Integrand size = 13, antiderivative size = 106 \[ \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx=-\frac {\left (a+b x^3\right )^9}{39 a x^{39}}+\frac {b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac {b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac {b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac {b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}} \]
-1/39*(b*x^3+a)^9/a/x^39+1/117*b*(b*x^3+a)^9/a^2/x^36-1/429*b^2*(b*x^3+a)^ 9/a^3/x^33+1/2145*b^3*(b*x^3+a)^9/a^4/x^30-1/19305*b^4*(b*x^3+a)^9/a^5/x^2 7
Time = 0.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx=-\frac {a^8}{39 x^{39}}-\frac {2 a^7 b}{9 x^{36}}-\frac {28 a^6 b^2}{33 x^{33}}-\frac {28 a^5 b^3}{15 x^{30}}-\frac {70 a^4 b^4}{27 x^{27}}-\frac {7 a^3 b^5}{3 x^{24}}-\frac {4 a^2 b^6}{3 x^{21}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{15 x^{15}} \]
-1/39*a^8/x^39 - (2*a^7*b)/(9*x^36) - (28*a^6*b^2)/(33*x^33) - (28*a^5*b^3 )/(15*x^30) - (70*a^4*b^4)/(27*x^27) - (7*a^3*b^5)/(3*x^24) - (4*a^2*b^6)/ (3*x^21) - (4*a*b^7)/(9*x^18) - b^8/(15*x^15)
Time = 0.21 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 55, 55, 55, 55, 48}
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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^8}{x^{42}}dx^3\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (-\frac {4 b \int \frac {\left (b x^3+a\right )^8}{x^{39}}dx^3}{13 a}-\frac {\left (a+b x^3\right )^9}{13 a x^{39}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (-\frac {4 b \left (-\frac {b \int \frac {\left (b x^3+a\right )^8}{x^{36}}dx^3}{4 a}-\frac {\left (a+b x^3\right )^9}{12 a x^{36}}\right )}{13 a}-\frac {\left (a+b x^3\right )^9}{13 a x^{39}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (-\frac {4 b \left (-\frac {b \left (-\frac {2 b \int \frac {\left (b x^3+a\right )^8}{x^{33}}dx^3}{11 a}-\frac {\left (a+b x^3\right )^9}{11 a x^{33}}\right )}{4 a}-\frac {\left (a+b x^3\right )^9}{12 a x^{36}}\right )}{13 a}-\frac {\left (a+b x^3\right )^9}{13 a x^{39}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (-\frac {4 b \left (-\frac {b \left (-\frac {2 b \left (-\frac {b \int \frac {\left (b x^3+a\right )^8}{x^{30}}dx^3}{10 a}-\frac {\left (a+b x^3\right )^9}{10 a x^{30}}\right )}{11 a}-\frac {\left (a+b x^3\right )^9}{11 a x^{33}}\right )}{4 a}-\frac {\left (a+b x^3\right )^9}{12 a x^{36}}\right )}{13 a}-\frac {\left (a+b x^3\right )^9}{13 a x^{39}}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{3} \left (-\frac {4 b \left (-\frac {b \left (-\frac {2 b \left (\frac {b \left (a+b x^3\right )^9}{90 a^2 x^{27}}-\frac {\left (a+b x^3\right )^9}{10 a x^{30}}\right )}{11 a}-\frac {\left (a+b x^3\right )^9}{11 a x^{33}}\right )}{4 a}-\frac {\left (a+b x^3\right )^9}{12 a x^{36}}\right )}{13 a}-\frac {\left (a+b x^3\right )^9}{13 a x^{39}}\right )\) |
(-1/13*(a + b*x^3)^9/(a*x^39) - (4*b*(-1/12*(a + b*x^3)^9/(a*x^36) - (b*(- 1/11*(a + b*x^3)^9/(a*x^33) - (2*b*(-1/10*(a + b*x^3)^9/(a*x^30) + (b*(a + b*x^3)^9)/(90*a^2*x^27)))/(11*a)))/(4*a)))/(13*a))/3
