Integrand size = 13, antiderivative size = 105 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=-\frac {a^8}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12}+70 a^4 b^4 \log (x) \] Output:
-1/12*a^8/x^12-8/9*a^7*b/x^9-14/3*a^6*b^2/x^6-56/3*a^5*b^3/x^3+56/3*a^3*b^ 5*x^3+14/3*a^2*b^6*x^6+8/9*a*b^7*x^9+1/12*b^8*x^12+70*a^4*b^4*ln(x)
Time = 0.00 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=-\frac {a^8}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12}+70 a^4 b^4 \log (x) \] Input:
Integrate[(a + b*x^3)^8/x^13,x]
Output:
-1/12*a^8/x^12 - (8*a^7*b)/(9*x^9) - (14*a^6*b^2)/(3*x^6) - (56*a^5*b^3)/( 3*x^3) + (56*a^3*b^5*x^3)/3 + (14*a^2*b^6*x^6)/3 + (8*a*b^7*x^9)/9 + (b^8* x^12)/12 + 70*a^4*b^4*Log[x]
Time = 0.37 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.98, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^8}{x^{15}}dx^3\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \int \left (b^8 x^9+8 a b^7 x^6+28 a^2 b^6 x^3+56 a^3 b^5+\frac {70 a^4 b^4}{x^3}+\frac {56 a^5 b^3}{x^6}+\frac {28 a^6 b^2}{x^9}+\frac {8 a^7 b}{x^{12}}+\frac {a^8}{x^{15}}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {a^8}{4 x^{12}}-\frac {8 a^7 b}{3 x^9}-\frac {14 a^6 b^2}{x^6}-\frac {56 a^5 b^3}{x^3}+70 a^4 b^4 \log \left (x^3\right )+56 a^3 b^5 x^3+14 a^2 b^6 x^6+\frac {8}{3} a b^7 x^9+\frac {b^8 x^{12}}{4}\right )\) |
Input:
Int[(a + b*x^3)^8/x^13,x]
Output:
(-1/4*a^8/x^12 - (8*a^7*b)/(3*x^9) - (14*a^6*b^2)/x^6 - (56*a^5*b^3)/x^3 + 56*a^3*b^5*x^3 + 14*a^2*b^6*x^6 + (8*a*b^7*x^9)/3 + (b^8*x^12)/4 + 70*a^4 *b^4*Log[x^3])/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 = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {a^{8}}{12 x^{12}}-\frac {8 a^{7} b}{9 x^{9}}-\frac {14 a^{6} b^{2}}{3 x^{6}}-\frac {56 a^{5} b^{3}}{3 x^{3}}+\frac {56 a^{3} b^{5} x^{3}}{3}+\frac {14 a^{2} x^{6} b^{6}}{3}+\frac {8 a \,b^{7} x^{9}}{9}+\frac {b^{8} x^{12}}{12}+70 a^{4} b^{4} \ln \left (x \right )\) | \(90\) |
norman | \(\frac {-\frac {1}{12} a^{8}+\frac {1}{12} b^{8} x^{24}+\frac {8}{9} a \,b^{7} x^{21}+\frac {14}{3} a^{2} b^{6} x^{18}+\frac {56}{3} a^{3} b^{5} x^{15}-\frac {56}{3} a^{5} b^{3} x^{9}-\frac {14}{3} a^{6} b^{2} x^{6}-\frac {8}{9} a^{7} b \,x^{3}}{x^{12}}+70 a^{4} b^{4} \ln \left (x \right )\) | \(92\) |
risch | \(\frac {b^{8} x^{12}}{12}+\frac {8 a \,b^{7} x^{9}}{9}+\frac {14 a^{2} x^{6} b^{6}}{3}+\frac {56 a^{3} b^{5} x^{3}}{3}+\frac {-\frac {56}{3} a^{5} b^{3} x^{9}-\frac {14}{3} a^{6} b^{2} x^{6}-\frac {8}{9} a^{7} b \,x^{3}-\frac {1}{12} a^{8}}{x^{12}}+70 a^{4} b^{4} \ln \left (x \right )\) | \(92\) |
parallelrisch | \(\frac {3 b^{8} x^{24}+32 a \,b^{7} x^{21}+168 a^{2} b^{6} x^{18}+672 a^{3} b^{5} x^{15}+2520 a^{4} b^{4} \ln \left (x \right ) x^{12}-672 a^{5} b^{3} x^{9}-168 a^{6} b^{2} x^{6}-32 a^{7} b \,x^{3}-3 a^{8}}{36 x^{12}}\) | \(95\) |
Input:
int((b*x^3+a)^8/x^13,x,method=_RETURNVERBOSE)
Output:
-1/12*a^8/x^12-8/9*a^7*b/x^9-14/3*a^6*b^2/x^6-56/3*a^5*b^3/x^3+56/3*a^3*b^ 5*x^3+14/3*a^2*x^6*b^6+8/9*a*b^7*x^9+1/12*b^8*x^12+70*a^4*b^4*ln(x)
