Integrand size = 13, antiderivative size = 106 \[ \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx=-\frac {a^8}{32 x^{32}}-\frac {4 a^7 b}{15 x^{30}}-\frac {a^6 b^2}{x^{28}}-\frac {28 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {28 a^3 b^5}{11 x^{22}}-\frac {7 a^2 b^6}{5 x^{20}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{16 x^{16}} \] Output:
-1/32*a^8/x^32-4/15*a^7*b/x^30-a^6*b^2/x^28-28/13*a^5*b^3/x^26-35/12*a^4*b ^4/x^24-28/11*a^3*b^5/x^22-7/5*a^2*b^6/x^20-4/9*a*b^7/x^18-1/16*b^8/x^16
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx=-\frac {a^8}{32 x^{32}}-\frac {4 a^7 b}{15 x^{30}}-\frac {a^6 b^2}{x^{28}}-\frac {28 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {28 a^3 b^5}{11 x^{22}}-\frac {7 a^2 b^6}{5 x^{20}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{16 x^{16}} \] Input:
Integrate[(a + b*x^2)^8/x^33,x]
Output:
-1/32*a^8/x^32 - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x ^26) - (35*a^4*b^4)/(12*x^24) - (28*a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^ 20) - (4*a*b^7)/(9*x^18) - b^8/(16*x^16)
Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {243, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^8}{x^{34}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^8}{x^{34}}+\frac {8 b a^7}{x^{32}}+\frac {28 b^2 a^6}{x^{30}}+\frac {56 b^3 a^5}{x^{28}}+\frac {70 b^4 a^4}{x^{26}}+\frac {56 b^5 a^3}{x^{24}}+\frac {28 b^6 a^2}{x^{22}}+\frac {8 b^7 a}{x^{20}}+\frac {b^8}{x^{18}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^8}{16 x^{32}}-\frac {8 a^7 b}{15 x^{30}}-\frac {2 a^6 b^2}{x^{28}}-\frac {56 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{6 x^{24}}-\frac {56 a^3 b^5}{11 x^{22}}-\frac {14 a^2 b^6}{5 x^{20}}-\frac {8 a b^7}{9 x^{18}}-\frac {b^8}{8 x^{16}}\right )\) |
Input:
Int[(a + b*x^2)^8/x^33,x]
Output:
(-1/16*a^8/x^32 - (8*a^7*b)/(15*x^30) - (2*a^6*b^2)/x^28 - (56*a^5*b^3)/(1 3*x^26) - (35*a^4*b^4)/(6*x^24) - (56*a^3*b^5)/(11*x^22) - (14*a^2*b^6)/(5 *x^20) - (8*a*b^7)/(9*x^18) - b^8/(8*x^16))/2
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_)^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 = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {a^{8}}{32 x^{32}}-\frac {4 a^{7} b}{15 x^{30}}-\frac {a^{6} b^{2}}{x^{28}}-\frac {28 a^{5} b^{3}}{13 x^{26}}-\frac {35 a^{4} b^{4}}{12 x^{24}}-\frac {28 a^{3} b^{5}}{11 x^{22}}-\frac {7 a^{2} b^{6}}{5 x^{20}}-\frac {4 a \,b^{7}}{9 x^{18}}-\frac {b^{8}}{16 x^{16}}\) | \(91\) |
norman | \(\frac {-\frac {1}{32} a^{8}-a^{6} b^{2} x^{4}-\frac {35}{12} a^{4} b^{4} x^{8}-\frac {4}{9} a \,b^{7} x^{14}-\frac {7}{5} a^{2} b^{6} x^{12}-\frac {4}{15} a^{7} b \,x^{2}-\frac {28}{11} a^{3} b^{5} x^{10}-\frac {1}{16} b^{8} x^{16}-\frac {28}{13} a^{5} b^{3} x^{6}}{x^{32}}\) | \(92\) |
risch | \(\frac {-\frac {1}{32} a^{8}-a^{6} b^{2} x^{4}-\frac {35}{12} a^{4} b^{4} x^{8}-\frac {4}{9} a \,b^{7} x^{14}-\frac {7}{5} a^{2} b^{6} x^{12}-\frac {4}{15} a^{7} b \,x^{2}-\frac {28}{11} a^{3} b^{5} x^{10}-\frac {1}{16} b^{8} x^{16}-\frac {28}{13} a^{5} b^{3} x^{6}}{x^{32}}\) | \(92\) |
gosper | \(-\frac {12870 b^{8} x^{16}+91520 a \,b^{7} x^{14}+288288 a^{2} b^{6} x^{12}+524160 a^{3} b^{5} x^{10}+600600 a^{4} b^{4} x^{8}+443520 a^{5} b^{3} x^{6}+205920 a^{6} b^{2} x^{4}+54912 a^{7} b \,x^{2}+6435 a^{8}}{205920 x^{32}}\) | \(93\) |
