Integrand size = 13, antiderivative size = 110 \[ \int x^{17} \left (a+b x^3\right )^8 \, dx=-\frac {a^5 \left (a+b x^3\right )^9}{27 b^6}+\frac {a^4 \left (a+b x^3\right )^{10}}{6 b^6}-\frac {10 a^3 \left (a+b x^3\right )^{11}}{33 b^6}+\frac {5 a^2 \left (a+b x^3\right )^{12}}{18 b^6}-\frac {5 a \left (a+b x^3\right )^{13}}{39 b^6}+\frac {\left (a+b x^3\right )^{14}}{42 b^6} \] Output:
-1/27*a^5*(b*x^3+a)^9/b^6+1/6*a^4*(b*x^3+a)^10/b^6-10/33*a^3*(b*x^3+a)^11/ b^6+5/18*a^2*(b*x^3+a)^12/b^6-5/39*a*(b*x^3+a)^13/b^6+1/42*(b*x^3+a)^14/b^ 6
Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98 \[ \int x^{17} \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^{18}}{18}+\frac {8}{21} a^7 b x^{21}+\frac {7}{6} a^6 b^2 x^{24}+\frac {56}{27} a^5 b^3 x^{27}+\frac {7}{3} a^4 b^4 x^{30}+\frac {56}{33} a^3 b^5 x^{33}+\frac {7}{9} a^2 b^6 x^{36}+\frac {8}{39} a b^7 x^{39}+\frac {b^8 x^{42}}{42} \] Input:
Integrate[x^17*(a + b*x^3)^8,x]
Output:
(a^8*x^18)/18 + (8*a^7*b*x^21)/21 + (7*a^6*b^2*x^24)/6 + (56*a^5*b^3*x^27) /27 + (7*a^4*b^4*x^30)/3 + (56*a^3*b^5*x^33)/33 + (7*a^2*b^6*x^36)/9 + (8* a*b^7*x^39)/39 + (b^8*x^42)/42
Time = 0.38 (sec) , antiderivative size = 114, 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.
\(\displaystyle \int x^{17} \left (a+b x^3\right )^8 \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int x^{15} \left (b x^3+a\right )^8dx^3\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \int \left (\frac {\left (b x^3+a\right )^{13}}{b^5}-\frac {5 a \left (b x^3+a\right )^{12}}{b^5}+\frac {10 a^2 \left (b x^3+a\right )^{11}}{b^5}-\frac {10 a^3 \left (b x^3+a\right )^{10}}{b^5}+\frac {5 a^4 \left (b x^3+a\right )^9}{b^5}-\frac {a^5 \left (b x^3+a\right )^8}{b^5}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {a^5 \left (a+b x^3\right )^9}{9 b^6}+\frac {a^4 \left (a+b x^3\right )^{10}}{2 b^6}-\frac {10 a^3 \left (a+b x^3\right )^{11}}{11 b^6}+\frac {5 a^2 \left (a+b x^3\right )^{12}}{6 b^6}+\frac {\left (a+b x^3\right )^{14}}{14 b^6}-\frac {5 a \left (a+b x^3\right )^{13}}{13 b^6}\right )\) |
Input:
Int[x^17*(a + b*x^3)^8,x]
Output:
(-1/9*(a^5*(a + b*x^3)^9)/b^6 + (a^4*(a + b*x^3)^10)/(2*b^6) - (10*a^3*(a + b*x^3)^11)/(11*b^6) + (5*a^2*(a + b*x^3)^12)/(6*b^6) - (5*a*(a + b*x^3)^ 13)/(13*b^6) + (a + b*x^3)^14/(14*b^6))/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 = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {1}{42} b^{8} x^{42}+\frac {56}{33} a^{3} b^{5} x^{33}+\frac {7}{6} a^{6} b^{2} x^{24}+\frac {7}{3} a^{4} b^{4} x^{30}+\frac {56}{27} a^{5} b^{3} x^{27}+\frac {7}{9} a^{2} b^{6} x^{36}+\frac {8}{39} a \,b^{7} x^{39}+\frac {8}{21} a^{7} b \,x^{21}+\frac {1}{18} a^{8} x^{18}\) | \(91\) |
