Integrand size = 11, antiderivative size = 106 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^2}{2}+\frac {8}{5} a^7 b x^5+\frac {7}{2} a^6 b^2 x^8+\frac {56}{11} a^5 b^3 x^{11}+5 a^4 b^4 x^{14}+\frac {56}{17} a^3 b^5 x^{17}+\frac {7}{5} a^2 b^6 x^{20}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{26}}{26} \] Output:
1/2*a^8*x^2+8/5*a^7*b*x^5+7/2*a^6*b^2*x^8+56/11*a^5*b^3*x^11+5*a^4*b^4*x^1 4+56/17*a^3*b^5*x^17+7/5*a^2*b^6*x^20+8/23*a*b^7*x^23+1/26*b^8*x^26
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^2}{2}+\frac {8}{5} a^7 b x^5+\frac {7}{2} a^6 b^2 x^8+\frac {56}{11} a^5 b^3 x^{11}+5 a^4 b^4 x^{14}+\frac {56}{17} a^3 b^5 x^{17}+\frac {7}{5} a^2 b^6 x^{20}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{26}}{26} \] Input:
Integrate[x*(a + b*x^3)^8,x]
Output:
(a^8*x^2)/2 + (8*a^7*b*x^5)/5 + (7*a^6*b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 + 5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/17 + (7*a^2*b^6*x^20)/5 + (8*a*b^7*x^2 3)/23 + (b^8*x^26)/26
Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {802, 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.
\(\displaystyle \int x \left (a+b x^3\right )^8 \, dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \int \left (a^8 x+8 a^7 b x^4+28 a^6 b^2 x^7+56 a^5 b^3 x^{10}+70 a^4 b^4 x^{13}+56 a^3 b^5 x^{16}+28 a^2 b^6 x^{19}+8 a b^7 x^{22}+b^8 x^{25}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^8 x^2}{2}+\frac {8}{5} a^7 b x^5+\frac {7}{2} a^6 b^2 x^8+\frac {56}{11} a^5 b^3 x^{11}+5 a^4 b^4 x^{14}+\frac {56}{17} a^3 b^5 x^{17}+\frac {7}{5} a^2 b^6 x^{20}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{26}}{26}\) |
Input:
Int[x*(a + b*x^3)^8,x]
Output:
(a^8*x^2)/2 + (8*a^7*b*x^5)/5 + (7*a^6*b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 + 5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/17 + (7*a^2*b^6*x^20)/5 + (8*a*b^7*x^2 3)/23 + (b^8*x^26)/26
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} a^{7} b \,x^{5}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} a^{5} b^{3} x^{11}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\) | \(91\) |
default | \(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} a^{7} b \,x^{5}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} a^{5} b^{3} x^{11}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\) | \(91\) |
norman | \(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} a^{7} b \,x^{5}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} a^{5} b^{3} x^{11}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\) | \(91\) |
risch | \(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} a^{7} b \,x^{5}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} a^{5} b^{3} x^{11}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\) | \(91\) |
parallelrisch | \(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} a^{7} b \,x^{5}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} a^{5} b^{3} x^{11}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\) | \(91\) |
orering | \(\frac {x^{2} \left (21505 b^{8} x^{24}+194480 a \,b^{7} x^{21}+782782 a^{2} b^{6} x^{18}+1841840 a^{3} b^{5} x^{15}+2795650 a^{4} b^{4} x^{12}+2846480 a^{5} b^{3} x^{9}+1956955 a^{6} b^{2} x^{6}+894608 a^{7} b \,x^{3}+279565 a^{8}\right )}{559130}\) | \(93\) |
Input:
int(x*(b*x^3+a)^8,x,method=_RETURNVERBOSE)
Output:
1/2*a^8*x^2+8/5*a^7*b*x^5+7/2*a^6*b^2*x^8+56/11*a^5*b^3*x^11+5*a^4*b^4*x^1 4+56/17*a^3*b^5*x^17+7/5*a^2*b^6*x^20+8/23*a*b^7*x^23+1/26*b^8*x^26
