3.14.19 \(\int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx\) [1319]

3.14.19.1 Optimal result

Integrand size = 22, antiderivative size = 90 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {49}{18 (2+3 x)^6}-\frac {154}{5 (2+3 x)^5}-\frac {1133}{4 (2+3 x)^4}-\frac {7480}{3 (2+3 x)^3}-\frac {46475}{2 (2+3 x)^2}-\frac {277750}{2+3 x}-\frac {75625}{3+5 x}+1615625 \log (2+3 x)-1615625 \log (3+5 x) \]

output

-49/18/(2+3*x)^6-154/5/(2+3*x)^5-1133/4/(2+3*x)^4-7480/3/(2+3*x)^3-46475/2 
/(2+3*x)^2-277750/(2+3*x)-75625/(3+5*x)+1615625*ln(2+3*x)-1615625*ln(3+5*x 
)
 

3.14.19.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {49}{18 (2+3 x)^6}-\frac {154}{5 (2+3 x)^5}-\frac {1133}{4 (2+3 x)^4}-\frac {7480}{3 (2+3 x)^3}-\frac {46475}{2 (2+3 x)^2}-\frac {277750}{2+3 x}-\frac {75625}{3+5 x}+1615625 \log (5 (2+3 x))-1615625 \log (3+5 x) \]

input

Integrate[(1 - 2*x)^2/((2 + 3*x)^7*(3 + 5*x)^2),x]
 

output

-49/(18*(2 + 3*x)^6) - 154/(5*(2 + 3*x)^5) - 1133/(4*(2 + 3*x)^4) - 7480/( 
3*(2 + 3*x)^3) - 46475/(2*(2 + 3*x)^2) - 277750/(2 + 3*x) - 75625/(3 + 5*x 
) + 1615625*Log[5*(2 + 3*x)] - 1615625*Log[3 + 5*x]
 

3.14.19.3 Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 90, 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.

\(\displaystyle \int \frac {(1-2 x)^2}{(3 x+2)^7 (5 x+3)^2} \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {8078125}{5 x+3}+\frac {378125}{(5 x+3)^2}+\frac {4846875}{3 x+2}+\frac {833250}{(3 x+2)^2}+\frac {139425}{(3 x+2)^3}+\frac {22440}{(3 x+2)^4}+\frac {3399}{(3 x+2)^5}+\frac {462}{(3 x+2)^6}+\frac {49}{(3 x+2)^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {277750}{3 x+2}-\frac {75625}{5 x+3}-\frac {46475}{2 (3 x+2)^2}-\frac {7480}{3 (3 x+2)^3}-\frac {1133}{4 (3 x+2)^4}-\frac {154}{5 (3 x+2)^5}-\frac {49}{18 (3 x+2)^6}+1615625 \log (3 x+2)-1615625 \log (5 x+3)\)

input

Int[(1 - 2*x)^2/((2 + 3*x)^7*(3 + 5*x)^2),x]
 

output

-49/(18*(2 + 3*x)^6) - 154/(5*(2 + 3*x)^5) - 1133/(4*(2 + 3*x)^4) - 7480/( 
3*(2 + 3*x)^3) - 46475/(2*(2 + 3*x)^2) - 277750/(2 + 3*x) - 75625/(3 + 5*x 
) + 1615625*Log[2 + 3*x] - 1615625*Log[3 + 5*x]
 

3.14.19.3.1 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]
 

3.14.19.4 Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70

method	result	size
norman	\(\frac {-392596875 x^{6}-\frac {26722518673}{90} x -\frac {9070209225}{4} x^{3}-\frac {5146799625}{2} x^{4}-\frac {4494718899}{4} x^{2}-\frac {3114601875}{2} x^{5}-\frac {980484959}{30}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+1615625 \ln \left (2+3 x \right )-1615625 \ln \left (3+5 x \right )\)	\(63\)
risch	\(\frac {-392596875 x^{6}-\frac {26722518673}{90} x -\frac {9070209225}{4} x^{3}-\frac {5146799625}{2} x^{4}-\frac {4494718899}{4} x^{2}-\frac {3114601875}{2} x^{5}-\frac {980484959}{30}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+1615625 \ln \left (2+3 x \right )-1615625 \ln \left (3+5 x \right )\)	\(64\)
default	\(-\frac {49}{18 \left (2+3 x \right )^{6}}-\frac {154}{5 \left (2+3 x \right )^{5}}-\frac {1133}{4 \left (2+3 x \right )^{4}}-\frac {7480}{3 \left (2+3 x \right )^{3}}-\frac {46475}{2 \left (2+3 x \right )^{2}}-\frac {277750}{2+3 x}-\frac {75625}{3+5 x}+1615625 \ln \left (2+3 x \right )-1615625 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(\frac {99264000320 x -29034720000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-6352896000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-11306790000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-595584000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-73703520000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-102514896000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-112230360000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-52011234000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{1920 \left (2+3 x \right )^{6} \left (3+5 x \right )}\)	\(185\)

input

int((1-2*x)^2/(2+3*x)^7/(3+5*x)^2,x,method=_RETURNVERBOSE)
 

output

(-392596875*x^6-26722518673/90*x-9070209225/4*x^3-5146799625/2*x^4-4494718 
899/4*x^2-3114601875/2*x^5-980484959/30)/(2+3*x)^6/(3+5*x)+1615625*ln(2+3* 
x)-1615625*ln(3+5*x)
 

