Integrand size = 13, antiderivative size = 129 \[ \int x^{20} \left (a+b x^3\right )^8 \, dx=\frac {a^6 \left (a+b x^3\right )^9}{27 b^7}-\frac {a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac {5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac {5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac {5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}-\frac {a \left (a+b x^3\right )^{14}}{7 b^7}+\frac {\left (a+b x^3\right )^{15}}{45 b^7} \] Output:
1/27*a^6*(b*x^3+a)^9/b^7-1/5*a^5*(b*x^3+a)^10/b^7+5/11*a^4*(b*x^3+a)^11/b^ 7-5/9*a^3*(b*x^3+a)^12/b^7+5/13*a^2*(b*x^3+a)^13/b^7-1/7*a*(b*x^3+a)^14/b^ 7+1/45*(b*x^3+a)^15/b^7
Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int x^{20} \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^{21}}{21}+\frac {1}{3} a^7 b x^{24}+\frac {28}{27} a^6 b^2 x^{27}+\frac {28}{15} a^5 b^3 x^{30}+\frac {70}{33} a^4 b^4 x^{33}+\frac {14}{9} a^3 b^5 x^{36}+\frac {28}{39} a^2 b^6 x^{39}+\frac {4}{21} a b^7 x^{42}+\frac {b^8 x^{45}}{45} \] Input:
Integrate[x^20*(a + b*x^3)^8,x]
Output:
(a^8*x^21)/21 + (a^7*b*x^24)/3 + (28*a^6*b^2*x^27)/27 + (28*a^5*b^3*x^30)/ 15 + (70*a^4*b^4*x^33)/33 + (14*a^3*b^5*x^36)/9 + (28*a^2*b^6*x^39)/39 + ( 4*a*b^7*x^42)/21 + (b^8*x^45)/45
Time = 0.42 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {798, 49, 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.
\(\displaystyle \int x^{20} \left (a+b x^3\right )^8 \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int x^{18} \left (b x^3+a\right )^8dx^3\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \int \left (\frac {\left (b x^3+a\right )^{14}}{b^6}-\frac {6 a \left (b x^3+a\right )^{13}}{b^6}+\frac {15 a^2 \left (b x^3+a\right )^{12}}{b^6}-\frac {20 a^3 \left (b x^3+a\right )^{11}}{b^6}+\frac {15 a^4 \left (b x^3+a\right )^{10}}{b^6}-\frac {6 a^5 \left (b x^3+a\right )^9}{b^6}+\frac {a^6 \left (b x^3+a\right )^8}{b^6}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {a^6 \left (a+b x^3\right )^9}{9 b^7}-\frac {3 a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac {15 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac {5 a^3 \left (a+b x^3\right )^{12}}{3 b^7}+\frac {15 a^2 \left (a+b x^3\right )^{13}}{13 b^7}+\frac {\left (a+b x^3\right )^{15}}{15 b^7}-\frac {3 a \left (a+b x^3\right )^{14}}{7 b^7}\right )\) |
Input:
Int[x^20*(a + b*x^3)^8,x]
Output:
((a^6*(a + b*x^3)^9)/(9*b^7) - (3*a^5*(a + b*x^3)^10)/(5*b^7) + (15*a^4*(a + b*x^3)^11)/(11*b^7) - (5*a^3*(a + b*x^3)^12)/(3*b^7) + (15*a^2*(a + b*x ^3)^13)/(13*b^7) - (3*a*(a + b*x^3)^14)/(7*b^7) + (a + b*x^3)^15/(15*b^7)) /3
