3.13.70 \(\int (1-2 x)^2 (2+3 x)^{10} (3+5 x)^3 \, dx\) [1270]

3.13.70.1 Optimal result
3.13.70.2 Mathematica [A] (verified)
3.13.70.3 Rubi [A] (verified)
3.13.70.4 Maple [A] (verified)
3.13.70.5 Fricas [A] (verification not implemented)
3.13.70.6 Sympy [A] (verification not implemented)
3.13.70.7 Maxima [A] (verification not implemented)
3.13.70.8 Giac [A] (verification not implemented)
3.13.70.9 Mupad [B] (verification not implemented)

3.13.70.1 Optimal result

Integrand size = 22, antiderivative size = 67 \[ \int (1-2 x)^2 (2+3 x)^{10} (3+5 x)^3 \, dx=-\frac {49 (2+3 x)^{11}}{8019}+\frac {763 (2+3 x)^{12}}{8748}-\frac {4099 (2+3 x)^{13}}{9477}+\frac {8285 (2+3 x)^{14}}{10206}-\frac {760 (2+3 x)^{15}}{2187}+\frac {125 (2+3 x)^{16}}{2916} \]

output
-49/8019*(2+3*x)^11+763/8748*(2+3*x)^12-4099/9477*(2+3*x)^13+8285/10206*(2 
+3*x)^14-760/2187*(2+3*x)^15+125/2916*(2+3*x)^16
 
3.13.70.2 Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.39 \[ \int (1-2 x)^2 (2+3 x)^{10} (3+5 x)^3 \, dx=27648 x+221184 x^2+1000704 x^3+2644160 x^4+3185792 x^5-\frac {10627328 x^6}{3}-\frac {154612896 x^7}{7}-40113468 x^8-26237700 x^9+36043704 x^{10}+\frac {1233925083 x^{11}}{11}+\frac {569034801 x^{12}}{4}+\frac {1417418757 x^{13}}{13}+\frac {734077485 x^{14}}{14}+14696640 x^{15}+\frac {7381125 x^{16}}{4} \]

input
Integrate[(1 - 2*x)^2*(2 + 3*x)^10*(3 + 5*x)^3,x]
 
output
27648*x + 221184*x^2 + 1000704*x^3 + 2644160*x^4 + 3185792*x^5 - (10627328 
*x^6)/3 - (154612896*x^7)/7 - 40113468*x^8 - 26237700*x^9 + 36043704*x^10 
+ (1233925083*x^11)/11 + (569034801*x^12)/4 + (1417418757*x^13)/13 + (7340 
77485*x^14)/14 + 14696640*x^15 + (7381125*x^16)/4
 
3.13.70.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 67, 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.

\(\displaystyle \int (1-2 x)^2 (3 x+2)^{10} (5 x+3)^3 \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {500}{243} (3 x+2)^{15}-\frac {3800}{243} (3 x+2)^{14}+\frac {8285}{243} (3 x+2)^{13}-\frac {4099}{243} (3 x+2)^{12}+\frac {763}{243} (3 x+2)^{11}-\frac {49}{243} (3 x+2)^{10}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {125 (3 x+2)^{16}}{2916}-\frac {760 (3 x+2)^{15}}{2187}+\frac {8285 (3 x+2)^{14}}{10206}-\frac {4099 (3 x+2)^{13}}{9477}+\frac {763 (3 x+2)^{12}}{8748}-\frac {49 (3 x+2)^{11}}{8019}\)

input
Int[(1 - 2*x)^2*(2 + 3*x)^10*(3 + 5*x)^3,x]
 
output
(-49*(2 + 3*x)^11)/8019 + (763*(2 + 3*x)^12)/8748 - (4099*(2 + 3*x)^13)/94 
77 + (8285*(2 + 3*x)^14)/10206 - (760*(2 + 3*x)^15)/2187 + (125*(2 + 3*x)^ 
16)/2916
 

3.13.70.3.1 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]
 
