, x^{4} + 121756635000 \, x^{3} + 28788432750 \, x^{2} + 7353062500 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (5 \, x + 3\right ) - 7353036900 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) - 25600 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) - 29974871850 \, x - 12393745441}{56942116 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]

input

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

output

1/56942116*(79408236600*x^4 + 121756635000*x^3 + 28788432750*x^2 + 7353062 
500*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(5*x + 3) - 7353 
036900*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(3*x + 2) - 2 
5600*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) - 299 
74871850*x - 12393745441)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 1 
6)
 

3.17.1.6 Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx=\frac {1031275800 x^{4} + 1581255000 x^{3} + 373875750 x^{2} - 389284050 x - 160957733}{119800296 x^{5} + 259567308 x^{4} + 159733728 x^{3} - 17748192 x^{2} - 47328512 x - 11832128} - \frac {6400 \log {\left (x - \frac {1}{2} \right )}}{14235529} + \frac {15625 \log {\left (x + \frac {3}{5} \right )}}{121} - \frac {15192225 \log {\left (x + \frac {2}{3} \right )}}{117649} \]

input

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

output

(1031275800*x**4 + 1581255000*x**3 + 373875750*x**2 - 389284050*x - 160957 
733)/(119800296*x**5 + 259567308*x**4 + 159733728*x**3 - 17748192*x**2 - 4 
7328512*x - 11832128) - 6400*log(x - 1/2)/14235529 + 15625*log(x + 3/5)/12 
1 - 15192225*log(x + 2/3)/117649
 

3.17.1.7 Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx=\frac {1031275800 \, x^{4} + 1581255000 \, x^{3} + 373875750 \, x^{2} - 389284050 \, x - 160957733}{739508 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac {15625}{121} \, \log \left (5 \, x + 3\right ) - \frac {15192225}{117649} \, \log \left (3 \, x + 2\right ) - \frac {6400}{14235529} \, \log \left (2 \, x - 1\right ) \]

input

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

output

1/739508*(1031275800*x^4 + 1581255000*x^3 + 373875750*x^2 - 389284050*x - 
160957733)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16) + 15625/121* 
log(5*x + 3) - 15192225/117649*log(3*x + 2) - 6400/14235529*log(2*x - 1)
 

3.17.1.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx=\frac {434043}{16807 \, {\left (3 \, x + 2\right )}} + \frac {192}{1294139 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}} + \frac {12393}{4802 \, {\left (3 \, x + 2\right )}^{2}} + \frac {117}{343 \, {\left (3 \, x + 2\right )}^{3}} + \frac {9}{196 \, {\left (3 \, x + 2\right )}^{4}} + \frac {15625}{121} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {6400}{14235529} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]

input

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

output

434043/16807/(3*x + 2) + 192/1294139/(7/(3*x + 2) - 2) + 12393/4802/(3*x + 
 2)^2 + 117/343/(3*x + 2)^3 + 9/196/(3*x + 2)^4 + 15625/121*log(abs(-1/(3* 
x + 2) + 5)) - 6400/14235529*log(abs(-7/(3*x + 2) + 2))
 

3.17.1.9 Mupad [B] (verification not implemented)

Time = 1.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx=\frac {15625\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {15192225\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {6400\,\ln \left (x-\frac {1}{2}\right )}{14235529}-\frac {\frac {1591475\,x^4}{184877}+\frac {7320625\,x^3}{554631}+\frac {2307875\,x^2}{739508}-\frac {64880675\,x}{19966716}-\frac {160957733}{119800296}}{-x^5-\frac {13\,x^4}{6}-\frac {4\,x^3}{3}+\frac {4\,x^2}{27}+\frac {32\,x}{81}+\frac {8}{81}} \]

input

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

output

(15625*log(x + 3/5))/121 - (15192225*log(x + 2/3))/117649 - (6400*log(x - 
1/2))/14235529 - ((2307875*x^2)/739508 - (64880675*x)/19966716 + (7320625* 
x^3)/554631 + (1591475*x^4)/184877 - 160957733/119800296)/((32*x)/81 + (4* 
x^2)/27 - (4*x^3)/3 - (13*x^4)/6 - x^5 + 8/81)