Integrand size = 13, antiderivative size = 106 \[ \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx=-\frac {\left (a+b x^2\right )^9}{26 a x^{26}}+\frac {b \left (a+b x^2\right )^9}{78 a^2 x^{24}}-\frac {b^2 \left (a+b x^2\right )^9}{286 a^3 x^{22}}+\frac {b^3 \left (a+b x^2\right )^9}{1430 a^4 x^{20}}-\frac {b^4 \left (a+b x^2\right )^9}{12870 a^5 x^{18}} \] Output:
-1/26*(b*x^2+a)^9/a/x^26+1/78*b*(b*x^2+a)^9/a^2/x^24-1/286*b^2*(b*x^2+a)^9 /a^3/x^22+1/1430*b^3*(b*x^2+a)^9/a^4/x^20-1/12870*b^4*(b*x^2+a)^9/a^5/x^18
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx=-\frac {a^8}{26 x^{26}}-\frac {a^7 b}{3 x^{24}}-\frac {14 a^6 b^2}{11 x^{22}}-\frac {14 a^5 b^3}{5 x^{20}}-\frac {35 a^4 b^4}{9 x^{18}}-\frac {7 a^3 b^5}{2 x^{16}}-\frac {2 a^2 b^6}{x^{14}}-\frac {2 a b^7}{3 x^{12}}-\frac {b^8}{10 x^{10}} \] Input:
Integrate[(a + b*x^2)^8/x^27,x]
Output:
-1/26*a^8/x^26 - (a^7*b)/(3*x^24) - (14*a^6*b^2)/(11*x^22) - (14*a^5*b^3)/ (5*x^20) - (35*a^4*b^4)/(9*x^18) - (7*a^3*b^5)/(2*x^16) - (2*a^2*b^6)/x^14 - (2*a*b^7)/(3*x^12) - b^8/(10*x^10)
Time = 0.19 (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 = {243, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^8}{x^{28}}dx^2\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^8}{x^{26}}dx^2}{13 a}-\frac {\left (a+b x^2\right )^9}{13 a x^{26}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {4 b \left (-\frac {b \int \frac {\left (b x^2+a\right )^8}{x^{24}}dx^2}{4 a}-\frac {\left (a+b x^2\right )^9}{12 a x^{24}}\right )}{13 a}-\frac {\left (a+b x^2\right )^9}{13 a x^{26}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {4 b \left (-\frac {b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^8}{x^{22}}dx^2}{11 a}-\frac {\left (a+b x^2\right )^9}{11 a x^{22}}\right )}{4 a}-\frac {\left (a+b x^2\right )^9}{12 a x^{24}}\right )}{13 a}-\frac {\left (a+b x^2\right )^9}{13 a x^{26}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {4 b \left (-\frac {b \left (-\frac {2 b \left (-\frac {b \int \frac {\left (b x^2+a\right )^8}{x^{20}}dx^2}{10 a}-\frac {\left (a+b x^2\right )^9}{10 a x^{20}}\right )}{11 a}-\frac {\left (a+b x^2\right )^9}{11 a x^{22}}\right )}{4 a}-\frac {\left (a+b x^2\right )^9}{12 a x^{24}}\right )}{13 a}-\frac {\left (a+b x^2\right )^9}{13 a x^{26}}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{2} \left (-\frac {4 b \left (-\frac {b \left (-\frac {2 b \left (\frac {b \left (a+b x^2\right )^9}{90 a^2 x^{18}}-\frac {\left (a+b x^2\right )^9}{10 a x^{20}}\right )}{11 a}-\frac {\left (a+b x^2\right )^9}{11 a x^{22}}\right )}{4 a}-\frac {\left (a+b x^2\right )^9}{12 a x^{24}}\right )}{13 a}-\frac {\left (a+b x^2\right )^9}{13 a x^{26}}\right )\) |
