3.13.89 \(\int \frac {(1-2 x)^2 (2+3 x)^7}{3+5 x} \, dx\) [1289]

3.13.89.1 Optimal result
3.13.89.2 Mathematica [A] (verified)
3.13.89.3 Rubi [A] (verified)
3.13.89.4 Maple [A] (verified)
3.13.89.5 Fricas [A] (verification not implemented)
3.13.89.6 Sympy [A] (verification not implemented)
3.13.89.7 Maxima [A] (verification not implemented)
3.13.89.8 Giac [A] (verification not implemented)
3.13.89.9 Mupad [B] (verification not implemented)

3.13.89.1 Optimal result

Integrand size = 22, antiderivative size = 72 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{3+5 x} \, dx=\frac {83333293 x}{1953125}+\frac {80555569 x^2}{781250}+\frac {1327159 x^3}{78125}-\frac {20577159 x^4}{62500}-\frac {7315947 x^5}{15625}+\frac {130383 x^6}{1250}+\frac {672867 x^7}{875}+\frac {16767 x^8}{25}+\frac {972 x^9}{5}+\frac {121 \log (3+5 x)}{9765625} \]

output
83333293/1953125*x+80555569/781250*x^2+1327159/78125*x^3-20577159/62500*x^ 
4-7315947/15625*x^5+130383/1250*x^6+672867/875*x^7+16767/25*x^8+972/5*x^9+ 
121/9765625*ln(3+5*x)
 
3.13.89.2 Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{3+5 x} \, dx=\frac {7880238537+58333305100 x+140972245750 x^2+23225282500 x^3-450125353125 x^4-640145362500 x^5+142606406250 x^6+1051354687500 x^7+916945312500 x^8+265781250000 x^9+16940 \log (3+5 x)}{1367187500} \]

input
Integrate[((1 - 2*x)^2*(2 + 3*x)^7)/(3 + 5*x),x]
 
output
(7880238537 + 58333305100*x + 140972245750*x^2 + 23225282500*x^3 - 4501253 
53125*x^4 - 640145362500*x^5 + 142606406250*x^6 + 1051354687500*x^7 + 9169 
45312500*x^8 + 265781250000*x^9 + 16940*Log[3 + 5*x])/1367187500
 
3.13.89.3 Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 72, 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {8748 x^8}{5}+\frac {134136 x^7}{25}+\frac {672867 x^6}{125}+\frac {391149 x^5}{625}-\frac {7315947 x^4}{3125}-\frac {20577159 x^3}{15625}+\frac {3981477 x^2}{78125}+\frac {80555569 x}{390625}+\frac {121}{1953125 (5 x+3)}+\frac {83333293}{1953125}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {972 x^9}{5}+\frac {16767 x^8}{25}+\frac {672867 x^7}{875}+\frac {130383 x^6}{1250}-\frac {7315947 x^5}{15625}-\frac {20577159 x^4}{62500}+\frac {1327159 x^3}{78125}+\frac {80555569 x^2}{781250}+\frac {83333293 x}{1953125}+\frac {121 \log (5 x+3)}{9765625}\)

input
Int[((1 - 2*x)^2*(2 + 3*x)^7)/(3 + 5*x),x]
 
output
(83333293*x)/1953125 + (80555569*x^2)/781250 + (1327159*x^3)/78125 - (2057 
7159*x^4)/62500 - (7315947*x^5)/15625 + (130383*x^6)/1250 + (672867*x^7)/8 
75 + (16767*x^8)/25 + (972*x^9)/5 + (121*Log[3 + 5*x])/9765625
 

3.13.89.3.1 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]
 
