Integrand size = 13, antiderivative size = 91 \[ \int x^{14} \left (a+b x^3\right )^8 \, dx=\frac {a^4 \left (a+b x^3\right )^9}{27 b^5}-\frac {2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac {2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}-\frac {a \left (a+b x^3\right )^{12}}{9 b^5}+\frac {\left (a+b x^3\right )^{13}}{39 b^5} \] Output:
1/27*a^4*(b*x^3+a)^9/b^5-2/15*a^3*(b*x^3+a)^10/b^5+2/11*a^2*(b*x^3+a)^11/b ^5-1/9*a*(b*x^3+a)^12/b^5+1/39*(b*x^3+a)^13/b^5
Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int x^{14} \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^{15}}{15}+\frac {4}{9} a^7 b x^{18}+\frac {4}{3} a^6 b^2 x^{21}+\frac {7}{3} a^5 b^3 x^{24}+\frac {70}{27} a^4 b^4 x^{27}+\frac {28}{15} a^3 b^5 x^{30}+\frac {28}{33} a^2 b^6 x^{33}+\frac {2}{9} a b^7 x^{36}+\frac {b^8 x^{39}}{39} \] Input:
Integrate[x^14*(a + b*x^3)^8,x]
Output:
(a^8*x^15)/15 + (4*a^7*b*x^18)/9 + (4*a^6*b^2*x^21)/3 + (7*a^5*b^3*x^24)/3 + (70*a^4*b^4*x^27)/27 + (28*a^3*b^5*x^30)/15 + (28*a^2*b^6*x^33)/33 + (2 *a*b^7*x^36)/9 + (b^8*x^39)/39
Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int x^{14} \left (a+b x^3\right )^8 \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int x^{12} \left (b x^3+a\right )^8dx^3\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \int \left (\frac {\left (b x^3+a\right )^{12}}{b^4}-\frac {4 a \left (b x^3+a\right )^{11}}{b^4}+\frac {6 a^2 \left (b x^3+a\right )^{10}}{b^4}-\frac {4 a^3 \left (b x^3+a\right )^9}{b^4}+\frac {a^4 \left (b x^3+a\right )^8}{b^4}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {a^4 \left (a+b x^3\right )^9}{9 b^5}-\frac {2 a^3 \left (a+b x^3\right )^{10}}{5 b^5}+\frac {6 a^2 \left (a+b x^3\right )^{11}}{11 b^5}+\frac {\left (a+b x^3\right )^{13}}{13 b^5}-\frac {a \left (a+b x^3\right )^{12}}{3 b^5}\right )\) |
Input:
Int[x^14*(a + b*x^3)^8,x]
Output:
((a^4*(a + b*x^3)^9)/(9*b^5) - (2*a^3*(a + b*x^3)^10)/(5*b^5) + (6*a^2*(a + b*x^3)^11)/(11*b^5) - (a*(a + b*x^3)^12)/(3*b^5) + (a + b*x^3)^13/(13*b^ 5))/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 = 0.42 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(\frac {4}{9} a^{7} b \,x^{18}+\frac {70}{27} a^{4} b^{4} x^{27}+\frac {28}{15} a^{3} b^{5} x^{30}+\frac {1}{15} a^{8} x^{15}+\frac {1}{39} b^{8} x^{39}+\frac {2}{9} a \,b^{7} x^{36}+\frac {7}{3} a^{5} b^{3} x^{24}+\frac {4}{3} a^{6} b^{2} x^{21}+\frac {28}{33} a^{2} b^{6} x^{33}\) | \(91\) |
