Integrand size = 13, antiderivative size = 105 \[ \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx=-\frac {a^8}{15 x^{15}}-\frac {2 a^7 b}{3 x^{12}}-\frac {28 a^6 b^2}{9 x^9}-\frac {28 a^5 b^3}{3 x^6}-\frac {70 a^4 b^4}{3 x^3}+\frac {28}{3} a^2 b^6 x^3+\frac {4}{3} a b^7 x^6+\frac {b^8 x^9}{9}+56 a^3 b^5 \log (x) \] Output:
-1/15*a^8/x^15-2/3*a^7*b/x^12-28/9*a^6*b^2/x^9-28/3*a^5*b^3/x^6-70/3*a^4*b ^4/x^3+28/3*a^2*b^6*x^3+4/3*a*b^7*x^6+1/9*b^8*x^9+56*a^3*b^5*ln(x)
Time = 0.01 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx=-\frac {a^8}{15 x^{15}}-\frac {2 a^7 b}{3 x^{12}}-\frac {28 a^6 b^2}{9 x^9}-\frac {28 a^5 b^3}{3 x^6}-\frac {70 a^4 b^4}{3 x^3}+\frac {28}{3} a^2 b^6 x^3+\frac {4}{3} a b^7 x^6+\frac {b^8 x^9}{9}+56 a^3 b^5 \log (x) \] Input:
Integrate[(a + b*x^3)^8/x^16,x]
Output:
-1/15*a^8/x^15 - (2*a^7*b)/(3*x^12) - (28*a^6*b^2)/(9*x^9) - (28*a^5*b^3)/ (3*x^6) - (70*a^4*b^4)/(3*x^3) + (28*a^2*b^6*x^3)/3 + (4*a*b^7*x^6)/3 + (b ^8*x^9)/9 + 56*a^3*b^5*Log[x]
Time = 0.37 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^8}{x^{18}}dx^3\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \int \left (\frac {a^8}{x^{18}}+\frac {8 b a^7}{x^{15}}+\frac {28 b^2 a^6}{x^{12}}+\frac {56 b^3 a^5}{x^9}+\frac {70 b^4 a^4}{x^6}+\frac {56 b^5 a^3}{x^3}+28 b^6 a^2+8 b^7 x^3 a+b^8 x^6\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {a^8}{5 x^{15}}-\frac {2 a^7 b}{x^{12}}-\frac {28 a^6 b^2}{3 x^9}-\frac {28 a^5 b^3}{x^6}-\frac {70 a^4 b^4}{x^3}+56 a^3 b^5 \log \left (x^3\right )+28 a^2 b^6 x^3+4 a b^7 x^6+\frac {b^8 x^9}{3}\right )\) |
Input:
Int[(a + b*x^3)^8/x^16,x]
Output:
(-1/5*a^8/x^15 - (2*a^7*b)/x^12 - (28*a^6*b^2)/(3*x^9) - (28*a^5*b^3)/x^6 - (70*a^4*b^4)/x^3 + 28*a^2*b^6*x^3 + 4*a*b^7*x^6 + (b^8*x^9)/3 + 56*a^3*b ^5*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 | \(-\frac {a^{8}}{15 x^{15}}-\frac {2 a^{7} b}{3 x^{12}}-\frac {28 a^{6} b^{2}}{9 x^{9}}-\frac {28 a^{5} b^{3}}{3 x^{6}}-\frac {70 a^{4} b^{4}}{3 x^{3}}+\frac {28 a^{2} b^{6} x^{3}}{3}+\frac {4 a \,b^{7} x^{6}}{3}+\frac {b^{8} x^{9}}{9}+56 a^{3} b^{5} \ln \left (x \right )\) | \(90\) |
norman | \(\frac {-\frac {1}{15} a^{8}+\frac {1}{9} b^{8} x^{24}+\frac {4}{3} a \,b^{7} x^{21}+\frac {28}{3} a^{2} b^{6} x^{18}-\frac {70}{3} a^{4} b^{4} x^{12}-\frac {28}{3} a^{5} b^{3} x^{9}-\frac {28}{9} a^{6} b^{2} x^{6}-\frac {2}{3} a^{7} b \,x^{3}}{x^{15}}+56 a^{3} b^{5} \ln \left (x \right )\) | \(92\) |
risch | \(\frac {b^{8} x^{9}}{9}+\frac {4 a \,b^{7} x^{6}}{3}+\frac {28 a^{2} b^{6} x^{3}}{3}+\frac {-\frac {1}{15} a^{8}-\frac {2}{3} a^{7} b \,x^{3}-\frac {28}{9} a^{6} b^{2} x^{6}-\frac {28}{3} a^{5} b^{3} x^{9}-\frac {70}{3} a^{4} b^{4} x^{12}}{x^{15}}+56 a^{3} b^{5} \ln \left (x \right )\) | \(92\) |
parallelrisch | \(\frac {5 b^{8} x^{24}+60 a \,b^{7} x^{21}+420 a^{2} b^{6} x^{18}+2520 a^{3} b^{5} \ln \left (x \right ) x^{15}-1050 a^{4} b^{4} x^{12}-420 a^{5} b^{3} x^{9}-140 a^{6} b^{2} x^{6}-30 a^{7} b \,x^{3}-3 a^{8}}{45 x^{15}}\) | \(95\) |
Input:
int((b*x^3+a)^8/x^16,x,method=_RETURNVERBOSE)
Output:
-1/15*a^8/x^15-2/3*a^7*b/x^12-28/9*a^6*b^2/x^9-28/3*a^5*b^3/x^6-70/3*a^4*b ^4/x^3+28/3*a^2*b^6*x^3+4/3*a*b^7*x^6+1/9*b^8*x^9+56*a^3*b^5*ln(x)