3.13.89.4 Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71

method result size
parallelrisch \(\frac {972 x^{9}}{5}+\frac {16767 x^{8}}{25}+\frac {672867 x^{7}}{875}+\frac {130383 x^{6}}{1250}-\frac {7315947 x^{5}}{15625}-\frac {20577159 x^{4}}{62500}+\frac {1327159 x^{3}}{78125}+\frac {80555569 x^{2}}{781250}+\frac {83333293 x}{1953125}+\frac {121 \ln \left (x +\frac {3}{5}\right )}{9765625}\) \(51\)
default \(\frac {83333293 x}{1953125}+\frac {80555569 x^{2}}{781250}+\frac {1327159 x^{3}}{78125}-\frac {20577159 x^{4}}{62500}-\frac {7315947 x^{5}}{15625}+\frac {130383 x^{6}}{1250}+\frac {672867 x^{7}}{875}+\frac {16767 x^{8}}{25}+\frac {972 x^{9}}{5}+\frac {121 \ln \left (3+5 x \right )}{9765625}\) \(53\)
norman \(\frac {83333293 x}{1953125}+\frac {80555569 x^{2}}{781250}+\frac {1327159 x^{3}}{78125}-\frac {20577159 x^{4}}{62500}-\frac {7315947 x^{5}}{15625}+\frac {130383 x^{6}}{1250}+\frac {672867 x^{7}}{875}+\frac {16767 x^{8}}{25}+\frac {972 x^{9}}{5}+\frac {121 \ln \left (3+5 x \right )}{9765625}\) \(53\)
risch \(\frac {83333293 x}{1953125}+\frac {80555569 x^{2}}{781250}+\frac {1327159 x^{3}}{78125}-\frac {20577159 x^{4}}{62500}-\frac {7315947 x^{5}}{15625}+\frac {130383 x^{6}}{1250}+\frac {672867 x^{7}}{875}+\frac {16767 x^{8}}{25}+\frac {972 x^{9}}{5}+\frac {121 \ln \left (3+5 x \right )}{9765625}\) \(53\)
meijerg \(\frac {121 \ln \left (1+\frac {5 x}{3}\right )}{9765625}+\frac {832 x}{5}-\frac {592 x \left (-5 x +6\right )}{25}-\frac {2772 x \left (\frac {100}{9} x^{2}-10 x +12\right )}{125}+\frac {30618 x \left (-\frac {625}{9} x^{3}+\frac {500}{9} x^{2}-50 x +60\right )}{3125}-\frac {66339 x \left (\frac {2500}{27} x^{4}-\frac {625}{9} x^{3}+\frac {500}{9} x^{2}-50 x +60\right )}{15625}-\frac {111537 x \left (-\frac {218750}{243} x^{5}+\frac {17500}{27} x^{4}-\frac {4375}{9} x^{3}+\frac {3500}{9} x^{2}-350 x +420\right )}{156250}+\frac {10451673 x \left (\frac {625000}{243} x^{6}-\frac {437500}{243} x^{5}+\frac {35000}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{9} x^{2}-700 x +840\right )}{21875000}-\frac {1948617 x \left (-\frac {2734375}{243} x^{7}+\frac {625000}{81} x^{6}-\frac {437500}{81} x^{5}+\frac {35000}{9} x^{4}-\frac {8750}{3} x^{3}+\frac {7000}{3} x^{2}-2100 x +2520\right )}{27343750}+\frac {1594323 x \left (\frac {109375000}{6561} x^{8}-\frac {2734375}{243} x^{7}+\frac {625000}{81} x^{6}-\frac {437500}{81} x^{5}+\frac {35000}{9} x^{4}-\frac {8750}{3} x^{3}+\frac {7000}{3} x^{2}-2100 x +2520\right )}{136718750}\) \(217\)

input
int((1-2*x)^2*(2+3*x)^7/(3+5*x),x,method=_RETURNVERBOSE)
 
output
972/5*x^9+16767/25*x^8+672867/875*x^7+130383/1250*x^6-7315947/15625*x^5-20 
577159/62500*x^4+1327159/78125*x^3+80555569/781250*x^2+83333293/1953125*x+ 
121/9765625*ln(x+3/5)
 
3.13.89.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{3+5 x} \, dx=\frac {972}{5} \, x^{9} + \frac {16767}{25} \, x^{8} + \frac {672867}{875} \, x^{7} + \frac {130383}{1250} \, x^{6} - \frac {7315947}{15625} \, x^{5} - \frac {20577159}{62500} \, x^{4} + \frac {1327159}{78125} \, x^{3} + \frac {80555569}{781250} \, x^{2} + \frac {83333293}{1953125} \, x + \frac {121}{9765625} \, \log \left (5 \, x + 3\right ) \]

input
integrate((1-2*x)^2*(2+3*x)^7/(3+5*x),x, algorithm="fricas")
 
output
972/5*x^9 + 16767/25*x^8 + 672867/875*x^7 + 130383/1250*x^6 - 7315947/1562 
5*x^5 - 20577159/62500*x^4 + 1327159/78125*x^3 + 80555569/781250*x^2 + 833 
33293/1953125*x + 121/9765625*log(5*x + 3)
 