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.41 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(\frac {4}{21} a \,b^{7} x^{42}+\frac {14}{9} a^{3} b^{5} x^{36}+\frac {1}{21} a^{8} x^{21}+\frac {70}{33} a^{4} b^{4} x^{33}+\frac {28}{39} a^{2} b^{6} x^{39}+\frac {1}{3} a^{7} b \,x^{24}+\frac {28}{27} a^{6} b^{2} x^{27}+\frac {28}{15} a^{5} b^{3} x^{30}+\frac {1}{45} b^{8} x^{45}\) | \(91\) |
default | \(\frac {4}{21} a \,b^{7} x^{42}+\frac {14}{9} a^{3} b^{5} x^{36}+\frac {1}{21} a^{8} x^{21}+\frac {70}{33} a^{4} b^{4} x^{33}+\frac {28}{39} a^{2} b^{6} x^{39}+\frac {1}{3} a^{7} b \,x^{24}+\frac {28}{27} a^{6} b^{2} x^{27}+\frac {28}{15} a^{5} b^{3} x^{30}+\frac {1}{45} b^{8} x^{45}\) | \(91\) |
risch | \(\frac {4}{21} a \,b^{7} x^{42}+\frac {14}{9} a^{3} b^{5} x^{36}+\frac {1}{21} a^{8} x^{21}+\frac {70}{33} a^{4} b^{4} x^{33}+\frac {28}{39} a^{2} b^{6} x^{39}+\frac {1}{3} a^{7} b \,x^{24}+\frac {28}{27} a^{6} b^{2} x^{27}+\frac {28}{15} a^{5} b^{3} x^{30}+\frac {1}{45} b^{8} x^{45}\) | \(91\) |
parallelrisch | \(\frac {4}{21} a \,b^{7} x^{42}+\frac {14}{9} a^{3} b^{5} x^{36}+\frac {1}{21} a^{8} x^{21}+\frac {70}{33} a^{4} b^{4} x^{33}+\frac {28}{39} a^{2} b^{6} x^{39}+\frac {1}{3} a^{7} b \,x^{24}+\frac {28}{27} a^{6} b^{2} x^{27}+\frac {28}{15} a^{5} b^{3} x^{30}+\frac {1}{45} b^{8} x^{45}\) | \(91\) |
orering | \(\frac {x^{21} \left (3003 b^{8} x^{24}+25740 a \,b^{7} x^{21}+97020 a^{2} b^{6} x^{18}+210210 a^{3} b^{5} x^{15}+286650 a^{4} b^{4} x^{12}+252252 a^{5} b^{3} x^{9}+140140 a^{6} b^{2} x^{6}+45045 a^{7} b \,x^{3}+6435 a^{8}\right )}{135135}\) | \(93\) |
Input:
int(x^20*(b*x^3+a)^8,x,method=_RETURNVERBOSE)
Output:
4/21*a*b^7*x^42+14/9*a^3*b^5*x^36+1/21*a^8*x^21+70/33*a^4*b^4*x^33+28/39*a ^2*b^6*x^39+1/3*a^7*b*x^24+28/27*a^6*b^2*x^27+28/15*a^5*b^3*x^30+1/45*b^8* x^45
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int x^{20} \left (a+b x^3\right )^8 \, dx=\frac {1}{45} \, b^{8} x^{45} + \frac {4}{21} \, a b^{7} x^{42} + \frac {28}{39} \, a^{2} b^{6} x^{39} + \frac {14}{9} \, a^{3} b^{5} x^{36} + \frac {70}{33} \, a^{4} b^{4} x^{33} + \frac {28}{15} \, a^{5} b^{3} x^{30} + \frac {28}{27} \, a^{6} b^{2} x^{27} + \frac {1}{3} \, a^{7} b x^{24} + \frac {1}{21} \, a^{8} x^{21} \] Input:
integrate(x^20*(b*x^3+a)^8,x, algorithm="fricas")
Output:
1/45*b^8*x^45 + 4/21*a*b^7*x^42 + 28/39*a^2*b^6*x^39 + 14/9*a^3*b^5*x^36 + 70/33*a^4*b^4*x^33 + 28/15*a^5*b^3*x^30 + 28/27*a^6*b^2*x^27 + 1/3*a^7*b* x^24 + 1/21*a^8*x^21
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81 \[ \int x^{20} \left (a+b x^3\right )^8 \, dx=\frac {a^{8} x^{21}}{21} + \frac {a^{7} b x^{24}}{3} + \frac {28 a^{6} b^{2} x^{27}}{27} + \frac {28 a^{5} b^{3} x^{30}}{15} + \frac {70 a^{4} b^{4} x^{33}}{33} + \frac {14 a^{3} b^{5} x^{36}}{9} + \frac {28 a^{2} b^{6} x^{39}}{39} + \frac {4 a b^{7} x^{42}}{21} + \frac {b^{8} x^{45}}{45} \] Input:
integrate(x**20*(b*x**3+a)**8,x)
Output:
a**8*x**21/21 + a**7*b*x**24/3 + 28*a**6*b**2*x**27/27 + 28*a**5*b**3*x**3 0/15 + 70*a**4*b**4*x**33/33 + 14*a**3*b**5*x**36/9 + 28*a**2*b**6*x**39/3 9 + 4*a*b**7*x**42/21 + b**8*x**45/45
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int x^{20} \left (a+b x^3\right )^8 \, dx=\frac {1}{45} \, b^{8} x^{45} + \frac {4}{21} \, a b^{7} x^{42} + \frac {28}{39} \, a^{2} b^{6} x^{39} + \frac {14}{9} \, a^{3} b^{5} x^{36} + \frac {70}{33} \, a^{4} b^{4} x^{33} + \frac {28}{15} \, a^{5} b^{3} x^{30} + \frac {28}{27} \, a^{6} b^{2} x^{27} + \frac {1}{3} \, a^{7} b x^{24} + \frac {1}{21} \, a^{8} x^{21} \] Input:
integrate(x^20*(b*x^3+a)^8,x, algorithm="maxima")
Output:
1/45*b^8*x^45 + 4/21*a*b^7*x^42 + 28/39*a^2*b^6*x^39 + 14/9*a^3*b^5*x^36 + 70/33*a^4*b^4*x^33 + 28/15*a^5*b^3*x^30 + 28/27*a^6*b^2*x^27 + 1/3*a^7*b* x^24 + 1/21*a^8*x^21
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int x^{20} \left (a+b x^3\right )^8 \, dx=\frac {1}{45} \, b^{8} x^{45} + \frac {4}{21} \, a b^{7} x^{42} + \frac {28}{39} \, a^{2} b^{6} x^{39} + \frac {14}{9} \, a^{3} b^{5} x^{36} + \frac {70}{33} \, a^{4} b^{4} x^{33} + \frac {28}{15} \, a^{5} b^{3} x^{30} + \frac {28}{27} \, a^{6} b^{2} x^{27} + \frac {1}{3} \, a^{7} b x^{24} + \frac {1}{21} \, a^{8} x^{21} \] Input:
integrate(x^20*(b*x^3+a)^8,x, algorithm="giac")
Output:
1/45*b^8*x^45 + 4/21*a*b^7*x^42 + 28/39*a^2*b^6*x^39 + 14/9*a^3*b^5*x^36 + 70/33*a^4*b^4*x^33 + 28/15*a^5*b^3*x^30 + 28/27*a^6*b^2*x^27 + 1/3*a^7*b* x^24 + 1/21*a^8*x^21
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int x^{20} \left (a+b x^3\right )^8 \, dx=\frac {a^8\,x^{21}}{21}+\frac {a^7\,b\,x^{24}}{3}+\frac {28\,a^6\,b^2\,x^{27}}{27}+\frac {28\,a^5\,b^3\,x^{30}}{15}+\frac {70\,a^4\,b^4\,x^{33}}{33}+\frac {14\,a^3\,b^5\,x^{36}}{9}+\frac {28\,a^2\,b^6\,x^{39}}{39}+\frac {4\,a\,b^7\,x^{42}}{21}+\frac {b^8\,x^{45}}{45} \] Input:
int(x^20*(a + b*x^3)^8,x)
Output:
(a^8*x^21)/21 + (b^8*x^45)/45 + (a^7*b*x^24)/3 + (4*a*b^7*x^42)/21 + (28*a ^6*b^2*x^27)/27 + (28*a^5*b^3*x^30)/15 + (70*a^4*b^4*x^33)/33 + (14*a^3*b^ 5*x^36)/9 + (28*a^2*b^6*x^39)/39
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71 \[ \int x^{20} \left (a+b x^3\right )^8 \, dx=\frac {x^{21} \left (3003 b^{8} x^{24}+25740 a \,b^{7} x^{21}+97020 a^{2} b^{6} x^{18}+210210 a^{3} b^{5} x^{15}+286650 a^{4} b^{4} x^{12}+252252 a^{5} b^{3} x^{9}+140140 a^{6} b^{2} x^{6}+45045 a^{7} b \,x^{3}+6435 a^{8}\right )}{135135} \] Input:
int(x^20*(b*x^3+a)^8,x)
Output:
(x**21*(6435*a**8 + 45045*a**7*b*x**3 + 140140*a**6*b**2*x**6 + 252252*a** 5*b**3*x**9 + 286650*a**4*b**4*x**12 + 210210*a**3*b**5*x**15 + 97020*a**2 *b**6*x**18 + 25740*a*b**7*x**21 + 3003*b**8*x**24))/135135