3.14.19.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.72 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {70667437500 \, x^{6} + 280314168750 \, x^{5} + 463211966250 \, x^{4} + 408159415125 \, x^{3} + 202262350455 \, x^{2} + 290812500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (5 \, x + 3\right ) - 290812500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (3 \, x + 2\right ) + 53445037346 \, x + 5882909754}{180 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} \]

input

integrate((1-2*x)^2/(2+3*x)^7/(3+5*x)^2,x, algorithm="fricas")
 

output

-1/180*(70667437500*x^6 + 280314168750*x^5 + 463211966250*x^4 + 4081594151 
25*x^3 + 202262350455*x^2 + 290812500*(3645*x^7 + 16767*x^6 + 33048*x^5 + 
36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(5*x + 3) - 290812500* 
(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 204 
8*x + 192)*log(3*x + 2) + 53445037346*x + 5882909754)/(3645*x^7 + 16767*x^ 
6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)
 

3.14.19.6 Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=\frac {- 70667437500 x^{6} - 280314168750 x^{5} - 463211966250 x^{4} - 408159415125 x^{3} - 202262350455 x^{2} - 53445037346 x - 5882909754}{656100 x^{7} + 3018060 x^{6} + 5948640 x^{5} + 6512400 x^{4} + 4276800 x^{3} + 1684800 x^{2} + 368640 x + 34560} - 1615625 \log {\left (x + \frac {3}{5} \right )} + 1615625 \log {\left (x + \frac {2}{3} \right )} \]

input

integrate((1-2*x)**2/(2+3*x)**7/(3+5*x)**2,x)
 

output

(-70667437500*x**6 - 280314168750*x**5 - 463211966250*x**4 - 408159415125* 
x**3 - 202262350455*x**2 - 53445037346*x - 5882909754)/(656100*x**7 + 3018 
060*x**6 + 5948640*x**5 + 6512400*x**4 + 4276800*x**3 + 1684800*x**2 + 368 
640*x + 34560) - 1615625*log(x + 3/5) + 1615625*log(x + 2/3)
 

3.14.19.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {70667437500 \, x^{6} + 280314168750 \, x^{5} + 463211966250 \, x^{4} + 408159415125 \, x^{3} + 202262350455 \, x^{2} + 53445037346 \, x + 5882909754}{180 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} - 1615625 \, \log \left (5 \, x + 3\right ) + 1615625 \, \log \left (3 \, x + 2\right ) \]

input

integrate((1-2*x)^2/(2+3*x)^7/(3+5*x)^2,x, algorithm="maxima")
 

output

-1/180*(70667437500*x^6 + 280314168750*x^5 + 463211966250*x^4 + 4081594151 
25*x^3 + 202262350455*x^2 + 53445037346*x + 5882909754)/(3645*x^7 + 16767* 
x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192) - 16156 
25*log(5*x + 3) + 1615625*log(3*x + 2)
 

3.14.19.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {75625}{5 \, x + 3} + \frac {625 \, {\left (\frac {22074930}{5 \, x + 3} + \frac {16294797}{{\left (5 \, x + 3\right )}^{2}} + \frac {6120660}{{\left (5 \, x + 3\right )}^{3}} + \frac {1179210}{{\left (5 \, x + 3\right )}^{4}} + \frac {94660}{{\left (5 \, x + 3\right )}^{5}} + 12117357\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{6}} + 1615625 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

input

integrate((1-2*x)^2/(2+3*x)^7/(3+5*x)^2,x, algorithm="giac")
 

output

-75625/(5*x + 3) + 625/4*(22074930/(5*x + 3) + 16294797/(5*x + 3)^2 + 6120 
660/(5*x + 3)^3 + 1179210/(5*x + 3)^4 + 94660/(5*x + 3)^5 + 12117357)/(1/( 
5*x + 3) + 3)^6 + 1615625*log(abs(-1/(5*x + 3) - 3))
 

3.14.19.9 Mupad [B] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=3231250\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {323125\,x^6}{3}+\frac {7690375\,x^5}{18}+\frac {114373325\,x^4}{162}+\frac {67186735\,x^3}{108}+\frac {499413211\,x^2}{1620}+\frac {26722518673\,x}{328050}+\frac {980484959}{109350}}{x^7+\frac {23\,x^6}{5}+\frac {136\,x^5}{15}+\frac {268\,x^4}{27}+\frac {176\,x^3}{27}+\frac {208\,x^2}{81}+\frac {2048\,x}{3645}+\frac {64}{1215}} \]

input

int((2*x - 1)^2/((3*x + 2)^7*(5*x + 3)^2),x)
 

output

3231250*atanh(30*x + 19) - ((26722518673*x)/328050 + (499413211*x^2)/1620 
+ (67186735*x^3)/108 + (114373325*x^4)/162 + (7690375*x^5)/18 + (323125*x^ 
6)/3 + 980484959/109350)/((2048*x)/3645 + (208*x^2)/81 + (176*x^3)/27 + (2 
68*x^4)/27 + (136*x^5)/15 + (23*x^6)/5 + x^7 + 64/1215)