default | \(\frac {1}{42} b^{8} x^{42}+\frac {56}{33} a^{3} b^{5} x^{33}+\frac {7}{6} a^{6} b^{2} x^{24}+\frac {7}{3} a^{4} b^{4} x^{30}+\frac {56}{27} a^{5} b^{3} x^{27}+\frac {7}{9} a^{2} b^{6} x^{36}+\frac {8}{39} a \,b^{7} x^{39}+\frac {8}{21} a^{7} b \,x^{21}+\frac {1}{18} a^{8} x^{18}\) | \(91\) |
risch | \(\frac {1}{42} b^{8} x^{42}+\frac {56}{33} a^{3} b^{5} x^{33}+\frac {7}{6} a^{6} b^{2} x^{24}+\frac {7}{3} a^{4} b^{4} x^{30}+\frac {56}{27} a^{5} b^{3} x^{27}+\frac {7}{9} a^{2} b^{6} x^{36}+\frac {8}{39} a \,b^{7} x^{39}+\frac {8}{21} a^{7} b \,x^{21}+\frac {1}{18} a^{8} x^{18}\) | \(91\) |
parallelrisch | \(\frac {1}{42} b^{8} x^{42}+\frac {56}{33} a^{3} b^{5} x^{33}+\frac {7}{6} a^{6} b^{2} x^{24}+\frac {7}{3} a^{4} b^{4} x^{30}+\frac {56}{27} a^{5} b^{3} x^{27}+\frac {7}{9} a^{2} b^{6} x^{36}+\frac {8}{39} a \,b^{7} x^{39}+\frac {8}{21} a^{7} b \,x^{21}+\frac {1}{18} a^{8} x^{18}\) | \(91\) |
orering | \(\frac {x^{18} \left (1287 b^{8} x^{24}+11088 a \,b^{7} x^{21}+42042 a^{2} b^{6} x^{18}+91728 a^{3} b^{5} x^{15}+126126 a^{4} b^{4} x^{12}+112112 a^{5} b^{3} x^{9}+63063 a^{6} b^{2} x^{6}+20592 a^{7} b \,x^{3}+3003 a^{8}\right )}{54054}\) | \(93\) |
Input:
int(x^17*(b*x^3+a)^8,x,method=_RETURNVERBOSE)
Output:
1/42*b^8*x^42+56/33*a^3*b^5*x^33+7/6*a^6*b^2*x^24+7/3*a^4*b^4*x^30+56/27*a ^5*b^3*x^27+7/9*a^2*b^6*x^36+8/39*a*b^7*x^39+8/21*a^7*b*x^21+1/18*a^8*x^18
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int x^{17} \left (a+b x^3\right )^8 \, dx=\frac {1}{42} \, b^{8} x^{42} + \frac {8}{39} \, a b^{7} x^{39} + \frac {7}{9} \, a^{2} b^{6} x^{36} + \frac {56}{33} \, a^{3} b^{5} x^{33} + \frac {7}{3} \, a^{4} b^{4} x^{30} + \frac {56}{27} \, a^{5} b^{3} x^{27} + \frac {7}{6} \, a^{6} b^{2} x^{24} + \frac {8}{21} \, a^{7} b x^{21} + \frac {1}{18} \, a^{8} x^{18} \] Input:
integrate(x^17*(b*x^3+a)^8,x, algorithm="fricas")
Output:
1/42*b^8*x^42 + 8/39*a*b^7*x^39 + 7/9*a^2*b^6*x^36 + 56/33*a^3*b^5*x^33 + 7/3*a^4*b^4*x^30 + 56/27*a^5*b^3*x^27 + 7/6*a^6*b^2*x^24 + 8/21*a^7*b*x^21 + 1/18*a^8*x^18
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97 \[ \int x^{17} \left (a+b x^3\right )^8 \, dx=\frac {a^{8} x^{18}}{18} + \frac {8 a^{7} b x^{21}}{21} + \frac {7 a^{6} b^{2} x^{24}}{6} + \frac {56 a^{5} b^{3} x^{27}}{27} + \frac {7 a^{4} b^{4} x^{30}}{3} + \frac {56 a^{3} b^{5} x^{33}}{33} + \frac {7 a^{2} b^{6} x^{36}}{9} + \frac {8 a b^{7} x^{39}}{39} + \frac {b^{8} x^{42}}{42} \] Input:
integrate(x**17*(b*x**3+a)**8,x)
Output:
a**8*x**18/18 + 8*a**7*b*x**21/21 + 7*a**6*b**2*x**24/6 + 56*a**5*b**3*x** 27/27 + 7*a**4*b**4*x**30/3 + 56*a**3*b**5*x**33/33 + 7*a**2*b**6*x**36/9 + 8*a*b**7*x**39/39 + b**8*x**42/42
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int x^{17} \left (a+b x^3\right )^8 \, dx=\frac {1}{42} \, b^{8} x^{42} + \frac {8}{39} \, a b^{7} x^{39} + \frac {7}{9} \, a^{2} b^{6} x^{36} + \frac {56}{33} \, a^{3} b^{5} x^{33} + \frac {7}{3} \, a^{4} b^{4} x^{30} + \frac {56}{27} \, a^{5} b^{3} x^{27} + \frac {7}{6} \, a^{6} b^{2} x^{24} + \frac {8}{21} \, a^{7} b x^{21} + \frac {1}{18} \, a^{8} x^{18} \] Input:
integrate(x^17*(b*x^3+a)^8,x, algorithm="maxima")
Output:
1/42*b^8*x^42 + 8/39*a*b^7*x^39 + 7/9*a^2*b^6*x^36 + 56/33*a^3*b^5*x^33 + 7/3*a^4*b^4*x^30 + 56/27*a^5*b^3*x^27 + 7/6*a^6*b^2*x^24 + 8/21*a^7*b*x^21 + 1/18*a^8*x^18
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int x^{17} \left (a+b x^3\right )^8 \, dx=\frac {1}{42} \, b^{8} x^{42} + \frac {8}{39} \, a b^{7} x^{39} + \frac {7}{9} \, a^{2} b^{6} x^{36} + \frac {56}{33} \, a^{3} b^{5} x^{33} + \frac {7}{3} \, a^{4} b^{4} x^{30} + \frac {56}{27} \, a^{5} b^{3} x^{27} + \frac {7}{6} \, a^{6} b^{2} x^{24} + \frac {8}{21} \, a^{7} b x^{21} + \frac {1}{18} \, a^{8} x^{18} \] Input:
integrate(x^17*(b*x^3+a)^8,x, algorithm="giac")
Output:
1/42*b^8*x^42 + 8/39*a*b^7*x^39 + 7/9*a^2*b^6*x^36 + 56/33*a^3*b^5*x^33 + 7/3*a^4*b^4*x^30 + 56/27*a^5*b^3*x^27 + 7/6*a^6*b^2*x^24 + 8/21*a^7*b*x^21 + 1/18*a^8*x^18
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int x^{17} \left (a+b x^3\right )^8 \, dx=\frac {a^8\,x^{18}}{18}+\frac {8\,a^7\,b\,x^{21}}{21}+\frac {7\,a^6\,b^2\,x^{24}}{6}+\frac {56\,a^5\,b^3\,x^{27}}{27}+\frac {7\,a^4\,b^4\,x^{30}}{3}+\frac {56\,a^3\,b^5\,x^{33}}{33}+\frac {7\,a^2\,b^6\,x^{36}}{9}+\frac {8\,a\,b^7\,x^{39}}{39}+\frac {b^8\,x^{42}}{42} \] Input:
int(x^17*(a + b*x^3)^8,x)
Output:
(a^8*x^18)/18 + (b^8*x^42)/42 + (8*a^7*b*x^21)/21 + (8*a*b^7*x^39)/39 + (7 *a^6*b^2*x^24)/6 + (56*a^5*b^3*x^27)/27 + (7*a^4*b^4*x^30)/3 + (56*a^3*b^5 *x^33)/33 + (7*a^2*b^6*x^36)/9
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int x^{17} \left (a+b x^3\right )^8 \, dx=\frac {x^{18} \left (1287 b^{8} x^{24}+11088 a \,b^{7} x^{21}+42042 a^{2} b^{6} x^{18}+91728 a^{3} b^{5} x^{15}+126126 a^{4} b^{4} x^{12}+112112 a^{5} b^{3} x^{9}+63063 a^{6} b^{2} x^{6}+20592 a^{7} b \,x^{3}+3003 a^{8}\right )}{54054} \] Input:
int(x^17*(b*x^3+a)^8,x)
Output:
(x**18*(3003*a**8 + 20592*a**7*b*x**3 + 63063*a**6*b**2*x**6 + 112112*a**5 *b**3*x**9 + 126126*a**4*b**4*x**12 + 91728*a**3*b**5*x**15 + 42042*a**2*b **6*x**18 + 11088*a*b**7*x**21 + 1287*b**8*x**24))/54054