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx=\frac {5 \, b^{8} x^{24} + 60 \, a b^{7} x^{21} + 420 \, a^{2} b^{6} x^{18} + 2520 \, a^{3} b^{5} x^{15} \log \left (x\right ) - 1050 \, a^{4} b^{4} x^{12} - 420 \, a^{5} b^{3} x^{9} - 140 \, a^{6} b^{2} x^{6} - 30 \, a^{7} b x^{3} - 3 \, a^{8}}{45 \, x^{15}} \] Input:
integrate((b*x^3+a)^8/x^16,x, algorithm="fricas")
Output:
1/45*(5*b^8*x^24 + 60*a*b^7*x^21 + 420*a^2*b^6*x^18 + 2520*a^3*b^5*x^15*lo g(x) - 1050*a^4*b^4*x^12 - 420*a^5*b^3*x^9 - 140*a^6*b^2*x^6 - 30*a^7*b*x^ 3 - 3*a^8)/x^15
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx=56 a^{3} b^{5} \log {\left (x \right )} + \frac {28 a^{2} b^{6} x^{3}}{3} + \frac {4 a b^{7} x^{6}}{3} + \frac {b^{8} x^{9}}{9} + \frac {- 3 a^{8} - 30 a^{7} b x^{3} - 140 a^{6} b^{2} x^{6} - 420 a^{5} b^{3} x^{9} - 1050 a^{4} b^{4} x^{12}}{45 x^{15}} \] Input:
integrate((b*x**3+a)**8/x**16,x)
Output:
56*a**3*b**5*log(x) + 28*a**2*b**6*x**3/3 + 4*a*b**7*x**6/3 + b**8*x**9/9 + (-3*a**8 - 30*a**7*b*x**3 - 140*a**6*b**2*x**6 - 420*a**5*b**3*x**9 - 10 50*a**4*b**4*x**12)/(45*x**15)
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx=\frac {1}{9} \, b^{8} x^{9} + \frac {4}{3} \, a b^{7} x^{6} + \frac {28}{3} \, a^{2} b^{6} x^{3} + \frac {56}{3} \, a^{3} b^{5} \log \left (x^{3}\right ) - \frac {1050 \, a^{4} b^{4} x^{12} + 420 \, a^{5} b^{3} x^{9} + 140 \, a^{6} b^{2} x^{6} + 30 \, a^{7} b x^{3} + 3 \, a^{8}}{45 \, x^{15}} \] Input:
integrate((b*x^3+a)^8/x^16,x, algorithm="maxima")
Output:
1/9*b^8*x^9 + 4/3*a*b^7*x^6 + 28/3*a^2*b^6*x^3 + 56/3*a^3*b^5*log(x^3) - 1 /45*(1050*a^4*b^4*x^12 + 420*a^5*b^3*x^9 + 140*a^6*b^2*x^6 + 30*a^7*b*x^3 + 3*a^8)/x^15
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx=\frac {1}{9} \, b^{8} x^{9} + \frac {4}{3} \, a b^{7} x^{6} + \frac {28}{3} \, a^{2} b^{6} x^{3} + 56 \, a^{3} b^{5} \log \left ({\left | x \right |}\right ) - \frac {1918 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 420 \, a^{5} b^{3} x^{9} + 140 \, a^{6} b^{2} x^{6} + 30 \, a^{7} b x^{3} + 3 \, a^{8}}{45 \, x^{15}} \] Input:
integrate((b*x^3+a)^8/x^16,x, algorithm="giac")
Output:
1/9*b^8*x^9 + 4/3*a*b^7*x^6 + 28/3*a^2*b^6*x^3 + 56*a^3*b^5*log(abs(x)) - 1/45*(1918*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 420*a^5*b^3*x^9 + 140*a^6*b^ 2*x^6 + 30*a^7*b*x^3 + 3*a^8)/x^15
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx=\frac {b^8\,x^9}{9}-\frac {\frac {a^8}{15}+\frac {2\,a^7\,b\,x^3}{3}+\frac {28\,a^6\,b^2\,x^6}{9}+\frac {28\,a^5\,b^3\,x^9}{3}+\frac {70\,a^4\,b^4\,x^{12}}{3}}{x^{15}}+\frac {4\,a\,b^7\,x^6}{3}+\frac {28\,a^2\,b^6\,x^3}{3}+56\,a^3\,b^5\,\ln \left (x\right ) \] Input:
int((a + b*x^3)^8/x^16,x)
Output:
(b^8*x^9)/9 - (a^8/15 + (2*a^7*b*x^3)/3 + (28*a^6*b^2*x^6)/9 + (28*a^5*b^3 *x^9)/3 + (70*a^4*b^4*x^12)/3)/x^15 + (4*a*b^7*x^6)/3 + (28*a^2*b^6*x^3)/3 + 56*a^3*b^5*log(x)
Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx=\frac {2520 \,\mathrm {log}\left (x \right ) a^{3} b^{5} x^{15}-3 a^{8}-30 a^{7} b \,x^{3}-140 a^{6} b^{2} x^{6}-420 a^{5} b^{3} x^{9}-1050 a^{4} b^{4} x^{12}+420 a^{2} b^{6} x^{18}+60 a \,b^{7} x^{21}+5 b^{8} x^{24}}{45 x^{15}} \] Input:
int((b*x^3+a)^8/x^16,x)
Output:
(2520*log(x)*a**3*b**5*x**15 - 3*a**8 - 30*a**7*b*x**3 - 140*a**6*b**2*x** 6 - 420*a**5*b**3*x**9 - 1050*a**4*b**4*x**12 + 420*a**2*b**6*x**18 + 60*a *b**7*x**21 + 5*b**8*x**24)/(45*x**15)