Integrand size = 13, antiderivative size = 105 \[ \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx=-\frac {a^8}{9 x^9}-\frac {4 a^7 b}{3 x^6}-\frac {28 a^6 b^2}{3 x^3}+\frac {70}{3} a^4 b^4 x^3+\frac {28}{3} a^3 b^5 x^6+\frac {28}{9} a^2 b^6 x^9+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{15}}{15}+56 a^5 b^3 \log (x) \] Output:
-1/9*a^8/x^9-4/3*a^7*b/x^6-28/3*a^6*b^2/x^3+70/3*a^4*b^4*x^3+28/3*a^3*b^5* x^6+28/9*a^2*b^6*x^9+2/3*a*b^7*x^12+1/15*b^8*x^15+56*a^5*b^3*ln(x)
Time = 0.01 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx=-\frac {a^8}{9 x^9}-\frac {4 a^7 b}{3 x^6}-\frac {28 a^6 b^2}{3 x^3}+\frac {70}{3} a^4 b^4 x^3+\frac {28}{3} a^3 b^5 x^6+\frac {28}{9} a^2 b^6 x^9+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{15}}{15}+56 a^5 b^3 \log (x) \] Input:
Integrate[(a + b*x^3)^8/x^10,x]
Output:
-1/9*a^8/x^9 - (4*a^7*b)/(3*x^6) - (28*a^6*b^2)/(3*x^3) + (70*a^4*b^4*x^3) /3 + (28*a^3*b^5*x^6)/3 + (28*a^2*b^6*x^9)/9 + (2*a*b^7*x^12)/3 + (b^8*x^1 5)/15 + 56*a^5*b^3*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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^8}{x^{12}}dx^3\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \int \left (b^8 x^{12}+8 a b^7 x^9+28 a^2 b^6 x^6+56 a^3 b^5 x^3+70 a^4 b^4+\frac {56 a^5 b^3}{x^3}+\frac {28 a^6 b^2}{x^6}+\frac {8 a^7 b}{x^9}+\frac {a^8}{x^{12}}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {a^8}{3 x^9}-\frac {4 a^7 b}{x^6}-\frac {28 a^6 b^2}{x^3}+56 a^5 b^3 \log \left (x^3\right )+70 a^4 b^4 x^3+28 a^3 b^5 x^6+\frac {28}{3} a^2 b^6 x^9+2 a b^7 x^{12}+\frac {b^8 x^{15}}{5}\right )\) |
Input:
Int[(a + b*x^3)^8/x^10,x]
Output:
(-1/3*a^8/x^9 - (4*a^7*b)/x^6 - (28*a^6*b^2)/x^3 + 70*a^4*b^4*x^3 + 28*a^3 *b^5*x^6 + (28*a^2*b^6*x^9)/3 + 2*a*b^7*x^12 + (b^8*x^15)/5 + 56*a^5*b^3*L og[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 | \(-\frac {a^{8}}{9 x^{9}}-\frac {4 a^{7} b}{3 x^{6}}-\frac {28 a^{6} b^{2}}{3 x^{3}}+\frac {70 a^{4} b^{4} x^{3}}{3}+\frac {28 a^{3} b^{5} x^{6}}{3}+\frac {28 a^{2} b^{6} x^{9}}{9}+\frac {2 a \,b^{7} x^{12}}{3}+\frac {b^{8} x^{15}}{15}+56 a^{5} b^{3} \ln \left (x \right )\) | \(90\) |
norman | \(\frac {-\frac {1}{9} a^{8}+\frac {1}{15} b^{8} x^{24}+\frac {2}{3} a \,b^{7} x^{21}+\frac {28}{9} a^{2} b^{6} x^{18}+\frac {28}{3} a^{3} b^{5} x^{15}+\frac {70}{3} a^{4} b^{4} x^{12}-\frac {28}{3} a^{6} b^{2} x^{6}-\frac {4}{3} a^{7} b \,x^{3}}{x^{9}}+56 a^{5} b^{3} \ln \left (x \right )\) | \(92\) |
risch | \(\frac {b^{8} x^{15}}{15}+\frac {2 a \,b^{7} x^{12}}{3}+\frac {28 a^{2} b^{6} x^{9}}{9}+\frac {28 a^{3} b^{5} x^{6}}{3}+\frac {70 a^{4} b^{4} x^{3}}{3}+\frac {-\frac {28}{3} a^{6} b^{2} x^{6}-\frac {4}{3} a^{7} b \,x^{3}-\frac {1}{9} a^{8}}{x^{9}}+56 a^{5} b^{3} \ln \left (x \right )\) | \(92\) |
parallelrisch | \(\frac {3 b^{8} x^{24}+30 a \,b^{7} x^{21}+140 a^{2} b^{6} x^{18}+420 a^{3} b^{5} x^{15}+1050 a^{4} b^{4} x^{12}+2520 a^{5} b^{3} \ln \left (x \right ) x^{9}-420 a^{6} b^{2} x^{6}-60 a^{7} b \,x^{3}-5 a^{8}}{45 x^{9}}\) | \(95\) |
Input:
int((b*x^3+a)^8/x^10,x,method=_RETURNVERBOSE)
Output:
-1/9*a^8/x^9-4/3*a^7*b/x^6-28/3*a^6*b^2/x^3+70/3*a^4*b^4*x^3+28/3*a^3*b^5* x^6+28/9*a^2*b^6*x^9+2/3*a*b^7*x^12+1/15*b^8*x^15+56*a^5*b^3*ln(x)