3.13.70.4 Maple [A] (verified)

Time = 2.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18

method result size
gosper \(\frac {x \left (22165518375 x^{15}+176536039680 x^{14}+629838482130 x^{13}+1309694931468 x^{12}+1708811507403 x^{11}+1347446190636 x^{10}+432956972448 x^{9}-315167252400 x^{8}-481842977616 x^{7}-265315729536 x^{6}-42551821312 x^{5}+38267733504 x^{4}+31761649920 x^{3}+12020456448 x^{2}+2656862208 x +332107776\right )}{12012}\) \(79\)
default \(27648 x +221184 x^{2}+1000704 x^{3}+2644160 x^{4}+3185792 x^{5}-\frac {10627328}{3} x^{6}-\frac {154612896}{7} x^{7}-40113468 x^{8}-26237700 x^{9}+36043704 x^{10}+\frac {1233925083}{11} x^{11}+\frac {569034801}{4} x^{12}+\frac {1417418757}{13} x^{13}+\frac {734077485}{14} x^{14}+14696640 x^{15}+\frac {7381125}{4} x^{16}\) \(80\)
norman \(27648 x +221184 x^{2}+1000704 x^{3}+2644160 x^{4}+3185792 x^{5}-\frac {10627328}{3} x^{6}-\frac {154612896}{7} x^{7}-40113468 x^{8}-26237700 x^{9}+36043704 x^{10}+\frac {1233925083}{11} x^{11}+\frac {569034801}{4} x^{12}+\frac {1417418757}{13} x^{13}+\frac {734077485}{14} x^{14}+14696640 x^{15}+\frac {7381125}{4} x^{16}\) \(80\)
risch \(27648 x +221184 x^{2}+1000704 x^{3}+2644160 x^{4}+3185792 x^{5}-\frac {10627328}{3} x^{6}-\frac {154612896}{7} x^{7}-40113468 x^{8}-26237700 x^{9}+36043704 x^{10}+\frac {1233925083}{11} x^{11}+\frac {569034801}{4} x^{12}+\frac {1417418757}{13} x^{13}+\frac {734077485}{14} x^{14}+14696640 x^{15}+\frac {7381125}{4} x^{16}\) \(80\)
parallelrisch \(27648 x +221184 x^{2}+1000704 x^{3}+2644160 x^{4}+3185792 x^{5}-\frac {10627328}{3} x^{6}-\frac {154612896}{7} x^{7}-40113468 x^{8}-26237700 x^{9}+36043704 x^{10}+\frac {1233925083}{11} x^{11}+\frac {569034801}{4} x^{12}+\frac {1417418757}{13} x^{13}+\frac {734077485}{14} x^{14}+14696640 x^{15}+\frac {7381125}{4} x^{16}\) \(80\)

input
int((1-2*x)^2*(2+3*x)^10*(3+5*x)^3,x,method=_RETURNVERBOSE)
 
output
1/12012*x*(22165518375*x^15+176536039680*x^14+629838482130*x^13+1309694931 
468*x^12+1708811507403*x^11+1347446190636*x^10+432956972448*x^9-3151672524 
00*x^8-481842977616*x^7-265315729536*x^6-42551821312*x^5+38267733504*x^4+3 
1761649920*x^3+12020456448*x^2+2656862208*x+332107776)
 
3.13.70.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18 \[ \int (1-2 x)^2 (2+3 x)^{10} (3+5 x)^3 \, dx=\frac {7381125}{4} \, x^{16} + 14696640 \, x^{15} + \frac {734077485}{14} \, x^{14} + \frac {1417418757}{13} \, x^{13} + \frac {569034801}{4} \, x^{12} + \frac {1233925083}{11} \, x^{11} + 36043704 \, x^{10} - 26237700 \, x^{9} - 40113468 \, x^{8} - \frac {154612896}{7} \, x^{7} - \frac {10627328}{3} \, x^{6} + 3185792 \, x^{5} + 2644160 \, x^{4} + 1000704 \, x^{3} + 221184 \, x^{2} + 27648 \, x \]

input
integrate((1-2*x)^2*(2+3*x)^10*(3+5*x)^3,x, algorithm="fricas")
 
output
7381125/4*x^16 + 14696640*x^15 + 734077485/14*x^14 + 1417418757/13*x^13 + 
569034801/4*x^12 + 1233925083/11*x^11 + 36043704*x^10 - 26237700*x^9 - 401 
13468*x^8 - 154612896/7*x^7 - 10627328/3*x^6 + 3185792*x^5 + 2644160*x^4 + 
 1000704*x^3 + 221184*x^2 + 27648*x
 
