Integrand size = 13, antiderivative size = 105 \[ \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx=-\frac {a^8}{21 x^{21}}-\frac {4 a^7 b}{9 x^{18}}-\frac {28 a^6 b^2}{15 x^{15}}-\frac {14 a^5 b^3}{3 x^{12}}-\frac {70 a^4 b^4}{9 x^9}-\frac {28 a^3 b^5}{3 x^6}-\frac {28 a^2 b^6}{3 x^3}+\frac {b^8 x^3}{3}+8 a b^7 \log (x) \] Output:
-1/21*a^8/x^21-4/9*a^7*b/x^18-28/15*a^6*b^2/x^15-14/3*a^5*b^3/x^12-70/9*a^ 4*b^4/x^9-28/3*a^3*b^5/x^6-28/3*a^2*b^6/x^3+1/3*b^8*x^3+8*a*b^7*ln(x)
Time = 0.01 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx=-\frac {a^8}{21 x^{21}}-\frac {4 a^7 b}{9 x^{18}}-\frac {28 a^6 b^2}{15 x^{15}}-\frac {14 a^5 b^3}{3 x^{12}}-\frac {70 a^4 b^4}{9 x^9}-\frac {28 a^3 b^5}{3 x^6}-\frac {28 a^2 b^6}{3 x^3}+\frac {b^8 x^3}{3}+8 a b^7 \log (x) \] Input:
Integrate[(a + b*x^3)^8/x^22,x]
Output:
-1/21*a^8/x^21 - (4*a^7*b)/(9*x^18) - (28*a^6*b^2)/(15*x^15) - (14*a^5*b^3 )/(3*x^12) - (70*a^4*b^4)/(9*x^9) - (28*a^3*b^5)/(3*x^6) - (28*a^2*b^6)/(3 *x^3) + (b^8*x^3)/3 + 8*a*b^7*Log[x]
Time = 0.37 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^8}{x^{24}}dx^3\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \int \left (\frac {a^8}{x^{24}}+\frac {8 b a^7}{x^{21}}+\frac {28 b^2 a^6}{x^{18}}+\frac {56 b^3 a^5}{x^{15}}+\frac {70 b^4 a^4}{x^{12}}+\frac {56 b^5 a^3}{x^9}+\frac {28 b^6 a^2}{x^6}+\frac {8 b^7 a}{x^3}+b^8\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {a^8}{7 x^{21}}-\frac {4 a^7 b}{3 x^{18}}-\frac {28 a^6 b^2}{5 x^{15}}-\frac {14 a^5 b^3}{x^{12}}-\frac {70 a^4 b^4}{3 x^9}-\frac {28 a^3 b^5}{x^6}-\frac {28 a^2 b^6}{x^3}+8 a b^7 \log \left (x^3\right )+b^8 x^3\right )\) |
Input:
Int[(a + b*x^3)^8/x^22,x]
Output:
(-1/7*a^8/x^21 - (4*a^7*b)/(3*x^18) - (28*a^6*b^2)/(5*x^15) - (14*a^5*b^3) /x^12 - (70*a^4*b^4)/(3*x^9) - (28*a^3*b^5)/x^6 - (28*a^2*b^6)/x^3 + b^8*x ^3 + 8*a*b^7*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 | \(-\frac {a^{8}}{21 x^{21}}-\frac {4 a^{7} b}{9 x^{18}}-\frac {28 a^{6} b^{2}}{15 x^{15}}-\frac {14 a^{5} b^{3}}{3 x^{12}}-\frac {70 a^{4} b^{4}}{9 x^{9}}-\frac {28 a^{3} b^{5}}{3 x^{6}}-\frac {28 a^{2} b^{6}}{3 x^{3}}+\frac {b^{8} x^{3}}{3}+8 a \,b^{7} \ln \left (x \right )\) | \(90\) |
norman | \(\frac {-\frac {1}{21} a^{8}+\frac {1}{3} b^{8} x^{24}-\frac {28}{3} a^{2} b^{6} x^{18}-\frac {28}{3} a^{3} b^{5} x^{15}-\frac {70}{9} a^{4} b^{4} x^{12}-\frac {14}{3} a^{5} b^{3} x^{9}-\frac {28}{15} a^{6} b^{2} x^{6}-\frac {4}{9} a^{7} b \,x^{3}}{x^{21}}+8 a \,b^{7} \ln \left (x \right )\) | \(92\) |
risch | \(\frac {b^{8} x^{3}}{3}+\frac {-\frac {1}{21} a^{8}-\frac {4}{9} a^{7} b \,x^{3}-\frac {28}{15} a^{6} b^{2} x^{6}-\frac {14}{3} a^{5} b^{3} x^{9}-\frac {70}{9} a^{4} b^{4} x^{12}-\frac {28}{3} a^{3} b^{5} x^{15}-\frac {28}{3} a^{2} b^{6} x^{18}}{x^{21}}+8 a \,b^{7} \ln \left (x \right )\) | \(92\) |
parallelrisch | \(\frac {105 b^{8} x^{24}+2520 a \,b^{7} \ln \left (x \right ) x^{21}-2940 a^{2} b^{6} x^{18}-2940 a^{3} b^{5} x^{15}-2450 a^{4} b^{4} x^{12}-1470 a^{5} b^{3} x^{9}-588 a^{6} b^{2} x^{6}-140 a^{7} b \,x^{3}-15 a^{8}}{315 x^{21}}\) | \(95\) |
Input:
int((b*x^3+a)^8/x^22,x,method=_RETURNVERBOSE)
Output:
-1/21*a^8/x^21-4/9*a^7*b/x^18-28/15*a^6*b^2/x^15-14/3*a^5*b^3/x^12-70/9*a^ 4*b^4/x^9-28/3*a^3*b^5/x^6-28/3*a^2*b^6/x^3+1/3*b^8*x^3+8*a*b^7*ln(x)