3.4.5.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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 = 3.61 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {28 a^{5} b^{3}}{15 x^{30}}-\frac {a^{8}}{39 x^{39}}-\frac {4 a^{2} b^{6}}{3 x^{21}}-\frac {b^{8}}{15 x^{15}}-\frac {4 a \,b^{7}}{9 x^{18}}-\frac {2 a^{7} b}{9 x^{36}}-\frac {7 a^{3} b^{5}}{3 x^{24}}-\frac {28 a^{6} b^{2}}{33 x^{33}}-\frac {70 a^{4} b^{4}}{27 x^{27}}\) | \(91\) |
risch | \(\frac {-\frac {2}{9} x^{3} b \,a^{7}-\frac {1}{39} a^{8}-\frac {28}{33} a^{6} b^{2} x^{6}-\frac {4}{9} a \,b^{7} x^{21}-\frac {1}{15} b^{8} x^{24}-\frac {28}{15} x^{9} b^{3} a^{5}-\frac {70}{27} a^{4} b^{4} x^{12}-\frac {7}{3} a^{3} b^{5} x^{15}-\frac {4}{3} a^{2} b^{6} x^{18}}{x^{39}}\) | \(92\) |
gosper | \(-\frac {1287 b^{8} x^{24}+8580 a \,b^{7} x^{21}+25740 a^{2} b^{6} x^{18}+45045 a^{3} b^{5} x^{15}+50050 a^{4} b^{4} x^{12}+36036 x^{9} b^{3} a^{5}+16380 a^{6} b^{2} x^{6}+4290 x^{3} b \,a^{7}+495 a^{8}}{19305 x^{39}}\) | \(93\) |
parallelrisch | \(\frac {-1287 b^{8} x^{24}-8580 a \,b^{7} x^{21}-25740 a^{2} b^{6} x^{18}-45045 a^{3} b^{5} x^{15}-50050 a^{4} b^{4} x^{12}-36036 x^{9} b^{3} a^{5}-16380 a^{6} b^{2} x^{6}-4290 x^{3} b \,a^{7}-495 a^{8}}{19305 x^{39}}\) | \(93\) |
-28/15*a^5*b^3/x^30-1/39*a^8/x^39-4/3*a^2*b^6/x^21-1/15*b^8/x^15-4/9*a*b^7 /x^18-2/9*a^7*b/x^36-7/3*a^3*b^5/x^24-28/33*a^6*b^2/x^33-70/27*a^4*b^4/x^2 7
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx=-\frac {1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \]
-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3 *b^5*x^15 + 50050*a^4*b^4*x^12 + 36036*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4 290*a^7*b*x^3 + 495*a^8)/x^39
Time = 0.70 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx=\frac {- 495 a^{8} - 4290 a^{7} b x^{3} - 16380 a^{6} b^{2} x^{6} - 36036 a^{5} b^{3} x^{9} - 50050 a^{4} b^{4} x^{12} - 45045 a^{3} b^{5} x^{15} - 25740 a^{2} b^{6} x^{18} - 8580 a b^{7} x^{21} - 1287 b^{8} x^{24}}{19305 x^{39}} \]
(-495*a**8 - 4290*a**7*b*x**3 - 16380*a**6*b**2*x**6 - 36036*a**5*b**3*x** 9 - 50050*a**4*b**4*x**12 - 45045*a**3*b**5*x**15 - 25740*a**2*b**6*x**18 - 8580*a*b**7*x**21 - 1287*b**8*x**24)/(19305*x**39)
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx=-\frac {1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \]
-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3 *b^5*x^15 + 50050*a^4*b^4*x^12 + 36036*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4 290*a^7*b*x^3 + 495*a^8)/x^39
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx=-\frac {1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \]
-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3 *b^5*x^15 + 50050*a^4*b^4*x^12 + 36036*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4 290*a^7*b*x^3 + 495*a^8)/x^39
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^3\right )^8}{x^{40}} \, dx=-\frac {\frac {a^8}{39}+\frac {2\,a^7\,b\,x^3}{9}+\frac {28\,a^6\,b^2\,x^6}{33}+\frac {28\,a^5\,b^3\,x^9}{15}+\frac {70\,a^4\,b^4\,x^{12}}{27}+\frac {7\,a^3\,b^5\,x^{15}}{3}+\frac {4\,a^2\,b^6\,x^{18}}{3}+\frac {4\,a\,b^7\,x^{21}}{9}+\frac {b^8\,x^{24}}{15}}{x^{39}} \]