Integrand size = 13, antiderivative size = 108 \[ \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx=-\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}} \] Output:
-1/45*a^8/x^45-4/21*a^7*b/x^42-28/39*a^6*b^2/x^39-14/9*a^5*b^3/x^36-70/33* a^4*b^4/x^33-28/15*a^3*b^5/x^30-28/27*a^2*b^6/x^27-1/3*a*b^7/x^24-1/21*b^8 /x^21
Time = 0.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx=-\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}} \] Input:
Integrate[(a + b*x^3)^8/x^46,x]
Output:
-1/45*a^8/x^45 - (4*a^7*b)/(21*x^42) - (28*a^6*b^2)/(39*x^39) - (14*a^5*b^ 3)/(9*x^36) - (70*a^4*b^4)/(33*x^33) - (28*a^3*b^5)/(15*x^30) - (28*a^2*b^ 6)/(27*x^27) - (a*b^7)/(3*x^24) - b^8/(21*x^21)
Time = 0.37 (sec) , antiderivative size = 110, 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, 53, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^8}{x^{48}}dx^3\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{3} \int \left (\frac {a^8}{x^{48}}+\frac {8 b a^7}{x^{45}}+\frac {28 b^2 a^6}{x^{42}}+\frac {56 b^3 a^5}{x^{39}}+\frac {70 b^4 a^4}{x^{36}}+\frac {56 b^5 a^3}{x^{33}}+\frac {28 b^6 a^2}{x^{30}}+\frac {8 b^7 a}{x^{27}}+\frac {b^8}{x^{24}}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {a^8}{15 x^{45}}-\frac {4 a^7 b}{7 x^{42}}-\frac {28 a^6 b^2}{13 x^{39}}-\frac {14 a^5 b^3}{3 x^{36}}-\frac {70 a^4 b^4}{11 x^{33}}-\frac {28 a^3 b^5}{5 x^{30}}-\frac {28 a^2 b^6}{9 x^{27}}-\frac {a b^7}{x^{24}}-\frac {b^8}{7 x^{21}}\right )\) |
Input:
Int[(a + b*x^3)^8/x^46,x]
Output:
(-1/15*a^8/x^45 - (4*a^7*b)/(7*x^42) - (28*a^6*b^2)/(13*x^39) - (14*a^5*b^ 3)/(3*x^36) - (70*a^4*b^4)/(11*x^33) - (28*a^3*b^5)/(5*x^30) - (28*a^2*b^6 )/(9*x^27) - (a*b^7)/x^24 - b^8/(7*x^21))/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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 = 91, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {a^{8}}{45 x^{45}}-\frac {4 a^{7} b}{21 x^{42}}-\frac {28 a^{6} b^{2}}{39 x^{39}}-\frac {14 a^{5} b^{3}}{9 x^{36}}-\frac {70 a^{4} b^{4}}{33 x^{33}}-\frac {28 a^{3} b^{5}}{15 x^{30}}-\frac {28 a^{2} b^{6}}{27 x^{27}}-\frac {a \,b^{7}}{3 x^{24}}-\frac {b^{8}}{21 x^{21}}\) | \(91\) |
risch | \(\frac {-\frac {1}{45} a^{8}-\frac {28}{15} a^{3} b^{5} x^{15}-\frac {1}{21} b^{8} x^{24}-\frac {4}{21} a^{7} b \,x^{3}-\frac {14}{9} a^{5} b^{3} x^{9}-\frac {28}{39} a^{6} b^{2} x^{6}-\frac {28}{27} a^{2} b^{6} x^{18}-\frac {1}{3} a \,b^{7} x^{21}-\frac {70}{33} a^{4} b^{4} x^{12}}{x^{45}}\) | \(92\) |
gosper | \(-\frac {6435 b^{8} x^{24}+45045 a \,b^{7} x^{21}+140140 a^{2} b^{6} x^{18}+252252 a^{3} b^{5} x^{15}+286650 a^{4} b^{4} x^{12}+210210 a^{5} b^{3} x^{9}+97020 a^{6} b^{2} x^{6}+25740 a^{7} b \,x^{3}+3003 a^{8}}{135135 x^{45}}\) | \(93\) |
parallelrisch | \(\frac {-6435 b^{8} x^{24}-45045 a \,b^{7} x^{21}-140140 a^{2} b^{6} x^{18}-252252 a^{3} b^{5} x^{15}-286650 a^{4} b^{4} x^{12}-210210 a^{5} b^{3} x^{9}-97020 a^{6} b^{2} x^{6}-25740 a^{7} b \,x^{3}-3003 a^{8}}{135135 x^{45}}\) | \(93\) |
orering | \(-\frac {6435 b^{8} x^{24}+45045 a \,b^{7} x^{21}+140140 a^{2} b^{6} x^{18}+252252 a^{3} b^{5} x^{15}+286650 a^{4} b^{4} x^{12}+210210 a^{5} b^{3} x^{9}+97020 a^{6} b^{2} x^{6}+25740 a^{7} b \,x^{3}+3003 a^{8}}{135135 x^{45}}\) | \(93\) |