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx=\frac {105 \, b^{8} x^{24} + 2520 \, a b^{7} x^{21} \log \left (x\right ) - 2940 \, a^{2} b^{6} x^{18} - 2940 \, a^{3} b^{5} x^{15} - 2450 \, a^{4} b^{4} x^{12} - 1470 \, a^{5} b^{3} x^{9} - 588 \, a^{6} b^{2} x^{6} - 140 \, a^{7} b x^{3} - 15 \, a^{8}}{315 \, x^{21}} \] Input:
integrate((b*x^3+a)^8/x^22,x, algorithm="fricas")
Output:
1/315*(105*b^8*x^24 + 2520*a*b^7*x^21*log(x) - 2940*a^2*b^6*x^18 - 2940*a^ 3*b^5*x^15 - 2450*a^4*b^4*x^12 - 1470*a^5*b^3*x^9 - 588*a^6*b^2*x^6 - 140* a^7*b*x^3 - 15*a^8)/x^21
Time = 0.46 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx=8 a b^{7} \log {\left (x \right )} + \frac {b^{8} x^{3}}{3} + \frac {- 15 a^{8} - 140 a^{7} b x^{3} - 588 a^{6} b^{2} x^{6} - 1470 a^{5} b^{3} x^{9} - 2450 a^{4} b^{4} x^{12} - 2940 a^{3} b^{5} x^{15} - 2940 a^{2} b^{6} x^{18}}{315 x^{21}} \] Input:
integrate((b*x**3+a)**8/x**22,x)
Output:
8*a*b**7*log(x) + b**8*x**3/3 + (-15*a**8 - 140*a**7*b*x**3 - 588*a**6*b** 2*x**6 - 1470*a**5*b**3*x**9 - 2450*a**4*b**4*x**12 - 2940*a**3*b**5*x**15 - 2940*a**2*b**6*x**18)/(315*x**21)
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx=\frac {1}{3} \, b^{8} x^{3} + \frac {8}{3} \, a b^{7} \log \left (x^{3}\right ) - \frac {2940 \, a^{2} b^{6} x^{18} + 2940 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 1470 \, a^{5} b^{3} x^{9} + 588 \, a^{6} b^{2} x^{6} + 140 \, a^{7} b x^{3} + 15 \, a^{8}}{315 \, x^{21}} \] Input:
integrate((b*x^3+a)^8/x^22,x, algorithm="maxima")
Output:
1/3*b^8*x^3 + 8/3*a*b^7*log(x^3) - 1/315*(2940*a^2*b^6*x^18 + 2940*a^3*b^5 *x^15 + 2450*a^4*b^4*x^12 + 1470*a^5*b^3*x^9 + 588*a^6*b^2*x^6 + 140*a^7*b *x^3 + 15*a^8)/x^21
Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx=\frac {1}{3} \, b^{8} x^{3} + 8 \, a b^{7} \log \left ({\left | x \right |}\right ) - \frac {2178 \, a b^{7} x^{21} + 2940 \, a^{2} b^{6} x^{18} + 2940 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 1470 \, a^{5} b^{3} x^{9} + 588 \, a^{6} b^{2} x^{6} + 140 \, a^{7} b x^{3} + 15 \, a^{8}}{315 \, x^{21}} \] Input:
integrate((b*x^3+a)^8/x^22,x, algorithm="giac")
Output:
1/3*b^8*x^3 + 8*a*b^7*log(abs(x)) - 1/315*(2178*a*b^7*x^21 + 2940*a^2*b^6* x^18 + 2940*a^3*b^5*x^15 + 2450*a^4*b^4*x^12 + 1470*a^5*b^3*x^9 + 588*a^6* b^2*x^6 + 140*a^7*b*x^3 + 15*a^8)/x^21
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx=-\frac {15\,a^8-105\,b^8\,x^{24}+140\,a^7\,b\,x^3+588\,a^6\,b^2\,x^6+1470\,a^5\,b^3\,x^9+2450\,a^4\,b^4\,x^{12}+2940\,a^3\,b^5\,x^{15}+2940\,a^2\,b^6\,x^{18}-2520\,a\,b^7\,x^{21}\,\ln \left (x\right )}{315\,x^{21}} \] Input:
int((a + b*x^3)^8/x^22,x)
Output:
-(15*a^8 - 105*b^8*x^24 + 140*a^7*b*x^3 + 588*a^6*b^2*x^6 + 1470*a^5*b^3*x ^9 + 2450*a^4*b^4*x^12 + 2940*a^3*b^5*x^15 + 2940*a^2*b^6*x^18 - 2520*a*b^ 7*x^21*log(x))/(315*x^21)
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx=\frac {2520 \,\mathrm {log}\left (x \right ) a \,b^{7} x^{21}-15 a^{8}-140 a^{7} b \,x^{3}-588 a^{6} b^{2} x^{6}-1470 a^{5} b^{3} x^{9}-2450 a^{4} b^{4} x^{12}-2940 a^{3} b^{5} x^{15}-2940 a^{2} b^{6} x^{18}+105 b^{8} x^{24}}{315 x^{21}} \] Input:
int((b*x^3+a)^8/x^22,x)
Output:
(2520*log(x)*a*b**7*x**21 - 15*a**8 - 140*a**7*b*x**3 - 588*a**6*b**2*x**6 - 1470*a**5*b**3*x**9 - 2450*a**4*b**4*x**12 - 2940*a**3*b**5*x**15 - 294 0*a**2*b**6*x**18 + 105*b**8*x**24)/(315*x**21)