Integrand size = 13, antiderivative size = 84 \[ \int \frac {\left (a+b x^2\right )^8}{x^{25}} \, dx=-\frac {\left (a+b x^2\right )^9}{24 a x^{24}}+\frac {b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac {b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}+\frac {b^3 \left (a+b x^2\right )^9}{3960 a^4 x^{18}} \]
-1/24*(b*x^2+a)^9/a/x^24+1/88*b*(b*x^2+a)^9/a^2/x^22-1/440*b^2*(b*x^2+a)^9 /a^3/x^20+1/3960*b^3*(b*x^2+a)^9/a^4/x^18
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b x^2\right )^8}{x^{25}} \, dx=-\frac {a^8}{24 x^{24}}-\frac {4 a^7 b}{11 x^{22}}-\frac {7 a^6 b^2}{5 x^{20}}-\frac {28 a^5 b^3}{9 x^{18}}-\frac {35 a^4 b^4}{8 x^{16}}-\frac {4 a^3 b^5}{x^{14}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {4 a b^7}{5 x^{10}}-\frac {b^8}{8 x^8} \]
-1/24*a^8/x^24 - (4*a^7*b)/(11*x^22) - (7*a^6*b^2)/(5*x^20) - (28*a^5*b^3) /(9*x^18) - (35*a^4*b^4)/(8*x^16) - (4*a^3*b^5)/x^14 - (7*a^2*b^6)/(3*x^12 ) - (4*a*b^7)/(5*x^10) - b^8/(8*x^8)
Time = 0.18 (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 = {243, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^8}{x^{25}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^8}{x^{26}}dx^2\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {b \int \frac {\left (b x^2+a\right )^8}{x^{24}}dx^2}{4 a}-\frac {\left (a+b x^2\right )^9}{12 a x^{24}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^8}{x^{22}}dx^2}{11 a}-\frac {\left (a+b x^2\right )^9}{11 a x^{22}}\right )}{4 a}-\frac {\left (a+b x^2\right )^9}{12 a x^{24}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {b \left (-\frac {2 b \left (-\frac {b \int \frac {\left (b x^2+a\right )^8}{x^{20}}dx^2}{10 a}-\frac {\left (a+b x^2\right )^9}{10 a x^{20}}\right )}{11 a}-\frac {\left (a+b x^2\right )^9}{11 a x^{22}}\right )}{4 a}-\frac {\left (a+b x^2\right )^9}{12 a x^{24}}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{2} \left (-\frac {b \left (-\frac {2 b \left (\frac {b \left (a+b x^2\right )^9}{90 a^2 x^{18}}-\frac {\left (a+b x^2\right )^9}{10 a x^{20}}\right )}{11 a}-\frac {\left (a+b x^2\right )^9}{11 a x^{22}}\right )}{4 a}-\frac {\left (a+b x^2\right )^9}{12 a x^{24}}\right )\) |
(-1/12*(a + b*x^2)^9/(a*x^24) - (b*(-1/11*(a + b*x^2)^9/(a*x^22) - (2*b*(- 1/10*(a + b*x^2)^9/(a*x^20) + (b*(a + b*x^2)^9)/(90*a^2*x^18)))/(11*a)))/( 4*a))/2
3.2.4.3.1 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_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 1.68 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\frac {35 a^{4} b^{4}}{8 x^{16}}-\frac {4 a^{3} b^{5}}{x^{14}}-\frac {7 a^{6} b^{2}}{5 x^{20}}-\frac {7 a^{2} b^{6}}{3 x^{12}}-\frac {4 a^{7} b}{11 x^{22}}-\frac {b^{8}}{8 x^{8}}-\frac {4 a \,b^{7}}{5 x^{10}}-\frac {28 a^{5} b^{3}}{9 x^{18}}-\frac {a^{8}}{24 x^{24}}\) | \(91\) |
