Integrand size = 13, antiderivative size = 104 \[ \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx=-\frac {a^8}{24 x^{24}}-\frac {8 a^7 b}{21 x^{21}}-\frac {14 a^6 b^2}{9 x^{18}}-\frac {56 a^5 b^3}{15 x^{15}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{3 x^6}-\frac {8 a b^7}{3 x^3}+b^8 \log (x) \] Output:
-1/24*a^8/x^24-8/21*a^7*b/x^21-14/9*a^6*b^2/x^18-56/15*a^5*b^3/x^15-35/6*a ^4*b^4/x^12-56/9*a^3*b^5/x^9-14/3*a^2*b^6/x^6-8/3*a*b^7/x^3+b^8*ln(x)
Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx=-\frac {a^8}{24 x^{24}}-\frac {8 a^7 b}{21 x^{21}}-\frac {14 a^6 b^2}{9 x^{18}}-\frac {56 a^5 b^3}{15 x^{15}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{3 x^6}-\frac {8 a b^7}{3 x^3}+b^8 \log (x) \] Input:
Integrate[(a + b*x^3)^8/x^25,x]
Output:
-1/24*a^8/x^24 - (8*a^7*b)/(21*x^21) - (14*a^6*b^2)/(9*x^18) - (56*a^5*b^3 )/(15*x^15) - (35*a^4*b^4)/(6*x^12) - (56*a^3*b^5)/(9*x^9) - (14*a^2*b^6)/ (3*x^6) - (8*a*b^7)/(3*x^3) + b^8*Log[x]
Time = 0.37 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^8}{x^{27}}dx^3\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} \int \left (\frac {a^8}{x^{27}}+\frac {8 b a^7}{x^{24}}+\frac {28 b^2 a^6}{x^{21}}+\frac {56 b^3 a^5}{x^{18}}+\frac {70 b^4 a^4}{x^{15}}+\frac {56 b^5 a^3}{x^{12}}+\frac {28 b^6 a^2}{x^9}+\frac {8 b^7 a}{x^6}+\frac {b^8}{x^3}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {a^8}{8 x^{24}}-\frac {8 a^7 b}{7 x^{21}}-\frac {14 a^6 b^2}{3 x^{18}}-\frac {56 a^5 b^3}{5 x^{15}}-\frac {35 a^4 b^4}{2 x^{12}}-\frac {56 a^3 b^5}{3 x^9}-\frac {14 a^2 b^6}{x^6}-\frac {8 a b^7}{x^3}+b^8 \log \left (x^3\right )\right )\) |
Input:
Int[(a + b*x^3)^8/x^25,x]
Output:
(-1/8*a^8/x^24 - (8*a^7*b)/(7*x^21) - (14*a^6*b^2)/(3*x^18) - (56*a^5*b^3) /(5*x^15) - (35*a^4*b^4)/(2*x^12) - (56*a^3*b^5)/(3*x^9) - (14*a^2*b^6)/x^ 6 - (8*a*b^7)/x^3 + b^8*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 = 0.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {a^{8}}{24 x^{24}}-\frac {8 a^{7} b}{21 x^{21}}-\frac {14 a^{6} b^{2}}{9 x^{18}}-\frac {56 a^{5} b^{3}}{15 x^{15}}-\frac {35 a^{4} b^{4}}{6 x^{12}}-\frac {56 a^{3} b^{5}}{9 x^{9}}-\frac {14 a^{2} b^{6}}{3 x^{6}}-\frac {8 a \,b^{7}}{3 x^{3}}+b^{8} \ln \left (x \right )\) | \(89\) |
norman | \(\frac {-\frac {1}{24} a^{8}-\frac {8}{3} a \,b^{7} x^{21}-\frac {14}{3} a^{2} b^{6} x^{18}-\frac {56}{9} a^{3} b^{5} x^{15}-\frac {35}{6} a^{4} b^{4} x^{12}-\frac {56}{15} a^{5} b^{3} x^{9}-\frac {14}{9} a^{6} b^{2} x^{6}-\frac {8}{21} a^{7} b \,x^{3}}{x^{24}}+b^{8} \ln \left (x \right )\) | \(91\) |
risch | \(\frac {-\frac {1}{24} a^{8}-\frac {8}{3} a \,b^{7} x^{21}-\frac {14}{3} a^{2} b^{6} x^{18}-\frac {56}{9} a^{3} b^{5} x^{15}-\frac {35}{6} a^{4} b^{4} x^{12}-\frac {56}{15} a^{5} b^{3} x^{9}-\frac {14}{9} a^{6} b^{2} x^{6}-\frac {8}{21} a^{7} b \,x^{3}}{x^{24}}+b^{8} \ln \left (x \right )\) | \(91\) |
parallelrisch | \(\frac {2520 b^{8} \ln \left (x \right ) x^{24}-6720 a \,b^{7} x^{21}-11760 a^{2} b^{6} x^{18}-15680 a^{3} b^{5} x^{15}-14700 a^{4} b^{4} x^{12}-9408 a^{5} b^{3} x^{9}-3920 a^{6} b^{2} x^{6}-960 a^{7} b \,x^{3}-105 a^{8}}{2520 x^{24}}\) | \(95\) |
Input:
int((b*x^3+a)^8/x^25,x,method=_RETURNVERBOSE)
Output:
-1/24*a^8/x^24-8/21*a^7*b/x^21-14/9*a^6*b^2/x^18-56/15*a^5*b^3/x^15-35/6*a ^4*b^4/x^12-56/9*a^3*b^5/x^9-14/3*a^2*b^6/x^6-8/3*a*b^7/x^3+b^8*ln(x)
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx=\frac {2520 \, b^{8} x^{24} \log \left (x\right ) - 6720 \, a b^{7} x^{21} - 11760 \, a^{2} b^{6} x^{18} - 15680 \, a^{3} b^{5} x^{15} - 14700 \, a^{4} b^{4} x^{12} - 9408 \, a^{5} b^{3} x^{9} - 3920 \, a^{6} b^{2} x^{6} - 960 \, a^{7} b x^{3} - 105 \, a^{8}}{2520 \, x^{24}} \] Input:
integrate((b*x^3+a)^8/x^25,x, algorithm="fricas")
Output:
1/2520*(2520*b^8*x^24*log(x) - 6720*a*b^7*x^21 - 11760*a^2*b^6*x^18 - 1568 0*a^3*b^5*x^15 - 14700*a^4*b^4*x^12 - 9408*a^5*b^3*x^9 - 3920*a^6*b^2*x^6 - 960*a^7*b*x^3 - 105*a^8)/x^24
Time = 0.57 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx=b^{8} \log {\left (x \right )} + \frac {- 105 a^{8} - 960 a^{7} b x^{3} - 3920 a^{6} b^{2} x^{6} - 9408 a^{5} b^{3} x^{9} - 14700 a^{4} b^{4} x^{12} - 15680 a^{3} b^{5} x^{15} - 11760 a^{2} b^{6} x^{18} - 6720 a b^{7} x^{21}}{2520 x^{24}} \] Input:
integrate((b*x**3+a)**8/x**25,x)
Output:
b**8*log(x) + (-105*a**8 - 960*a**7*b*x**3 - 3920*a**6*b**2*x**6 - 9408*a* *5*b**3*x**9 - 14700*a**4*b**4*x**12 - 15680*a**3*b**5*x**15 - 11760*a**2* b**6*x**18 - 6720*a*b**7*x**21)/(2520*x**24)
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx=\frac {1}{3} \, b^{8} \log \left (x^{3}\right ) - \frac {6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \] Input:
integrate((b*x^3+a)^8/x^25,x, algorithm="maxima")
Output:
1/3*b^8*log(x^3) - 1/2520*(6720*a*b^7*x^21 + 11760*a^2*b^6*x^18 + 15680*a^ 3*b^5*x^15 + 14700*a^4*b^4*x^12 + 9408*a^5*b^3*x^9 + 3920*a^6*b^2*x^6 + 96 0*a^7*b*x^3 + 105*a^8)/x^24
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx=b^{8} \log \left ({\left | x \right |}\right ) - \frac {2283 \, b^{8} x^{24} + 6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \] Input:
integrate((b*x^3+a)^8/x^25,x, algorithm="giac")
Output:
b^8*log(abs(x)) - 1/2520*(2283*b^8*x^24 + 6720*a*b^7*x^21 + 11760*a^2*b^6* x^18 + 15680*a^3*b^5*x^15 + 14700*a^4*b^4*x^12 + 9408*a^5*b^3*x^9 + 3920*a ^6*b^2*x^6 + 960*a^7*b*x^3 + 105*a^8)/x^24
Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx=b^8\,\ln \left (x\right )-\frac {\frac {a^8}{24}+\frac {8\,a^7\,b\,x^3}{21}+\frac {14\,a^6\,b^2\,x^6}{9}+\frac {56\,a^5\,b^3\,x^9}{15}+\frac {35\,a^4\,b^4\,x^{12}}{6}+\frac {56\,a^3\,b^5\,x^{15}}{9}+\frac {14\,a^2\,b^6\,x^{18}}{3}+\frac {8\,a\,b^7\,x^{21}}{3}}{x^{24}} \] Input:
int((a + b*x^3)^8/x^25,x)
Output:
b^8*log(x) - (a^8/24 + (8*a^7*b*x^3)/21 + (8*a*b^7*x^21)/3 + (14*a^6*b^2*x ^6)/9 + (56*a^5*b^3*x^9)/15 + (35*a^4*b^4*x^12)/6 + (56*a^3*b^5*x^15)/9 + (14*a^2*b^6*x^18)/3)/x^24
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx=\frac {2520 \,\mathrm {log}\left (x \right ) b^{8} x^{24}-105 a^{8}-960 a^{7} b \,x^{3}-3920 a^{6} b^{2} x^{6}-9408 a^{5} b^{3} x^{9}-14700 a^{4} b^{4} x^{12}-15680 a^{3} b^{5} x^{15}-11760 a^{2} b^{6} x^{18}-6720 a \,b^{7} x^{21}}{2520 x^{24}} \] Input:
int((b*x^3+a)^8/x^25,x)
Output:
(2520*log(x)*b**8*x**24 - 105*a**8 - 960*a**7*b*x**3 - 3920*a**6*b**2*x**6 - 9408*a**5*b**3*x**9 - 14700*a**4*b**4*x**12 - 15680*a**3*b**5*x**15 - 1 1760*a**2*b**6*x**18 - 6720*a*b**7*x**21)/(2520*x**24)