default | \(\frac {4}{9} a^{7} b \,x^{18}+\frac {70}{27} a^{4} b^{4} x^{27}+\frac {28}{15} a^{3} b^{5} x^{30}+\frac {1}{15} a^{8} x^{15}+\frac {1}{39} b^{8} x^{39}+\frac {2}{9} a \,b^{7} x^{36}+\frac {7}{3} a^{5} b^{3} x^{24}+\frac {4}{3} a^{6} b^{2} x^{21}+\frac {28}{33} a^{2} b^{6} x^{33}\) | \(91\) |
risch | \(\frac {4}{9} a^{7} b \,x^{18}+\frac {70}{27} a^{4} b^{4} x^{27}+\frac {28}{15} a^{3} b^{5} x^{30}+\frac {1}{15} a^{8} x^{15}+\frac {1}{39} b^{8} x^{39}+\frac {2}{9} a \,b^{7} x^{36}+\frac {7}{3} a^{5} b^{3} x^{24}+\frac {4}{3} a^{6} b^{2} x^{21}+\frac {28}{33} a^{2} b^{6} x^{33}\) | \(91\) |
parallelrisch | \(\frac {4}{9} a^{7} b \,x^{18}+\frac {70}{27} a^{4} b^{4} x^{27}+\frac {28}{15} a^{3} b^{5} x^{30}+\frac {1}{15} a^{8} x^{15}+\frac {1}{39} b^{8} x^{39}+\frac {2}{9} a \,b^{7} x^{36}+\frac {7}{3} a^{5} b^{3} x^{24}+\frac {4}{3} a^{6} b^{2} x^{21}+\frac {28}{33} a^{2} b^{6} x^{33}\) | \(91\) |
orering | \(\frac {x^{15} \left (495 b^{8} x^{24}+4290 a \,b^{7} x^{21}+16380 a^{2} b^{6} x^{18}+36036 a^{3} b^{5} x^{15}+50050 a^{4} b^{4} x^{12}+45045 a^{5} b^{3} x^{9}+25740 a^{6} b^{2} x^{6}+8580 a^{7} b \,x^{3}+1287 a^{8}\right )}{19305}\) | \(93\) |
Input:
int(x^14*(b*x^3+a)^8,x,method=_RETURNVERBOSE)
Output:
4/9*a^7*b*x^18+70/27*a^4*b^4*x^27+28/15*a^3*b^5*x^30+1/15*a^8*x^15+1/39*b^ 8*x^39+2/9*a*b^7*x^36+7/3*a^5*b^3*x^24+4/3*a^6*b^2*x^21+28/33*a^2*b^6*x^33
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int x^{14} \left (a+b x^3\right )^8 \, dx=\frac {1}{39} \, b^{8} x^{39} + \frac {2}{9} \, a b^{7} x^{36} + \frac {28}{33} \, a^{2} b^{6} x^{33} + \frac {28}{15} \, a^{3} b^{5} x^{30} + \frac {70}{27} \, a^{4} b^{4} x^{27} + \frac {7}{3} \, a^{5} b^{3} x^{24} + \frac {4}{3} \, a^{6} b^{2} x^{21} + \frac {4}{9} \, a^{7} b x^{18} + \frac {1}{15} \, a^{8} x^{15} \] Input:
integrate(x^14*(b*x^3+a)^8,x, algorithm="fricas")
Output:
1/39*b^8*x^39 + 2/9*a*b^7*x^36 + 28/33*a^2*b^6*x^33 + 28/15*a^3*b^5*x^30 + 70/27*a^4*b^4*x^27 + 7/3*a^5*b^3*x^24 + 4/3*a^6*b^2*x^21 + 4/9*a^7*b*x^18 + 1/15*a^8*x^15
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.18 \[ \int x^{14} \left (a+b x^3\right )^8 \, dx=\frac {a^{8} x^{15}}{15} + \frac {4 a^{7} b x^{18}}{9} + \frac {4 a^{6} b^{2} x^{21}}{3} + \frac {7 a^{5} b^{3} x^{24}}{3} + \frac {70 a^{4} b^{4} x^{27}}{27} + \frac {28 a^{3} b^{5} x^{30}}{15} + \frac {28 a^{2} b^{6} x^{33}}{33} + \frac {2 a b^{7} x^{36}}{9} + \frac {b^{8} x^{39}}{39} \] Input:
integrate(x**14*(b*x**3+a)**8,x)
Output:
a**8*x**15/15 + 4*a**7*b*x**18/9 + 4*a**6*b**2*x**21/3 + 7*a**5*b**3*x**24 /3 + 70*a**4*b**4*x**27/27 + 28*a**3*b**5*x**30/15 + 28*a**2*b**6*x**33/33 + 2*a*b**7*x**36/9 + b**8*x**39/39