3.13.89.6 Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{3+5 x} \, dx=\frac {972 x^{9}}{5} + \frac {16767 x^{8}}{25} + \frac {672867 x^{7}}{875} + \frac {130383 x^{6}}{1250} - \frac {7315947 x^{5}}{15625} - \frac {20577159 x^{4}}{62500} + \frac {1327159 x^{3}}{78125} + \frac {80555569 x^{2}}{781250} + \frac {83333293 x}{1953125} + \frac {121 \log {\left (5 x + 3 \right )}}{9765625} \]

input
integrate((1-2*x)**2*(2+3*x)**7/(3+5*x),x)
 
output
972*x**9/5 + 16767*x**8/25 + 672867*x**7/875 + 130383*x**6/1250 - 7315947* 
x**5/15625 - 20577159*x**4/62500 + 1327159*x**3/78125 + 80555569*x**2/7812 
50 + 83333293*x/1953125 + 121*log(5*x + 3)/9765625
 
3.13.89.7 Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{3+5 x} \, dx=\frac {972}{5} \, x^{9} + \frac {16767}{25} \, x^{8} + \frac {672867}{875} \, x^{7} + \frac {130383}{1250} \, x^{6} - \frac {7315947}{15625} \, x^{5} - \frac {20577159}{62500} \, x^{4} + \frac {1327159}{78125} \, x^{3} + \frac {80555569}{781250} \, x^{2} + \frac {83333293}{1953125} \, x + \frac {121}{9765625} \, \log \left (5 \, x + 3\right ) \]

input
integrate((1-2*x)^2*(2+3*x)^7/(3+5*x),x, algorithm="maxima")
 
output
972/5*x^9 + 16767/25*x^8 + 672867/875*x^7 + 130383/1250*x^6 - 7315947/1562 
5*x^5 - 20577159/62500*x^4 + 1327159/78125*x^3 + 80555569/781250*x^2 + 833 
33293/1953125*x + 121/9765625*log(5*x + 3)
 
3.13.89.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{3+5 x} \, dx=\frac {972}{5} \, x^{9} + \frac {16767}{25} \, x^{8} + \frac {672867}{875} \, x^{7} + \frac {130383}{1250} \, x^{6} - \frac {7315947}{15625} \, x^{5} - \frac {20577159}{62500} \, x^{4} + \frac {1327159}{78125} \, x^{3} + \frac {80555569}{781250} \, x^{2} + \frac {83333293}{1953125} \, x + \frac {121}{9765625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

input
integrate((1-2*x)^2*(2+3*x)^7/(3+5*x),x, algorithm="giac")
 
output
972/5*x^9 + 16767/25*x^8 + 672867/875*x^7 + 130383/1250*x^6 - 7315947/1562 
5*x^5 - 20577159/62500*x^4 + 1327159/78125*x^3 + 80555569/781250*x^2 + 833 
33293/1953125*x + 121/9765625*log(abs(5*x + 3))
 
3.13.89.9 Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^2 (2+3 x)^7}{3+5 x} \, dx=\frac {83333293\,x}{1953125}+\frac {121\,\ln \left (x+\frac {3}{5}\right )}{9765625}+\frac {80555569\,x^2}{781250}+\frac {1327159\,x^3}{78125}-\frac {20577159\,x^4}{62500}-\frac {7315947\,x^5}{15625}+\frac {130383\,x^6}{1250}+\frac {672867\,x^7}{875}+\frac {16767\,x^8}{25}+\frac {972\,x^9}{5} \]

input
int(((2*x - 1)^2*(3*x + 2)^7)/(5*x + 3),x)
 
output
(83333293*x)/1953125 + (121*log(x + 3/5))/9765625 + (80555569*x^2)/781250 
+ (1327159*x^3)/78125 - (20577159*x^4)/62500 - (7315947*x^5)/15625 + (1303 
83*x^6)/1250 + (672867*x^7)/875 + (16767*x^8)/25 + (972*x^9)/5