3.13.70.6 Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.34 \[ \int (1-2 x)^2 (2+3 x)^{10} (3+5 x)^3 \, dx=\frac {7381125 x^{16}}{4} + 14696640 x^{15} + \frac {734077485 x^{14}}{14} + \frac {1417418757 x^{13}}{13} + \frac {569034801 x^{12}}{4} + \frac {1233925083 x^{11}}{11} + 36043704 x^{10} - 26237700 x^{9} - 40113468 x^{8} - \frac {154612896 x^{7}}{7} - \frac {10627328 x^{6}}{3} + 3185792 x^{5} + 2644160 x^{4} + 1000704 x^{3} + 221184 x^{2} + 27648 x \]

input
integrate((1-2*x)**2*(2+3*x)**10*(3+5*x)**3,x)
 
output
7381125*x**16/4 + 14696640*x**15 + 734077485*x**14/14 + 1417418757*x**13/1 
3 + 569034801*x**12/4 + 1233925083*x**11/11 + 36043704*x**10 - 26237700*x* 
*9 - 40113468*x**8 - 154612896*x**7/7 - 10627328*x**6/3 + 3185792*x**5 + 2 
644160*x**4 + 1000704*x**3 + 221184*x**2 + 27648*x
 
3.13.70.7 Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18 \[ \int (1-2 x)^2 (2+3 x)^{10} (3+5 x)^3 \, dx=\frac {7381125}{4} \, x^{16} + 14696640 \, x^{15} + \frac {734077485}{14} \, x^{14} + \frac {1417418757}{13} \, x^{13} + \frac {569034801}{4} \, x^{12} + \frac {1233925083}{11} \, x^{11} + 36043704 \, x^{10} - 26237700 \, x^{9} - 40113468 \, x^{8} - \frac {154612896}{7} \, x^{7} - \frac {10627328}{3} \, x^{6} + 3185792 \, x^{5} + 2644160 \, x^{4} + 1000704 \, x^{3} + 221184 \, x^{2} + 27648 \, x \]

input
integrate((1-2*x)^2*(2+3*x)^10*(3+5*x)^3,x, algorithm="maxima")
 
output
7381125/4*x^16 + 14696640*x^15 + 734077485/14*x^14 + 1417418757/13*x^13 + 
569034801/4*x^12 + 1233925083/11*x^11 + 36043704*x^10 - 26237700*x^9 - 401 
13468*x^8 - 154612896/7*x^7 - 10627328/3*x^6 + 3185792*x^5 + 2644160*x^4 + 
 1000704*x^3 + 221184*x^2 + 27648*x
 
3.13.70.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18 \[ \int (1-2 x)^2 (2+3 x)^{10} (3+5 x)^3 \, dx=\frac {7381125}{4} \, x^{16} + 14696640 \, x^{15} + \frac {734077485}{14} \, x^{14} + \frac {1417418757}{13} \, x^{13} + \frac {569034801}{4} \, x^{12} + \frac {1233925083}{11} \, x^{11} + 36043704 \, x^{10} - 26237700 \, x^{9} - 40113468 \, x^{8} - \frac {154612896}{7} \, x^{7} - \frac {10627328}{3} \, x^{6} + 3185792 \, x^{5} + 2644160 \, x^{4} + 1000704 \, x^{3} + 221184 \, x^{2} + 27648 \, x \]

input
integrate((1-2*x)^2*(2+3*x)^10*(3+5*x)^3,x, algorithm="giac")
 
output
7381125/4*x^16 + 14696640*x^15 + 734077485/14*x^14 + 1417418757/13*x^13 + 
569034801/4*x^12 + 1233925083/11*x^11 + 36043704*x^10 - 26237700*x^9 - 401 
13468*x^8 - 154612896/7*x^7 - 10627328/3*x^6 + 3185792*x^5 + 2644160*x^4 + 
 1000704*x^3 + 221184*x^2 + 27648*x
 
3.13.70.9 Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18 \[ \int (1-2 x)^2 (2+3 x)^{10} (3+5 x)^3 \, dx=\frac {7381125\,x^{16}}{4}+14696640\,x^{15}+\frac {734077485\,x^{14}}{14}+\frac {1417418757\,x^{13}}{13}+\frac {569034801\,x^{12}}{4}+\frac {1233925083\,x^{11}}{11}+36043704\,x^{10}-26237700\,x^9-40113468\,x^8-\frac {154612896\,x^7}{7}-\frac {10627328\,x^6}{3}+3185792\,x^5+2644160\,x^4+1000704\,x^3+221184\,x^2+27648\,x \]

input
int((2*x - 1)^2*(3*x + 2)^10*(5*x + 3)^3,x)
 
output
27648*x + 221184*x^2 + 1000704*x^3 + 2644160*x^4 + 3185792*x^5 - (10627328 
*x^6)/3 - (154612896*x^7)/7 - 40113468*x^8 - 26237700*x^9 + 36043704*x^10 
+ (1233925083*x^11)/11 + (569034801*x^12)/4 + (1417418757*x^13)/13 + (7340 
77485*x^14)/14 + 14696640*x^15 + (7381125*x^16)/4