parallelrisch | \(\frac {-12870 b^{8} x^{16}-91520 a \,b^{7} x^{14}-288288 a^{2} b^{6} x^{12}-524160 a^{3} b^{5} x^{10}-600600 a^{4} b^{4} x^{8}-443520 a^{5} b^{3} x^{6}-205920 a^{6} b^{2} x^{4}-54912 a^{7} b \,x^{2}-6435 a^{8}}{205920 x^{32}}\) | \(93\) |
orering | \(-\frac {12870 b^{8} x^{16}+91520 a \,b^{7} x^{14}+288288 a^{2} b^{6} x^{12}+524160 a^{3} b^{5} x^{10}+600600 a^{4} b^{4} x^{8}+443520 a^{5} b^{3} x^{6}+205920 a^{6} b^{2} x^{4}+54912 a^{7} b \,x^{2}+6435 a^{8}}{205920 x^{32}}\) | \(93\) |
Input:
int((b*x^2+a)^8/x^33,x,method=_RETURNVERBOSE)
Output:
-1/32*a^8/x^32-4/15*a^7*b/x^30-a^6*b^2/x^28-28/13*a^5*b^3/x^26-35/12*a^4*b ^4/x^24-28/11*a^3*b^5/x^22-7/5*a^2*b^6/x^20-4/9*a*b^7/x^18-1/16*b^8/x^16
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx=-\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \] Input:
integrate((b*x^2+a)^8/x^33,x, algorithm="fricas")
Output:
-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 52416 0*a^3*b^5*x^10 + 600600*a^4*b^4*x^8 + 443520*a^5*b^3*x^6 + 205920*a^6*b^2* x^4 + 54912*a^7*b*x^2 + 6435*a^8)/x^32
Time = 0.69 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx=\frac {- 6435 a^{8} - 54912 a^{7} b x^{2} - 205920 a^{6} b^{2} x^{4} - 443520 a^{5} b^{3} x^{6} - 600600 a^{4} b^{4} x^{8} - 524160 a^{3} b^{5} x^{10} - 288288 a^{2} b^{6} x^{12} - 91520 a b^{7} x^{14} - 12870 b^{8} x^{16}}{205920 x^{32}} \] Input:
integrate((b*x**2+a)**8/x**33,x)
Output:
(-6435*a**8 - 54912*a**7*b*x**2 - 205920*a**6*b**2*x**4 - 443520*a**5*b**3 *x**6 - 600600*a**4*b**4*x**8 - 524160*a**3*b**5*x**10 - 288288*a**2*b**6* x**12 - 91520*a*b**7*x**14 - 12870*b**8*x**16)/(205920*x**32)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx=-\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \] Input:
integrate((b*x^2+a)^8/x^33,x, algorithm="maxima")
Output:
-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 52416 0*a^3*b^5*x^10 + 600600*a^4*b^4*x^8 + 443520*a^5*b^3*x^6 + 205920*a^6*b^2* x^4 + 54912*a^7*b*x^2 + 6435*a^8)/x^32
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx=-\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \] Input:
integrate((b*x^2+a)^8/x^33,x, algorithm="giac")
Output:
-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 52416 0*a^3*b^5*x^10 + 600600*a^4*b^4*x^8 + 443520*a^5*b^3*x^6 + 205920*a^6*b^2* x^4 + 54912*a^7*b*x^2 + 6435*a^8)/x^32
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx=-\frac {\frac {a^8}{32}+\frac {4\,a^7\,b\,x^2}{15}+a^6\,b^2\,x^4+\frac {28\,a^5\,b^3\,x^6}{13}+\frac {35\,a^4\,b^4\,x^8}{12}+\frac {28\,a^3\,b^5\,x^{10}}{11}+\frac {7\,a^2\,b^6\,x^{12}}{5}+\frac {4\,a\,b^7\,x^{14}}{9}+\frac {b^8\,x^{16}}{16}}{x^{32}} \] Input:
int((a + b*x^2)^8/x^33,x)
Output:
-(a^8/32 + (b^8*x^16)/16 + (4*a^7*b*x^2)/15 + (4*a*b^7*x^14)/9 + a^6*b^2*x ^4 + (28*a^5*b^3*x^6)/13 + (35*a^4*b^4*x^8)/12 + (28*a^3*b^5*x^10)/11 + (7 *a^2*b^6*x^12)/5)/x^32
Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^8}{x^{33}} \, dx=\frac {-12870 b^{8} x^{16}-91520 a \,b^{7} x^{14}-288288 a^{2} b^{6} x^{12}-524160 a^{3} b^{5} x^{10}-600600 a^{4} b^{4} x^{8}-443520 a^{5} b^{3} x^{6}-205920 a^{6} b^{2} x^{4}-54912 a^{7} b \,x^{2}-6435 a^{8}}{205920 x^{32}} \] Input:
int((b*x^2+a)^8/x^33,x)
Output:
( - 6435*a**8 - 54912*a**7*b*x**2 - 205920*a**6*b**2*x**4 - 443520*a**5*b* *3*x**6 - 600600*a**4*b**4*x**8 - 524160*a**3*b**5*x**10 - 288288*a**2*b** 6*x**12 - 91520*a*b**7*x**14 - 12870*b**8*x**16)/(205920*x**32)