Integrand size = 13, antiderivative size = 84 \[ \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac {b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}} \]
-1/36*(b*x^3+a)^9/a/x^36+1/132*b*(b*x^3+a)^9/a^2/x^33-1/660*b^2*(b*x^3+a)^ 9/a^3/x^30+1/5940*b^3*(b*x^3+a)^9/a^4/x^27
Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx=-\frac {a^8}{36 x^{36}}-\frac {8 a^7 b}{33 x^{33}}-\frac {14 a^6 b^2}{15 x^{30}}-\frac {56 a^5 b^3}{27 x^{27}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {8 a^3 b^5}{3 x^{21}}-\frac {14 a^2 b^6}{9 x^{18}}-\frac {8 a b^7}{15 x^{15}}-\frac {b^8}{12 x^{12}} \]
-1/36*a^8/x^36 - (8*a^7*b)/(33*x^33) - (14*a^6*b^2)/(15*x^30) - (56*a^5*b^ 3)/(27*x^27) - (35*a^4*b^4)/(12*x^24) - (8*a^3*b^5)/(3*x^21) - (14*a^2*b^6 )/(9*x^18) - (8*a*b^7)/(15*x^15) - b^8/(12*x^12)
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {798, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^8}{x^{39}}dx^3\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (-\frac {b \int \frac {\left (b x^3+a\right )^8}{x^{36}}dx^3}{4 a}-\frac {\left (a+b x^3\right )^9}{12 a x^{36}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (-\frac {b \left (-\frac {2 b \int \frac {\left (b x^3+a\right )^8}{x^{33}}dx^3}{11 a}-\frac {\left (a+b x^3\right )^9}{11 a x^{33}}\right )}{4 a}-\frac {\left (a+b x^3\right )^9}{12 a x^{36}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (-\frac {b \left (-\frac {2 b \left (-\frac {b \int \frac {\left (b x^3+a\right )^8}{x^{30}}dx^3}{10 a}-\frac {\left (a+b x^3\right )^9}{10 a x^{30}}\right )}{11 a}-\frac {\left (a+b x^3\right )^9}{11 a x^{33}}\right )}{4 a}-\frac {\left (a+b x^3\right )^9}{12 a x^{36}}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{3} \left (-\frac {b \left (-\frac {2 b \left (\frac {b \left (a+b x^3\right )^9}{90 a^2 x^{27}}-\frac {\left (a+b x^3\right )^9}{10 a x^{30}}\right )}{11 a}-\frac {\left (a+b x^3\right )^9}{11 a x^{33}}\right )}{4 a}-\frac {\left (a+b x^3\right )^9}{12 a x^{36}}\right )\) |
(-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))/3
3.4.4.3.1 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 = 3.57 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\frac {b^{8}}{12 x^{12}}-\frac {14 a^{6} b^{2}}{15 x^{30}}-\frac {8 a^{3} b^{5}}{3 x^{21}}-\frac {8 a \,b^{7}}{15 x^{15}}-\frac {14 a^{2} b^{6}}{9 x^{18}}-\frac {a^{8}}{36 x^{36}}-\frac {35 a^{4} b^{4}}{12 x^{24}}-\frac {8 a^{7} b}{33 x^{33}}-\frac {56 a^{5} b^{3}}{27 x^{27}}\) | \(91\) |
norman | \(\frac {-\frac {56}{27} x^{9} b^{3} a^{5}-\frac {1}{36} a^{8}-\frac {8}{33} x^{3} b \,a^{7}-\frac {14}{15} a^{6} b^{2} x^{6}-\frac {35}{12} a^{4} b^{4} x^{12}-\frac {8}{3} a^{3} b^{5} x^{15}-\frac {1}{12} b^{8} x^{24}-\frac {14}{9} a^{2} b^{6} x^{18}-\frac {8}{15} a \,b^{7} x^{21}}{x^{36}}\) | \(92\) |
risch | \(\frac {-\frac {56}{27} x^{9} b^{3} a^{5}-\frac {1}{36} a^{8}-\frac {8}{33} x^{3} b \,a^{7}-\frac {14}{15} a^{6} b^{2} x^{6}-\frac {35}{12} a^{4} b^{4} x^{12}-\frac {8}{3} a^{3} b^{5} x^{15}-\frac {1}{12} b^{8} x^{24}-\frac {14}{9} a^{2} b^{6} x^{18}-\frac {8}{15} a \,b^{7} x^{21}}{x^{36}}\) | \(92\) |
gosper | \(-\frac {495 b^{8} x^{24}+3168 a \,b^{7} x^{21}+9240 a^{2} b^{6} x^{18}+15840 a^{3} b^{5} x^{15}+17325 a^{4} b^{4} x^{12}+12320 x^{9} b^{3} a^{5}+5544 a^{6} b^{2} x^{6}+1440 x^{3} b \,a^{7}+165 a^{8}}{5940 x^{36}}\) | \(93\) |
parallelrisch | \(\frac {-495 b^{8} x^{24}-3168 a \,b^{7} x^{21}-9240 a^{2} b^{6} x^{18}-15840 a^{3} b^{5} x^{15}-17325 a^{4} b^{4} x^{12}-12320 x^{9} b^{3} a^{5}-5544 a^{6} b^{2} x^{6}-1440 x^{3} b \,a^{7}-165 a^{8}}{5940 x^{36}}\) | \(93\) |
-1/12*b^8/x^12-14/15*a^6*b^2/x^30-8/3*a^3*b^5/x^21-8/15*a*b^7/x^15-14/9*a^ 2*b^6/x^18-1/36*a^8/x^36-35/12*a^4*b^4/x^24-8/33*a^7*b/x^33-56/27*a^5*b^3/ x^27
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx=-\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]
-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^ 5*x^15 + 17325*a^4*b^4*x^12 + 12320*a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440* a^7*b*x^3 + 165*a^8)/x^36
Time = 0.66 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx=\frac {- 165 a^{8} - 1440 a^{7} b x^{3} - 5544 a^{6} b^{2} x^{6} - 12320 a^{5} b^{3} x^{9} - 17325 a^{4} b^{4} x^{12} - 15840 a^{3} b^{5} x^{15} - 9240 a^{2} b^{6} x^{18} - 3168 a b^{7} x^{21} - 495 b^{8} x^{24}}{5940 x^{36}} \]
(-165*a**8 - 1440*a**7*b*x**3 - 5544*a**6*b**2*x**6 - 12320*a**5*b**3*x**9 - 17325*a**4*b**4*x**12 - 15840*a**3*b**5*x**15 - 9240*a**2*b**6*x**18 - 3168*a*b**7*x**21 - 495*b**8*x**24)/(5940*x**36)
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx=-\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]
-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^ 5*x^15 + 17325*a^4*b^4*x^12 + 12320*a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440* a^7*b*x^3 + 165*a^8)/x^36
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx=-\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]
-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^ 5*x^15 + 17325*a^4*b^4*x^12 + 12320*a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440* a^7*b*x^3 + 165*a^8)/x^36
Time = 5.47 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx=-\frac {\frac {a^8}{36}+\frac {8\,a^7\,b\,x^3}{33}+\frac {14\,a^6\,b^2\,x^6}{15}+\frac {56\,a^5\,b^3\,x^9}{27}+\frac {35\,a^4\,b^4\,x^{12}}{12}+\frac {8\,a^3\,b^5\,x^{15}}{3}+\frac {14\,a^2\,b^6\,x^{18}}{9}+\frac {8\,a\,b^7\,x^{21}}{15}+\frac {b^8\,x^{24}}{12}}{x^{36}} \]