norman | \(\frac {-\frac {1}{24} a^{8}-4 x^{10} b^{5} a^{3}-\frac {7}{3} x^{12} b^{6} a^{2}-\frac {4}{5} x^{14} b^{7} a -\frac {1}{8} b^{8} x^{16}-\frac {35}{8} x^{8} b^{4} a^{4}-\frac {4}{11} a^{7} b \,x^{2}-\frac {7}{5} x^{4} b^{2} a^{6}-\frac {28}{9} x^{6} b^{3} a^{5}}{x^{24}}\) | \(92\) |
risch | \(\frac {-\frac {1}{24} a^{8}-4 x^{10} b^{5} a^{3}-\frac {7}{3} x^{12} b^{6} a^{2}-\frac {4}{5} x^{14} b^{7} a -\frac {1}{8} b^{8} x^{16}-\frac {35}{8} x^{8} b^{4} a^{4}-\frac {4}{11} a^{7} b \,x^{2}-\frac {7}{5} x^{4} b^{2} a^{6}-\frac {28}{9} x^{6} b^{3} a^{5}}{x^{24}}\) | \(92\) |
gosper | \(-\frac {495 b^{8} x^{16}+3168 x^{14} b^{7} a +9240 x^{12} b^{6} a^{2}+15840 x^{10} b^{5} a^{3}+17325 x^{8} b^{4} a^{4}+12320 x^{6} b^{3} a^{5}+5544 x^{4} b^{2} a^{6}+1440 a^{7} b \,x^{2}+165 a^{8}}{3960 x^{24}}\) | \(93\) |
parallelrisch | \(\frac {-495 b^{8} x^{16}-3168 x^{14} b^{7} a -9240 x^{12} b^{6} a^{2}-15840 x^{10} b^{5} a^{3}-17325 x^{8} b^{4} a^{4}-12320 x^{6} b^{3} a^{5}-5544 x^{4} b^{2} a^{6}-1440 a^{7} b \,x^{2}-165 a^{8}}{3960 x^{24}}\) | \(93\) |
-35/8*a^4*b^4/x^16-4*a^3*b^5/x^14-7/5*a^6*b^2/x^20-7/3*a^2*b^6/x^12-4/11*a ^7*b/x^22-1/8*b^8/x^8-4/5*a*b^7/x^10-28/9*a^5*b^3/x^18-1/24*a^8/x^24
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^8}{x^{25}} \, dx=-\frac {495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \]
-1/3960*(495*b^8*x^16 + 3168*a*b^7*x^14 + 9240*a^2*b^6*x^12 + 15840*a^3*b^ 5*x^10 + 17325*a^4*b^4*x^8 + 12320*a^5*b^3*x^6 + 5544*a^6*b^2*x^4 + 1440*a ^7*b*x^2 + 165*a^8)/x^24
Time = 0.55 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^2\right )^8}{x^{25}} \, dx=\frac {- 165 a^{8} - 1440 a^{7} b x^{2} - 5544 a^{6} b^{2} x^{4} - 12320 a^{5} b^{3} x^{6} - 17325 a^{4} b^{4} x^{8} - 15840 a^{3} b^{5} x^{10} - 9240 a^{2} b^{6} x^{12} - 3168 a b^{7} x^{14} - 495 b^{8} x^{16}}{3960 x^{24}} \]
(-165*a**8 - 1440*a**7*b*x**2 - 5544*a**6*b**2*x**4 - 12320*a**5*b**3*x**6 - 17325*a**4*b**4*x**8 - 15840*a**3*b**5*x**10 - 9240*a**2*b**6*x**12 - 3 168*a*b**7*x**14 - 495*b**8*x**16)/(3960*x**24)
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^8}{x^{25}} \, dx=-\frac {495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \]
-1/3960*(495*b^8*x^16 + 3168*a*b^7*x^14 + 9240*a^2*b^6*x^12 + 15840*a^3*b^ 5*x^10 + 17325*a^4*b^4*x^8 + 12320*a^5*b^3*x^6 + 5544*a^6*b^2*x^4 + 1440*a ^7*b*x^2 + 165*a^8)/x^24
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^8}{x^{25}} \, dx=-\frac {495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \]
-1/3960*(495*b^8*x^16 + 3168*a*b^7*x^14 + 9240*a^2*b^6*x^12 + 15840*a^3*b^ 5*x^10 + 17325*a^4*b^4*x^8 + 12320*a^5*b^3*x^6 + 5544*a^6*b^2*x^4 + 1440*a ^7*b*x^2 + 165*a^8)/x^24
Time = 4.66 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^8}{x^{25}} \, dx=-\frac {\frac {a^8}{24}+\frac {4\,a^7\,b\,x^2}{11}+\frac {7\,a^6\,b^2\,x^4}{5}+\frac {28\,a^5\,b^3\,x^6}{9}+\frac {35\,a^4\,b^4\,x^8}{8}+4\,a^3\,b^5\,x^{10}+\frac {7\,a^2\,b^6\,x^{12}}{3}+\frac {4\,a\,b^7\,x^{14}}{5}+\frac {b^8\,x^{16}}{8}}{x^{24}} \]