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {1}{26} \, b^{8} x^{26} + \frac {8}{23} \, a b^{7} x^{23} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {56}{17} \, a^{3} b^{5} x^{17} + 5 \, a^{4} b^{4} x^{14} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {7}{2} \, a^{6} b^{2} x^{8} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{2} \, a^{8} x^{2} \] Input:
integrate(x*(b*x^3+a)^8,x, algorithm="fricas")
Output:
1/26*b^8*x^26 + 8/23*a*b^7*x^23 + 7/5*a^2*b^6*x^20 + 56/17*a^3*b^5*x^17 + 5*a^4*b^4*x^14 + 56/11*a^5*b^3*x^11 + 7/2*a^6*b^2*x^8 + 8/5*a^7*b*x^5 + 1/ 2*a^8*x^2
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^{8} x^{2}}{2} + \frac {8 a^{7} b x^{5}}{5} + \frac {7 a^{6} b^{2} x^{8}}{2} + \frac {56 a^{5} b^{3} x^{11}}{11} + 5 a^{4} b^{4} x^{14} + \frac {56 a^{3} b^{5} x^{17}}{17} + \frac {7 a^{2} b^{6} x^{20}}{5} + \frac {8 a b^{7} x^{23}}{23} + \frac {b^{8} x^{26}}{26} \] Input:
integrate(x*(b*x**3+a)**8,x)
Output:
a**8*x**2/2 + 8*a**7*b*x**5/5 + 7*a**6*b**2*x**8/2 + 56*a**5*b**3*x**11/11 + 5*a**4*b**4*x**14 + 56*a**3*b**5*x**17/17 + 7*a**2*b**6*x**20/5 + 8*a*b **7*x**23/23 + b**8*x**26/26
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {1}{26} \, b^{8} x^{26} + \frac {8}{23} \, a b^{7} x^{23} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {56}{17} \, a^{3} b^{5} x^{17} + 5 \, a^{4} b^{4} x^{14} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {7}{2} \, a^{6} b^{2} x^{8} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{2} \, a^{8} x^{2} \] Input:
integrate(x*(b*x^3+a)^8,x, algorithm="maxima")
Output:
1/26*b^8*x^26 + 8/23*a*b^7*x^23 + 7/5*a^2*b^6*x^20 + 56/17*a^3*b^5*x^17 + 5*a^4*b^4*x^14 + 56/11*a^5*b^3*x^11 + 7/2*a^6*b^2*x^8 + 8/5*a^7*b*x^5 + 1/ 2*a^8*x^2
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {1}{26} \, b^{8} x^{26} + \frac {8}{23} \, a b^{7} x^{23} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {56}{17} \, a^{3} b^{5} x^{17} + 5 \, a^{4} b^{4} x^{14} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {7}{2} \, a^{6} b^{2} x^{8} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{2} \, a^{8} x^{2} \] Input:
integrate(x*(b*x^3+a)^8,x, algorithm="giac")
Output:
1/26*b^8*x^26 + 8/23*a*b^7*x^23 + 7/5*a^2*b^6*x^20 + 56/17*a^3*b^5*x^17 + 5*a^4*b^4*x^14 + 56/11*a^5*b^3*x^11 + 7/2*a^6*b^2*x^8 + 8/5*a^7*b*x^5 + 1/ 2*a^8*x^2
Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^8\,x^2}{2}+\frac {8\,a^7\,b\,x^5}{5}+\frac {7\,a^6\,b^2\,x^8}{2}+\frac {56\,a^5\,b^3\,x^{11}}{11}+5\,a^4\,b^4\,x^{14}+\frac {56\,a^3\,b^5\,x^{17}}{17}+\frac {7\,a^2\,b^6\,x^{20}}{5}+\frac {8\,a\,b^7\,x^{23}}{23}+\frac {b^8\,x^{26}}{26} \] Input:
int(x*(a + b*x^3)^8,x)
Output:
(a^8*x^2)/2 + (b^8*x^26)/26 + (8*a^7*b*x^5)/5 + (8*a*b^7*x^23)/23 + (7*a^6 *b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 + 5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/17 + (7*a^2*b^6*x^20)/5
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {x^{2} \left (21505 b^{8} x^{24}+194480 a \,b^{7} x^{21}+782782 a^{2} b^{6} x^{18}+1841840 a^{3} b^{5} x^{15}+2795650 a^{4} b^{4} x^{12}+2846480 a^{5} b^{3} x^{9}+1956955 a^{6} b^{2} x^{6}+894608 a^{7} b \,x^{3}+279565 a^{8}\right )}{559130} \] Input:
int(x*(b*x^3+a)^8,x)
Output:
(x**2*(279565*a**8 + 894608*a**7*b*x**3 + 1956955*a**6*b**2*x**6 + 2846480 *a**5*b**3*x**9 + 2795650*a**4*b**4*x**12 + 1841840*a**3*b**5*x**15 + 7827 82*a**2*b**6*x**18 + 194480*a*b**7*x**21 + 21505*b**8*x**24))/559130