3.14.22 \(\int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx\) [1322]

3.14.22.1 Optimal result

Integrand size = 22, antiderivative size = 80 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx=\frac {9251661 x}{1953125}+\frac {1390203 x^2}{390625}-\frac {162612 x^3}{15625}-\frac {193833 x^4}{12500}+\frac {104247 x^5}{15625}+\frac {13608 x^6}{625}+\frac {8748 x^7}{875}-\frac {121}{19531250 (3+5 x)^2}-\frac {2497}{9765625 (3+5 x)}+\frac {21949 \log (3+5 x)}{9765625} \]

output

9251661/1953125*x+1390203/390625*x^2-162612/15625*x^3-193833/12500*x^4+104 
247/15625*x^5+13608/625*x^6+8748/875*x^7-121/19531250/(3+5*x)^2-2497/97656 
25/(3+5*x)+21949/9765625*ln(3+5*x)
 

3.14.22.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx=\frac {13601177777+103624499690 x+275860261575 x^2+179818432500 x^3-496018096875 x^4-909633768750 x^5+11543765625 x^6+1244084062500 x^7+1154250000000 x^8+341718750000 x^9+3072860 (3+5 x)^2 \log (3+5 x)}{1367187500 (3+5 x)^2} \]

input

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

output

(13601177777 + 103624499690*x + 275860261575*x^2 + 179818432500*x^3 - 4960 
18096875*x^4 - 909633768750*x^5 + 11543765625*x^6 + 1244084062500*x^7 + 11 
54250000000*x^8 + 341718750000*x^9 + 3072860*(3 + 5*x)^2*Log[3 + 5*x])/(13 
67187500*(3 + 5*x)^2)
 

3.14.22.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 80, 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {8748 x^6}{125}+\frac {81648 x^5}{625}+\frac {104247 x^4}{3125}-\frac {193833 x^3}{3125}-\frac {487836 x^2}{15625}+\frac {2780406 x}{390625}+\frac {21949}{1953125 (5 x+3)}+\frac {2497}{1953125 (5 x+3)^2}+\frac {121}{1953125 (5 x+3)^3}+\frac {9251661}{1953125}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8748 x^7}{875}+\frac {13608 x^6}{625}+\frac {104247 x^5}{15625}-\frac {193833 x^4}{12500}-\frac {162612 x^3}{15625}+\frac {1390203 x^2}{390625}+\frac {9251661 x}{1953125}-\frac {2497}{9765625 (5 x+3)}-\frac {121}{19531250 (5 x+3)^2}+\frac {21949 \log (5 x+3)}{9765625}\)

input

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

output

(9251661*x)/1953125 + (1390203*x^2)/390625 - (162612*x^3)/15625 - (193833* 
x^4)/12500 + (104247*x^5)/15625 + (13608*x^6)/625 + (8748*x^7)/875 - 121/( 
19531250*(3 + 5*x)^2) - 2497/(9765625*(3 + 5*x)) + (21949*Log[3 + 5*x])/97 
65625
 

3.14.22.3.1 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]
 

3.14.22.4 Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {8748 x^{7}}{875}+\frac {13608 x^{6}}{625}+\frac {104247 x^{5}}{15625}-\frac {193833 x^{4}}{12500}-\frac {162612 x^{3}}{15625}+\frac {1390203 x^{2}}{390625}+\frac {9251661 x}{1953125}+\frac {-\frac {2497 x}{1953125}-\frac {15103}{19531250}}{\left (3+5 x \right )^{2}}+\frac {21949 \ln \left (3+5 x \right )}{9765625}\)	\(57\)
default	\(\frac {9251661 x}{1953125}+\frac {1390203 x^{2}}{390625}-\frac {162612 x^{3}}{15625}-\frac {193833 x^{4}}{12500}+\frac {104247 x^{5}}{15625}+\frac {13608 x^{6}}{625}+\frac {8748 x^{7}}{875}-\frac {121}{19531250 \left (3+5 x \right )^{2}}-\frac {2497}{9765625 \left (3+5 x \right )}+\frac {21949 \ln \left (3+5 x \right )}{9765625}\)	\(61\)
norman	\(\frac {\frac {249802459}{5859375} x +\frac {1224407377}{7031250} x^{2}+\frac {10275339}{78125} x^{3}-\frac {22675113}{62500} x^{4}-\frac {20791629}{31250} x^{5}+\frac {105543}{12500} x^{6}+\frac {3981069}{4375} x^{7}+\frac {147744}{175} x^{8}+\frac {8748}{35} x^{9}}{\left (3+5 x \right )^{2}}+\frac {21949 \ln \left (3+5 x \right )}{9765625}\)	\(62\)
parallelrisch	\(\frac {615093750000 x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}-1637340783750 x^{5}-892832574375 x^{4}[phone] \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] x^{3}[phone] \ln \left (x +\frac {3}{5}\right ) x [phone] x^{2}[phone] \ln \left (x +\frac {3}{5}\right )[phone] x}{2460937500 \left (3+5 x \right )^{2}}\)	\(76\)
meijerg	\(\frac {64 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {416 x^{2}}{27 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {592 x \left (15 x +6\right )}{225 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {21949 \ln \left (1+\frac {5 x}{3}\right )}{9765625}-\frac {924 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {20412 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {14742 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {37179 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{31250 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {3483891 x \left (\frac {125000}{729} x^{6}-\frac {43750}{243} x^{5}+\frac {17500}{81} x^{4}-\frac {8750}{27} x^{3}+\frac {7000}{9} x^{2}+2100 x +840\right )}{3125000 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {433026 x \left (-\frac {390625}{729} x^{7}+\frac {125000}{243} x^{6}-\frac {43750}{81} x^{5}+\frac {17500}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{3} x^{2}+6300 x +2520\right )}{1953125 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {3188646 x \left (\frac {3906250}{6561} x^{8}-\frac {390625}{729} x^{7}+\frac {125000}{243} x^{6}-\frac {43750}{81} x^{5}+\frac {17500}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{3} x^{2}+6300 x +2520\right )}{68359375 \left (1+\frac {5 x}{3}\right )^{2}}\)	\(297\)