Input:
int((b*x^3+a)^8/x^46,x,method=_RETURNVERBOSE)
Output:
-1/45*a^8/x^45-4/21*a^7*b/x^42-28/39*a^6*b^2/x^39-14/9*a^5*b^3/x^36-70/33* a^4*b^4/x^33-28/15*a^3*b^5/x^30-28/27*a^2*b^6/x^27-1/3*a*b^7/x^24-1/21*b^8 /x^21
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx=-\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \] Input:
integrate((b*x^3+a)^8/x^46,x, algorithm="fricas")
Output:
-1/135135*(6435*b^8*x^24 + 45045*a*b^7*x^21 + 140140*a^2*b^6*x^18 + 252252 *a^3*b^5*x^15 + 286650*a^4*b^4*x^12 + 210210*a^5*b^3*x^9 + 97020*a^6*b^2*x ^6 + 25740*a^7*b*x^3 + 3003*a^8)/x^45
Time = 0.82 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx=\frac {- 3003 a^{8} - 25740 a^{7} b x^{3} - 97020 a^{6} b^{2} x^{6} - 210210 a^{5} b^{3} x^{9} - 286650 a^{4} b^{4} x^{12} - 252252 a^{3} b^{5} x^{15} - 140140 a^{2} b^{6} x^{18} - 45045 a b^{7} x^{21} - 6435 b^{8} x^{24}}{135135 x^{45}} \] Input:
integrate((b*x**3+a)**8/x**46,x)
Output:
(-3003*a**8 - 25740*a**7*b*x**3 - 97020*a**6*b**2*x**6 - 210210*a**5*b**3* x**9 - 286650*a**4*b**4*x**12 - 252252*a**3*b**5*x**15 - 140140*a**2*b**6* x**18 - 45045*a*b**7*x**21 - 6435*b**8*x**24)/(135135*x**45)
Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx=-\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \] Input:
integrate((b*x^3+a)^8/x^46,x, algorithm="maxima")
Output:
-1/135135*(6435*b^8*x^24 + 45045*a*b^7*x^21 + 140140*a^2*b^6*x^18 + 252252 *a^3*b^5*x^15 + 286650*a^4*b^4*x^12 + 210210*a^5*b^3*x^9 + 97020*a^6*b^2*x ^6 + 25740*a^7*b*x^3 + 3003*a^8)/x^45
Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx=-\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \] Input:
integrate((b*x^3+a)^8/x^46,x, algorithm="giac")
Output:
-1/135135*(6435*b^8*x^24 + 45045*a*b^7*x^21 + 140140*a^2*b^6*x^18 + 252252 *a^3*b^5*x^15 + 286650*a^4*b^4*x^12 + 210210*a^5*b^3*x^9 + 97020*a^6*b^2*x ^6 + 25740*a^7*b*x^3 + 3003*a^8)/x^45
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx=-\frac {\frac {a^8}{45}+\frac {4\,a^7\,b\,x^3}{21}+\frac {28\,a^6\,b^2\,x^6}{39}+\frac {14\,a^5\,b^3\,x^9}{9}+\frac {70\,a^4\,b^4\,x^{12}}{33}+\frac {28\,a^3\,b^5\,x^{15}}{15}+\frac {28\,a^2\,b^6\,x^{18}}{27}+\frac {a\,b^7\,x^{21}}{3}+\frac {b^8\,x^{24}}{21}}{x^{45}} \] Input:
int((a + b*x^3)^8/x^46,x)
Output:
-(a^8/45 + (b^8*x^24)/21 + (4*a^7*b*x^3)/21 + (a*b^7*x^21)/3 + (28*a^6*b^2 *x^6)/39 + (14*a^5*b^3*x^9)/9 + (70*a^4*b^4*x^12)/33 + (28*a^3*b^5*x^15)/1 5 + (28*a^2*b^6*x^18)/27)/x^45
Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x^{46}} \, dx=\frac {-6435 b^{8} x^{24}-45045 a \,b^{7} x^{21}-140140 a^{2} b^{6} x^{18}-252252 a^{3} b^{5} x^{15}-286650 a^{4} b^{4} x^{12}-210210 a^{5} b^{3} x^{9}-97020 a^{6} b^{2} x^{6}-25740 a^{7} b \,x^{3}-3003 a^{8}}{135135 x^{45}} \] Input:
int((b*x^3+a)^8/x^46,x)
Output:
( - 3003*a**8 - 25740*a**7*b*x**3 - 97020*a**6*b**2*x**6 - 210210*a**5*b** 3*x**9 - 286650*a**4*b**4*x**12 - 252252*a**3*b**5*x**15 - 140140*a**2*b** 6*x**18 - 45045*a*b**7*x**21 - 6435*b**8*x**24)/(135135*x**45)