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {3 \, b^{8} x^{24} + 32 \, a b^{7} x^{21} + 168 \, a^{2} b^{6} x^{18} + 672 \, a^{3} b^{5} x^{15} + 2520 \, a^{4} b^{4} x^{12} \log \left (x\right ) - 672 \, a^{5} b^{3} x^{9} - 168 \, a^{6} b^{2} x^{6} - 32 \, a^{7} b x^{3} - 3 \, a^{8}}{36 \, x^{12}} \] Input:
integrate((b*x^3+a)^8/x^13,x, algorithm="fricas")
Output:
1/36*(3*b^8*x^24 + 32*a*b^7*x^21 + 168*a^2*b^6*x^18 + 672*a^3*b^5*x^15 + 2 520*a^4*b^4*x^12*log(x) - 672*a^5*b^3*x^9 - 168*a^6*b^2*x^6 - 32*a^7*b*x^3 - 3*a^8)/x^12
Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=70 a^{4} b^{4} \log {\left (x \right )} + \frac {56 a^{3} b^{5} x^{3}}{3} + \frac {14 a^{2} b^{6} x^{6}}{3} + \frac {8 a b^{7} x^{9}}{9} + \frac {b^{8} x^{12}}{12} + \frac {- 3 a^{8} - 32 a^{7} b x^{3} - 168 a^{6} b^{2} x^{6} - 672 a^{5} b^{3} x^{9}}{36 x^{12}} \] Input:
integrate((b*x**3+a)**8/x**13,x)
Output:
70*a**4*b**4*log(x) + 56*a**3*b**5*x**3/3 + 14*a**2*b**6*x**6/3 + 8*a*b**7 *x**9/9 + b**8*x**12/12 + (-3*a**8 - 32*a**7*b*x**3 - 168*a**6*b**2*x**6 - 672*a**5*b**3*x**9)/(36*x**12)
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {1}{12} \, b^{8} x^{12} + \frac {8}{9} \, a b^{7} x^{9} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {56}{3} \, a^{3} b^{5} x^{3} + \frac {70}{3} \, a^{4} b^{4} \log \left (x^{3}\right ) - \frac {672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \] Input:
integrate((b*x^3+a)^8/x^13,x, algorithm="maxima")
Output:
1/12*b^8*x^12 + 8/9*a*b^7*x^9 + 14/3*a^2*b^6*x^6 + 56/3*a^3*b^5*x^3 + 70/3 *a^4*b^4*log(x^3) - 1/36*(672*a^5*b^3*x^9 + 168*a^6*b^2*x^6 + 32*a^7*b*x^3 + 3*a^8)/x^12
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {1}{12} \, b^{8} x^{12} + \frac {8}{9} \, a b^{7} x^{9} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} \log \left ({\left | x \right |}\right ) - \frac {1750 \, a^{4} b^{4} x^{12} + 672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \] Input:
integrate((b*x^3+a)^8/x^13,x, algorithm="giac")
Output:
1/12*b^8*x^12 + 8/9*a*b^7*x^9 + 14/3*a^2*b^6*x^6 + 56/3*a^3*b^5*x^3 + 70*a ^4*b^4*log(abs(x)) - 1/36*(1750*a^4*b^4*x^12 + 672*a^5*b^3*x^9 + 168*a^6*b ^2*x^6 + 32*a^7*b*x^3 + 3*a^8)/x^12
Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {b^8\,x^{12}}{12}-\frac {\frac {a^8}{12}+\frac {8\,a^7\,b\,x^3}{9}+\frac {14\,a^6\,b^2\,x^6}{3}+\frac {56\,a^5\,b^3\,x^9}{3}}{x^{12}}+\frac {8\,a\,b^7\,x^9}{9}+\frac {56\,a^3\,b^5\,x^3}{3}+\frac {14\,a^2\,b^6\,x^6}{3}+70\,a^4\,b^4\,\ln \left (x\right ) \] Input:
int((a + b*x^3)^8/x^13,x)
Output:
(b^8*x^12)/12 - (a^8/12 + (8*a^7*b*x^3)/9 + (14*a^6*b^2*x^6)/3 + (56*a^5*b ^3*x^9)/3)/x^12 + (8*a*b^7*x^9)/9 + (56*a^3*b^5*x^3)/3 + (14*a^2*b^6*x^6)/ 3 + 70*a^4*b^4*log(x)
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {2520 \,\mathrm {log}\left (x \right ) a^{4} b^{4} x^{12}-3 a^{8}-32 a^{7} b \,x^{3}-168 a^{6} b^{2} x^{6}-672 a^{5} b^{3} x^{9}+672 a^{3} b^{5} x^{15}+168 a^{2} b^{6} x^{18}+32 a \,b^{7} x^{21}+3 b^{8} x^{24}}{36 x^{12}} \] Input:
int((b*x^3+a)^8/x^13,x)
Output:
(2520*log(x)*a**4*b**4*x**12 - 3*a**8 - 32*a**7*b*x**3 - 168*a**6*b**2*x** 6 - 672*a**5*b**3*x**9 + 672*a**3*b**5*x**15 + 168*a**2*b**6*x**18 + 32*a* b**7*x**21 + 3*b**8*x**24)/(36*x**12)