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int x^{14} \left (a+b x^3\right )^8 \, dx=\frac {1}{39} \, b^{8} x^{39} + \frac {2}{9} \, a b^{7} x^{36} + \frac {28}{33} \, a^{2} b^{6} x^{33} + \frac {28}{15} \, a^{3} b^{5} x^{30} + \frac {70}{27} \, a^{4} b^{4} x^{27} + \frac {7}{3} \, a^{5} b^{3} x^{24} + \frac {4}{3} \, a^{6} b^{2} x^{21} + \frac {4}{9} \, a^{7} b x^{18} + \frac {1}{15} \, a^{8} x^{15} \] Input:
integrate(x^14*(b*x^3+a)^8,x, algorithm="maxima")
Output:
1/39*b^8*x^39 + 2/9*a*b^7*x^36 + 28/33*a^2*b^6*x^33 + 28/15*a^3*b^5*x^30 + 70/27*a^4*b^4*x^27 + 7/3*a^5*b^3*x^24 + 4/3*a^6*b^2*x^21 + 4/9*a^7*b*x^18 + 1/15*a^8*x^15
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int x^{14} \left (a+b x^3\right )^8 \, dx=\frac {1}{39} \, b^{8} x^{39} + \frac {2}{9} \, a b^{7} x^{36} + \frac {28}{33} \, a^{2} b^{6} x^{33} + \frac {28}{15} \, a^{3} b^{5} x^{30} + \frac {70}{27} \, a^{4} b^{4} x^{27} + \frac {7}{3} \, a^{5} b^{3} x^{24} + \frac {4}{3} \, a^{6} b^{2} x^{21} + \frac {4}{9} \, a^{7} b x^{18} + \frac {1}{15} \, a^{8} x^{15} \] Input:
integrate(x^14*(b*x^3+a)^8,x, algorithm="giac")
Output:
1/39*b^8*x^39 + 2/9*a*b^7*x^36 + 28/33*a^2*b^6*x^33 + 28/15*a^3*b^5*x^30 + 70/27*a^4*b^4*x^27 + 7/3*a^5*b^3*x^24 + 4/3*a^6*b^2*x^21 + 4/9*a^7*b*x^18 + 1/15*a^8*x^15
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int x^{14} \left (a+b x^3\right )^8 \, dx=\frac {a^8\,x^{15}}{15}+\frac {4\,a^7\,b\,x^{18}}{9}+\frac {4\,a^6\,b^2\,x^{21}}{3}+\frac {7\,a^5\,b^3\,x^{24}}{3}+\frac {70\,a^4\,b^4\,x^{27}}{27}+\frac {28\,a^3\,b^5\,x^{30}}{15}+\frac {28\,a^2\,b^6\,x^{33}}{33}+\frac {2\,a\,b^7\,x^{36}}{9}+\frac {b^8\,x^{39}}{39} \] Input:
int(x^14*(a + b*x^3)^8,x)
Output:
(a^8*x^15)/15 + (b^8*x^39)/39 + (4*a^7*b*x^18)/9 + (2*a*b^7*x^36)/9 + (4*a ^6*b^2*x^21)/3 + (7*a^5*b^3*x^24)/3 + (70*a^4*b^4*x^27)/27 + (28*a^3*b^5*x ^30)/15 + (28*a^2*b^6*x^33)/33
Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01 \[ \int x^{14} \left (a+b x^3\right )^8 \, dx=\frac {x^{15} \left (495 b^{8} x^{24}+4290 a \,b^{7} x^{21}+16380 a^{2} b^{6} x^{18}+36036 a^{3} b^{5} x^{15}+50050 a^{4} b^{4} x^{12}+45045 a^{5} b^{3} x^{9}+25740 a^{6} b^{2} x^{6}+8580 a^{7} b \,x^{3}+1287 a^{8}\right )}{19305} \] Input:
int(x^14*(b*x^3+a)^8,x)
Output:
(x**15*(1287*a**8 + 8580*a**7*b*x**3 + 25740*a**6*b**2*x**6 + 45045*a**5*b **3*x**9 + 50050*a**4*b**4*x**12 + 36036*a**3*b**5*x**15 + 16380*a**2*b**6 *x**18 + 4290*a*b**7*x**21 + 495*b**8*x**24))/19305