Input:
Int[(a + b*x^2)^8/x^27,x]
Output:
(-1/13*(a + b*x^2)^9/(a*x^26) - (4*b*(-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)))/(13*a))/2
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 = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {14 a^{6} b^{2}}{11 x^{22}}-\frac {a^{7} b}{3 x^{24}}-\frac {b^{8}}{10 x^{10}}-\frac {a^{8}}{26 x^{26}}-\frac {2 a^{2} b^{6}}{x^{14}}-\frac {14 a^{5} b^{3}}{5 x^{20}}-\frac {35 a^{4} b^{4}}{9 x^{18}}-\frac {7 a^{3} b^{5}}{2 x^{16}}-\frac {2 a \,b^{7}}{3 x^{12}}\) | \(91\) |
norman | \(\frac {-\frac {1}{26} a^{8}-2 a^{2} b^{6} x^{12}-\frac {2}{3} a \,b^{7} x^{14}-\frac {1}{3} a^{7} b \,x^{2}-\frac {1}{10} b^{8} x^{16}-\frac {7}{2} a^{3} b^{5} x^{10}-\frac {35}{9} a^{4} b^{4} x^{8}-\frac {14}{11} a^{6} b^{2} x^{4}-\frac {14}{5} a^{5} b^{3} x^{6}}{x^{26}}\) | \(92\) |
risch | \(\frac {-\frac {1}{26} a^{8}-2 a^{2} b^{6} x^{12}-\frac {2}{3} a \,b^{7} x^{14}-\frac {1}{3} a^{7} b \,x^{2}-\frac {1}{10} b^{8} x^{16}-\frac {7}{2} a^{3} b^{5} x^{10}-\frac {35}{9} a^{4} b^{4} x^{8}-\frac {14}{11} a^{6} b^{2} x^{4}-\frac {14}{5} a^{5} b^{3} x^{6}}{x^{26}}\) | \(92\) |
gosper | \(-\frac {1287 b^{8} x^{16}+8580 a \,b^{7} x^{14}+25740 a^{2} b^{6} x^{12}+45045 a^{3} b^{5} x^{10}+50050 a^{4} b^{4} x^{8}+36036 a^{5} b^{3} x^{6}+16380 a^{6} b^{2} x^{4}+4290 a^{7} b \,x^{2}+495 a^{8}}{12870 x^{26}}\) | \(93\) |
parallelrisch | \(\frac {-1287 b^{8} x^{16}-8580 a \,b^{7} x^{14}-25740 a^{2} b^{6} x^{12}-45045 a^{3} b^{5} x^{10}-50050 a^{4} b^{4} x^{8}-36036 a^{5} b^{3} x^{6}-16380 a^{6} b^{2} x^{4}-4290 a^{7} b \,x^{2}-495 a^{8}}{12870 x^{26}}\) | \(93\) |
orering | \(-\frac {1287 b^{8} x^{16}+8580 a \,b^{7} x^{14}+25740 a^{2} b^{6} x^{12}+45045 a^{3} b^{5} x^{10}+50050 a^{4} b^{4} x^{8}+36036 a^{5} b^{3} x^{6}+16380 a^{6} b^{2} x^{4}+4290 a^{7} b \,x^{2}+495 a^{8}}{12870 x^{26}}\) | \(93\) |
Input:
int((b*x^2+a)^8/x^27,x,method=_RETURNVERBOSE)
Output:
-14/11*a^6*b^2/x^22-1/3*a^7*b/x^24-1/10*b^8/x^10-1/26*a^8/x^26-2*a^2*b^6/x ^14-14/5*a^5*b^3/x^20-35/9*a^4*b^4/x^18-7/2*a^3*b^5/x^16-2/3*a*b^7/x^12
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx=-\frac {1287 \, b^{8} x^{16} + 8580 \, a b^{7} x^{14} + 25740 \, a^{2} b^{6} x^{12} + 45045 \, a^{3} b^{5} x^{10} + 50050 \, a^{4} b^{4} x^{8} + 36036 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 4290 \, a^{7} b x^{2} + 495 \, a^{8}}{12870 \, x^{26}} \] Input:
integrate((b*x^2+a)^8/x^27,x, algorithm="fricas")
Output:
-1/12870*(1287*b^8*x^16 + 8580*a*b^7*x^14 + 25740*a^2*b^6*x^12 + 45045*a^3 *b^5*x^10 + 50050*a^4*b^4*x^8 + 36036*a^5*b^3*x^6 + 16380*a^6*b^2*x^4 + 42 90*a^7*b*x^2 + 495*a^8)/x^26