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx=\frac {3 \, b^{8} x^{24} + 30 \, a b^{7} x^{21} + 140 \, a^{2} b^{6} x^{18} + 420 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 2520 \, a^{5} b^{3} x^{9} \log \left (x\right ) - 420 \, a^{6} b^{2} x^{6} - 60 \, a^{7} b x^{3} - 5 \, a^{8}}{45 \, x^{9}} \] Input:
integrate((b*x^3+a)^8/x^10,x, algorithm="fricas")
Output:
1/45*(3*b^8*x^24 + 30*a*b^7*x^21 + 140*a^2*b^6*x^18 + 420*a^3*b^5*x^15 + 1 050*a^4*b^4*x^12 + 2520*a^5*b^3*x^9*log(x) - 420*a^6*b^2*x^6 - 60*a^7*b*x^ 3 - 5*a^8)/x^9
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx=56 a^{5} b^{3} \log {\left (x \right )} + \frac {70 a^{4} b^{4} x^{3}}{3} + \frac {28 a^{3} b^{5} x^{6}}{3} + \frac {28 a^{2} b^{6} x^{9}}{9} + \frac {2 a b^{7} x^{12}}{3} + \frac {b^{8} x^{15}}{15} + \frac {- a^{8} - 12 a^{7} b x^{3} - 84 a^{6} b^{2} x^{6}}{9 x^{9}} \] Input:
integrate((b*x**3+a)**8/x**10,x)
Output:
56*a**5*b**3*log(x) + 70*a**4*b**4*x**3/3 + 28*a**3*b**5*x**6/3 + 28*a**2* b**6*x**9/9 + 2*a*b**7*x**12/3 + b**8*x**15/15 + (-a**8 - 12*a**7*b*x**3 - 84*a**6*b**2*x**6)/(9*x**9)
Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx=\frac {1}{15} \, b^{8} x^{15} + \frac {2}{3} \, a b^{7} x^{12} + \frac {28}{9} \, a^{2} b^{6} x^{9} + \frac {28}{3} \, a^{3} b^{5} x^{6} + \frac {70}{3} \, a^{4} b^{4} x^{3} + \frac {56}{3} \, a^{5} b^{3} \log \left (x^{3}\right ) - \frac {84 \, a^{6} b^{2} x^{6} + 12 \, a^{7} b x^{3} + a^{8}}{9 \, x^{9}} \] Input:
integrate((b*x^3+a)^8/x^10,x, algorithm="maxima")
Output:
1/15*b^8*x^15 + 2/3*a*b^7*x^12 + 28/9*a^2*b^6*x^9 + 28/3*a^3*b^5*x^6 + 70/ 3*a^4*b^4*x^3 + 56/3*a^5*b^3*log(x^3) - 1/9*(84*a^6*b^2*x^6 + 12*a^7*b*x^3 + a^8)/x^9
Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx=\frac {1}{15} \, b^{8} x^{15} + \frac {2}{3} \, a b^{7} x^{12} + \frac {28}{9} \, a^{2} b^{6} x^{9} + \frac {28}{3} \, a^{3} b^{5} x^{6} + \frac {70}{3} \, a^{4} b^{4} x^{3} + 56 \, a^{5} b^{3} \log \left ({\left | x \right |}\right ) - \frac {308 \, a^{5} b^{3} x^{9} + 84 \, a^{6} b^{2} x^{6} + 12 \, a^{7} b x^{3} + a^{8}}{9 \, x^{9}} \] Input:
integrate((b*x^3+a)^8/x^10,x, algorithm="giac")
Output:
1/15*b^8*x^15 + 2/3*a*b^7*x^12 + 28/9*a^2*b^6*x^9 + 28/3*a^3*b^5*x^6 + 70/ 3*a^4*b^4*x^3 + 56*a^5*b^3*log(abs(x)) - 1/9*(308*a^5*b^3*x^9 + 84*a^6*b^2 *x^6 + 12*a^7*b*x^3 + a^8)/x^9
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx=\frac {b^8\,x^{15}}{15}-\frac {\frac {a^8}{9}+\frac {4\,a^7\,b\,x^3}{3}+\frac {28\,a^6\,b^2\,x^6}{3}}{x^9}+\frac {2\,a\,b^7\,x^{12}}{3}+\frac {70\,a^4\,b^4\,x^3}{3}+\frac {28\,a^3\,b^5\,x^6}{3}+\frac {28\,a^2\,b^6\,x^9}{9}+56\,a^5\,b^3\,\ln \left (x\right ) \] Input:
int((a + b*x^3)^8/x^10,x)
Output:
(b^8*x^15)/15 - (a^8/9 + (4*a^7*b*x^3)/3 + (28*a^6*b^2*x^6)/3)/x^9 + (2*a* b^7*x^12)/3 + (70*a^4*b^4*x^3)/3 + (28*a^3*b^5*x^6)/3 + (28*a^2*b^6*x^9)/9 + 56*a^5*b^3*log(x)
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx=\frac {2520 \,\mathrm {log}\left (x \right ) a^{5} b^{3} x^{9}-5 a^{8}-60 a^{7} b \,x^{3}-420 a^{6} b^{2} x^{6}+1050 a^{4} b^{4} x^{12}+420 a^{3} b^{5} x^{15}+140 a^{2} b^{6} x^{18}+30 a \,b^{7} x^{21}+3 b^{8} x^{24}}{45 x^{9}} \] Input:
int((b*x^3+a)^8/x^10,x)
Output:
(2520*log(x)*a**5*b**3*x**9 - 5*a**8 - 60*a**7*b*x**3 - 420*a**6*b**2*x**6 + 1050*a**4*b**4*x**12 + 420*a**3*b**5*x**15 + 140*a**2*b**6*x**18 + 30*a *b**7*x**21 + 3*b**8*x**24)/(45*x**9)