input

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

output

8748/875*x^7+13608/625*x^6+104247/15625*x^5-193833/12500*x^4-162612/15625* 
x^3+1390203/390625*x^2+9251661/1953125*x+25*(-2497/48828125*x-15103/488281 
250)/(3+5*x)^2+21949/9765625*ln(3+5*x)
 

3.14.22.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx=\frac {68343750000 \, x^{9} + 230850000000 \, x^{8} + 248816812500 \, x^{7} + 2308753125 \, x^{6} - 181926753750 \, x^{5} - 99203619375 \, x^{4} + 35963686500 \, x^{3} + 47615255100 \, x^{2} + 614572 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 11656743280 \, x - 211442}{273437500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

input

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

output

1/273437500*(68343750000*x^9 + 230850000000*x^8 + 248816812500*x^7 + 23087 
53125*x^6 - 181926753750*x^5 - 99203619375*x^4 + 35963686500*x^3 + 4761525 
5100*x^2 + 614572*(25*x^2 + 30*x + 9)*log(5*x + 3) + 11656743280*x - 21144 
2)/(25*x^2 + 30*x + 9)
 

3.14.22.6 Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx=\frac {8748 x^{7}}{875} + \frac {13608 x^{6}}{625} + \frac {104247 x^{5}}{15625} - \frac {193833 x^{4}}{12500} - \frac {162612 x^{3}}{15625} + \frac {1390203 x^{2}}{390625} + \frac {9251661 x}{1953125} + \frac {- 24970 x - 15103}{488281250 x^{2} + 585937500 x + 175781250} + \frac {21949 \log {\left (5 x + 3 \right )}}{9765625} \]

input

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

output

8748*x**7/875 + 13608*x**6/625 + 104247*x**5/15625 - 193833*x**4/12500 - 1 
62612*x**3/15625 + 1390203*x**2/390625 + 9251661*x/1953125 + (-24970*x - 1 
5103)/(488281250*x**2 + 585937500*x + 175781250) + 21949*log(5*x + 3)/9765 
625
 

3.14.22.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx=\frac {8748}{875} \, x^{7} + \frac {13608}{625} \, x^{6} + \frac {104247}{15625} \, x^{5} - \frac {193833}{12500} \, x^{4} - \frac {162612}{15625} \, x^{3} + \frac {1390203}{390625} \, x^{2} + \frac {9251661}{1953125} \, x - \frac {11 \, {\left (2270 \, x + 1373\right )}}{19531250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {21949}{9765625} \, \log \left (5 \, x + 3\right ) \]

input

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

output

8748/875*x^7 + 13608/625*x^6 + 104247/15625*x^5 - 193833/12500*x^4 - 16261 
2/15625*x^3 + 1390203/390625*x^2 + 9251661/1953125*x - 11/19531250*(2270*x 
 + 1373)/(25*x^2 + 30*x + 9) + 21949/9765625*log(5*x + 3)
 

3.14.22.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx=\frac {8748}{875} \, x^{7} + \frac {13608}{625} \, x^{6} + \frac {104247}{15625} \, x^{5} - \frac {193833}{12500} \, x^{4} - \frac {162612}{15625} \, x^{3} + \frac {1390203}{390625} \, x^{2} + \frac {9251661}{1953125} \, x - \frac {11 \, {\left (2270 \, x + 1373\right )}}{19531250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {21949}{9765625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

input

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

output

8748/875*x^7 + 13608/625*x^6 + 104247/15625*x^5 - 193833/12500*x^4 - 16261 
2/15625*x^3 + 1390203/390625*x^2 + 9251661/1953125*x - 11/19531250*(2270*x 
 + 1373)/(5*x + 3)^2 + 21949/9765625*log(abs(5*x + 3))
 

3.14.22.9 Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx=\frac {9251661\,x}{1953125}+\frac {21949\,\ln \left (x+\frac {3}{5}\right )}{9765625}-\frac {\frac {2497\,x}{48828125}+\frac {15103}{488281250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\frac {1390203\,x^2}{390625}-\frac {162612\,x^3}{15625}-\frac {193833\,x^4}{12500}+\frac {104247\,x^5}{15625}+\frac {13608\,x^6}{625}+\frac {8748\,x^7}{875} \]

input

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

output

(9251661*x)/1953125 + (21949*log(x + 3/5))/9765625 - ((2497*x)/48828125 + 
15103/488281250)/((6*x)/5 + x^2 + 9/25) + (1390203*x^2)/390625 - (162612*x 
^3)/15625 - (193833*x^4)/12500 + (104247*x^5)/15625 + (13608*x^6)/625 + (8 
748*x^7)/875