Time = 0.65 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx=\frac {- 495 a^{8} - 4290 a^{7} b x^{2} - 16380 a^{6} b^{2} x^{4} - 36036 a^{5} b^{3} x^{6} - 50050 a^{4} b^{4} x^{8} - 45045 a^{3} b^{5} x^{10} - 25740 a^{2} b^{6} x^{12} - 8580 a b^{7} x^{14} - 1287 b^{8} x^{16}}{12870 x^{26}} \] Input:
integrate((b*x**2+a)**8/x**27,x)
Output:
(-495*a**8 - 4290*a**7*b*x**2 - 16380*a**6*b**2*x**4 - 36036*a**5*b**3*x** 6 - 50050*a**4*b**4*x**8 - 45045*a**3*b**5*x**10 - 25740*a**2*b**6*x**12 - 8580*a*b**7*x**14 - 1287*b**8*x**16)/(12870*x**26)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx=-\frac {1287 \, b^{8} x^{16} + 8580 \, a b^{7} x^{14} + 25740 \, a^{2} b^{6} x^{12} + 45045 \, a^{3} b^{5} x^{10} + 50050 \, a^{4} b^{4} x^{8} + 36036 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 4290 \, a^{7} b x^{2} + 495 \, a^{8}}{12870 \, x^{26}} \] Input:
integrate((b*x^2+a)^8/x^27,x, algorithm="maxima")
Output:
-1/12870*(1287*b^8*x^16 + 8580*a*b^7*x^14 + 25740*a^2*b^6*x^12 + 45045*a^3 *b^5*x^10 + 50050*a^4*b^4*x^8 + 36036*a^5*b^3*x^6 + 16380*a^6*b^2*x^4 + 42 90*a^7*b*x^2 + 495*a^8)/x^26
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx=-\frac {1287 \, b^{8} x^{16} + 8580 \, a b^{7} x^{14} + 25740 \, a^{2} b^{6} x^{12} + 45045 \, a^{3} b^{5} x^{10} + 50050 \, a^{4} b^{4} x^{8} + 36036 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 4290 \, a^{7} b x^{2} + 495 \, a^{8}}{12870 \, x^{26}} \] Input:
integrate((b*x^2+a)^8/x^27,x, algorithm="giac")
Output:
-1/12870*(1287*b^8*x^16 + 8580*a*b^7*x^14 + 25740*a^2*b^6*x^12 + 45045*a^3 *b^5*x^10 + 50050*a^4*b^4*x^8 + 36036*a^5*b^3*x^6 + 16380*a^6*b^2*x^4 + 42 90*a^7*b*x^2 + 495*a^8)/x^26
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx=-\frac {\frac {a^8}{26}+\frac {a^7\,b\,x^2}{3}+\frac {14\,a^6\,b^2\,x^4}{11}+\frac {14\,a^5\,b^3\,x^6}{5}+\frac {35\,a^4\,b^4\,x^8}{9}+\frac {7\,a^3\,b^5\,x^{10}}{2}+2\,a^2\,b^6\,x^{12}+\frac {2\,a\,b^7\,x^{14}}{3}+\frac {b^8\,x^{16}}{10}}{x^{26}} \] Input:
int((a + b*x^2)^8/x^27,x)
Output:
-(a^8/26 + (b^8*x^16)/10 + (a^7*b*x^2)/3 + (2*a*b^7*x^14)/3 + (14*a^6*b^2* x^4)/11 + (14*a^5*b^3*x^6)/5 + (35*a^4*b^4*x^8)/9 + (7*a^3*b^5*x^10)/2 + 2 *a^2*b^6*x^12)/x^26
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{27}} \, dx=\frac {-1287 b^{8} x^{16}-8580 a \,b^{7} x^{14}-25740 a^{2} b^{6} x^{12}-45045 a^{3} b^{5} x^{10}-50050 a^{4} b^{4} x^{8}-36036 a^{5} b^{3} x^{6}-16380 a^{6} b^{2} x^{4}-4290 a^{7} b \,x^{2}-495 a^{8}}{12870 x^{26}} \] Input:
int((b*x^2+a)^8/x^27,x)
Output:
( - 495*a**8 - 4290*a**7*b*x**2 - 16380*a**6*b**2*x**4 - 36036*a**5*b**3*x **6 - 50050*a**4*b**4*x**8 - 45045*a**3*b**5*x**10 - 25740*a**2*b**6*x**12 - 8580*a*b**7*x**14 - 1287*b**8*x**16)/(12870*x**26)