Integrand size = 13, antiderivative size = 91 \[ \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx=-\frac {a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac {a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac {3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac {a}{3 b^5 \left (a+b x^2\right )^6}-\frac {1}{10 b^5 \left (a+b x^2\right )^5} \] Output:
-1/18*a^4/b^5/(b*x^2+a)^9+1/4*a^3/b^5/(b*x^2+a)^8-3/7*a^2/b^5/(b*x^2+a)^7+ 1/3*a/b^5/(b*x^2+a)^6-1/10/b^5/(b*x^2+a)^5
Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.63 \[ \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx=-\frac {a^4+9 a^3 b x^2+36 a^2 b^2 x^4+84 a b^3 x^6+126 b^4 x^8}{1260 b^5 \left (a+b x^2\right )^9} \] Input:
Integrate[x^9/(a + b*x^2)^10,x]
Output:
-1/1260*(a^4 + 9*a^3*b*x^2 + 36*a^2*b^2*x^4 + 84*a*b^3*x^6 + 126*b^4*x^8)/ (b^5*(a + b*x^2)^9)
Time = 0.22 (sec) , antiderivative size = 95, 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.
\(\displaystyle \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {x^8}{\left (b x^2+a\right )^{10}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^4}{b^4 \left (b x^2+a\right )^{10}}-\frac {4 a^3}{b^4 \left (b x^2+a\right )^9}+\frac {6 a^2}{b^4 \left (b x^2+a\right )^8}-\frac {4 a}{b^4 \left (b x^2+a\right )^7}+\frac {1}{b^4 \left (b x^2+a\right )^6}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^4}{9 b^5 \left (a+b x^2\right )^9}+\frac {a^3}{2 b^5 \left (a+b x^2\right )^8}-\frac {6 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac {2 a}{3 b^5 \left (a+b x^2\right )^6}-\frac {1}{5 b^5 \left (a+b x^2\right )^5}\right )\) |
Input:
Int[x^9/(a + b*x^2)^10,x]
Output:
(-1/9*a^4/(b^5*(a + b*x^2)^9) + a^3/(2*b^5*(a + b*x^2)^8) - (6*a^2)/(7*b^5 *(a + b*x^2)^7) + (2*a)/(3*b^5*(a + b*x^2)^6) - 1/(5*b^5*(a + b*x^2)^5))/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 = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(-\frac {126 b^{4} x^{8}+84 a \,b^{3} x^{6}+36 a^{2} b^{2} x^{4}+9 a^{3} b \,x^{2}+a^{4}}{1260 \left (b \,x^{2}+a \right )^{9} b^{5}}\) | \(56\) |
orering | \(-\frac {126 b^{4} x^{8}+84 a \,b^{3} x^{6}+36 a^{2} b^{2} x^{4}+9 a^{3} b \,x^{2}+a^{4}}{1260 \left (b \,x^{2}+a \right )^{9} b^{5}}\) | \(56\) |
norman | \(\frac {-\frac {a^{4}}{1260 b^{5}}-\frac {a^{3} x^{2}}{140 b^{4}}-\frac {a^{2} x^{4}}{35 b^{3}}-\frac {a \,x^{6}}{15 b^{2}}-\frac {x^{8}}{10 b}}{\left (b \,x^{2}+a \right )^{9}}\) | \(59\) |
risch | \(\frac {-\frac {a^{4}}{1260 b^{5}}-\frac {a^{3} x^{2}}{140 b^{4}}-\frac {a^{2} x^{4}}{35 b^{3}}-\frac {a \,x^{6}}{15 b^{2}}-\frac {x^{8}}{10 b}}{\left (b \,x^{2}+a \right )^{9}}\) | \(59\) |
parallelrisch | \(\frac {-126 b^{8} x^{8}-84 a \,b^{7} x^{6}-36 a^{2} b^{6} x^{4}-9 a^{3} b^{5} x^{2}-a^{4} b^{4}}{1260 b^{9} \left (b \,x^{2}+a \right )^{9}}\) | \(63\) |
default | \(-\frac {a^{4}}{18 b^{5} \left (b \,x^{2}+a \right )^{9}}+\frac {a^{3}}{4 b^{5} \left (b \,x^{2}+a \right )^{8}}-\frac {3 a^{2}}{7 b^{5} \left (b \,x^{2}+a \right )^{7}}+\frac {a}{3 b^{5} \left (b \,x^{2}+a \right )^{6}}-\frac {1}{10 b^{5} \left (b \,x^{2}+a \right )^{5}}\) | \(82\) |
Input:
int(x^9/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
Output:
-1/1260*(126*b^4*x^8+84*a*b^3*x^6+36*a^2*b^2*x^4+9*a^3*b*x^2+a^4)/(b*x^2+a )^9/b^5
Time = 0.06 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.60 \[ \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx=-\frac {126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \, {\left (b^{14} x^{18} + 9 \, a b^{13} x^{16} + 36 \, a^{2} b^{12} x^{14} + 84 \, a^{3} b^{11} x^{12} + 126 \, a^{4} b^{10} x^{10} + 126 \, a^{5} b^{9} x^{8} + 84 \, a^{6} b^{8} x^{6} + 36 \, a^{7} b^{7} x^{4} + 9 \, a^{8} b^{6} x^{2} + a^{9} b^{5}\right )}} \] Input:
integrate(x^9/(b*x^2+a)^10,x, algorithm="fricas")
Output:
-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/ (b^14*x^18 + 9*a*b^13*x^16 + 36*a^2*b^12*x^14 + 84*a^3*b^11*x^12 + 126*a^4 *b^10*x^10 + 126*a^5*b^9*x^8 + 84*a^6*b^8*x^6 + 36*a^7*b^7*x^4 + 9*a^8*b^6 *x^2 + a^9*b^5)
Time = 0.67 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.70 \[ \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx=\frac {- a^{4} - 9 a^{3} b x^{2} - 36 a^{2} b^{2} x^{4} - 84 a b^{3} x^{6} - 126 b^{4} x^{8}}{1260 a^{9} b^{5} + 11340 a^{8} b^{6} x^{2} + 45360 a^{7} b^{7} x^{4} + 105840 a^{6} b^{8} x^{6} + 158760 a^{5} b^{9} x^{8} + 158760 a^{4} b^{10} x^{10} + 105840 a^{3} b^{11} x^{12} + 45360 a^{2} b^{12} x^{14} + 11340 a b^{13} x^{16} + 1260 b^{14} x^{18}} \] Input:
integrate(x**9/(b*x**2+a)**10,x)
Output:
(-a**4 - 9*a**3*b*x**2 - 36*a**2*b**2*x**4 - 84*a*b**3*x**6 - 126*b**4*x** 8)/(1260*a**9*b**5 + 11340*a**8*b**6*x**2 + 45360*a**7*b**7*x**4 + 105840* a**6*b**8*x**6 + 158760*a**5*b**9*x**8 + 158760*a**4*b**10*x**10 + 105840* a**3*b**11*x**12 + 45360*a**2*b**12*x**14 + 11340*a*b**13*x**16 + 1260*b** 14*x**18)
Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.60 \[ \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx=-\frac {126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \, {\left (b^{14} x^{18} + 9 \, a b^{13} x^{16} + 36 \, a^{2} b^{12} x^{14} + 84 \, a^{3} b^{11} x^{12} + 126 \, a^{4} b^{10} x^{10} + 126 \, a^{5} b^{9} x^{8} + 84 \, a^{6} b^{8} x^{6} + 36 \, a^{7} b^{7} x^{4} + 9 \, a^{8} b^{6} x^{2} + a^{9} b^{5}\right )}} \] Input:
integrate(x^9/(b*x^2+a)^10,x, algorithm="maxima")
Output:
-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/ (b^14*x^18 + 9*a*b^13*x^16 + 36*a^2*b^12*x^14 + 84*a^3*b^11*x^12 + 126*a^4 *b^10*x^10 + 126*a^5*b^9*x^8 + 84*a^6*b^8*x^6 + 36*a^7*b^7*x^4 + 9*a^8*b^6 *x^2 + a^9*b^5)
Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.60 \[ \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx=-\frac {126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \, {\left (b x^{2} + a\right )}^{9} b^{5}} \] Input:
integrate(x^9/(b*x^2+a)^10,x, algorithm="giac")
Output:
-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/ ((b*x^2 + a)^9*b^5)
Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.63 \[ \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx=-\frac {a^4+9\,a^3\,b\,x^2+36\,a^2\,b^2\,x^4+84\,a\,b^3\,x^6+126\,b^4\,x^8}{1260\,a^9\,b^5+11340\,a^8\,b^6\,x^2+45360\,a^7\,b^7\,x^4+105840\,a^6\,b^8\,x^6+158760\,a^5\,b^9\,x^8+158760\,a^4\,b^{10}\,x^{10}+105840\,a^3\,b^{11}\,x^{12}+45360\,a^2\,b^{12}\,x^{14}+11340\,a\,b^{13}\,x^{16}+1260\,b^{14}\,x^{18}} \] Input:
int(x^9/(a + b*x^2)^10,x)
Output:
-(a^4 + 126*b^4*x^8 + 9*a^3*b*x^2 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4)/(1260*a ^9*b^5 + 1260*b^14*x^18 + 11340*a*b^13*x^16 + 11340*a^8*b^6*x^2 + 45360*a^ 7*b^7*x^4 + 105840*a^6*b^8*x^6 + 158760*a^5*b^9*x^8 + 158760*a^4*b^10*x^10 + 105840*a^3*b^11*x^12 + 45360*a^2*b^12*x^14)
Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.59 \[ \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx=\frac {-126 b^{4} x^{8}-84 a \,b^{3} x^{6}-36 a^{2} b^{2} x^{4}-9 a^{3} b \,x^{2}-a^{4}}{1260 b^{5} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:
int(x^9/(b*x^2+a)^10,x)
Output:
( - a**4 - 9*a**3*b*x**2 - 36*a**2*b**2*x**4 - 84*a*b**3*x**6 - 126*b**4*x **8)/(1260*b**5*(a**9 + 9*a**8*b*x**2 + 36*a**7*b**2*x**4 + 84*a**6*b**3*x **6 + 126*a**5*b**4*x**8 + 126*a**4*b**5*x**10 + 84*a**3*b**6*x**12 + 36*a **2*b**7*x**14 + 9*a*b**8*x**16 + b**9*x**18))