Integrand size = 13, antiderivative size = 58 \[ \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx=\frac {x^{14}}{18 a \left (a+b x^2\right )^9}+\frac {x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac {x^{14}}{504 a^3 \left (a+b x^2\right )^7} \] Output:
1/18*x^14/a/(b*x^2+a)^9+1/72*x^14/a^2/(b*x^2+a)^8+1/504*x^14/a^3/(b*x^2+a) ^7
Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.36 \[ \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {a^6+9 a^5 b x^2+36 a^4 b^2 x^4+84 a^3 b^3 x^6+126 a^2 b^4 x^8+126 a b^5 x^{10}+84 b^6 x^{12}}{504 b^7 \left (a+b x^2\right )^9} \] Input:
Integrate[x^13/(a + b*x^2)^10,x]
Output:
-1/504*(a^6 + 9*a^5*b*x^2 + 36*a^4*b^2*x^4 + 84*a^3*b^3*x^6 + 126*a^2*b^4* x^8 + 126*a*b^5*x^10 + 84*b^6*x^12)/(b^7*(a + b*x^2)^9)
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {243, 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.
\(\displaystyle \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {x^{12}}{\left (b x^2+a\right )^{10}}dx^2\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \int \frac {x^{12}}{\left (b x^2+a\right )^9}dx^2}{9 a}+\frac {x^{14}}{9 a \left (a+b x^2\right )^9}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {\int \frac {x^{12}}{\left (b x^2+a\right )^8}dx^2}{8 a}+\frac {x^{14}}{8 a \left (a+b x^2\right )^8}\right )}{9 a}+\frac {x^{14}}{9 a \left (a+b x^2\right )^9}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {x^{14}}{56 a^2 \left (a+b x^2\right )^7}+\frac {x^{14}}{8 a \left (a+b x^2\right )^8}\right )}{9 a}+\frac {x^{14}}{9 a \left (a+b x^2\right )^9}\right )\) |
Input:
Int[x^13/(a + b*x^2)^10,x]
Output:
(x^14/(9*a*(a + b*x^2)^9) + (2*(x^14/(8*a*(a + b*x^2)^8) + x^14/(56*a^2*(a + b*x^2)^7)))/(9*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 = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.34
method | result | size |
gosper | \(-\frac {84 b^{6} x^{12}+126 a \,b^{5} x^{10}+126 a^{2} b^{4} x^{8}+84 a^{3} x^{6} b^{3}+36 a^{4} b^{2} x^{4}+9 a^{5} b \,x^{2}+a^{6}}{504 \left (b \,x^{2}+a \right )^{9} b^{7}}\) | \(78\) |
orering | \(-\frac {84 b^{6} x^{12}+126 a \,b^{5} x^{10}+126 a^{2} b^{4} x^{8}+84 a^{3} x^{6} b^{3}+36 a^{4} b^{2} x^{4}+9 a^{5} b \,x^{2}+a^{6}}{504 \left (b \,x^{2}+a \right )^{9} b^{7}}\) | \(78\) |
norman | \(\frac {-\frac {a^{6}}{504 b^{7}}-\frac {a^{5} x^{2}}{56 b^{6}}-\frac {a^{4} x^{4}}{14 b^{5}}-\frac {a^{3} x^{6}}{6 b^{4}}-\frac {a^{2} x^{8}}{4 b^{3}}-\frac {a \,x^{10}}{4 b^{2}}-\frac {x^{12}}{6 b}}{\left (b \,x^{2}+a \right )^{9}}\) | \(81\) |
risch | \(\frac {-\frac {a^{6}}{504 b^{7}}-\frac {a^{5} x^{2}}{56 b^{6}}-\frac {a^{4} x^{4}}{14 b^{5}}-\frac {a^{3} x^{6}}{6 b^{4}}-\frac {a^{2} x^{8}}{4 b^{3}}-\frac {a \,x^{10}}{4 b^{2}}-\frac {x^{12}}{6 b}}{\left (b \,x^{2}+a \right )^{9}}\) | \(81\) |
parallelrisch | \(\frac {-84 b^{8} x^{12}-126 a \,b^{7} x^{10}-126 a^{2} b^{6} x^{8}-84 a^{3} b^{5} x^{6}-36 a^{4} x^{4} b^{4}-9 a^{5} b^{3} x^{2}-a^{6} b^{2}}{504 b^{9} \left (b \,x^{2}+a \right )^{9}}\) | \(85\) |
default | \(-\frac {a^{6}}{18 b^{7} \left (b \,x^{2}+a \right )^{9}}-\frac {3 a^{2}}{2 b^{7} \left (b \,x^{2}+a \right )^{5}}+\frac {3 a}{4 b^{7} \left (b \,x^{2}+a \right )^{4}}-\frac {15 a^{4}}{14 b^{7} \left (b \,x^{2}+a \right )^{7}}+\frac {3 a^{5}}{8 b^{7} \left (b \,x^{2}+a \right )^{8}}-\frac {1}{6 b^{7} \left (b \,x^{2}+a \right )^{3}}+\frac {5 a^{3}}{3 b^{7} \left (b \,x^{2}+a \right )^{6}}\) | \(116\) |
Input:
int(x^13/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
Output:
-1/504*(84*b^6*x^12+126*a*b^5*x^10+126*a^2*b^4*x^8+84*a^3*b^3*x^6+36*a^4*b ^2*x^4+9*a^5*b*x^2+a^6)/(b*x^2+a)^9/b^7
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (52) = 104\).
Time = 0.06 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.90 \[ \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \, {\left (b^{16} x^{18} + 9 \, a b^{15} x^{16} + 36 \, a^{2} b^{14} x^{14} + 84 \, a^{3} b^{13} x^{12} + 126 \, a^{4} b^{12} x^{10} + 126 \, a^{5} b^{11} x^{8} + 84 \, a^{6} b^{10} x^{6} + 36 \, a^{7} b^{9} x^{4} + 9 \, a^{8} b^{8} x^{2} + a^{9} b^{7}\right )}} \] Input:
integrate(x^13/(b*x^2+a)^10,x, algorithm="fricas")
Output:
-1/504*(84*b^6*x^12 + 126*a*b^5*x^10 + 126*a^2*b^4*x^8 + 84*a^3*b^3*x^6 + 36*a^4*b^2*x^4 + 9*a^5*b*x^2 + a^6)/(b^16*x^18 + 9*a*b^15*x^16 + 36*a^2*b^ 14*x^14 + 84*a^3*b^13*x^12 + 126*a^4*b^12*x^10 + 126*a^5*b^11*x^8 + 84*a^6 *b^10*x^6 + 36*a^7*b^9*x^4 + 9*a^8*b^8*x^2 + a^9*b^7)
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (48) = 96\).
Time = 0.73 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.07 \[ \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx=\frac {- a^{6} - 9 a^{5} b x^{2} - 36 a^{4} b^{2} x^{4} - 84 a^{3} b^{3} x^{6} - 126 a^{2} b^{4} x^{8} - 126 a b^{5} x^{10} - 84 b^{6} x^{12}}{504 a^{9} b^{7} + 4536 a^{8} b^{8} x^{2} + 18144 a^{7} b^{9} x^{4} + 42336 a^{6} b^{10} x^{6} + 63504 a^{5} b^{11} x^{8} + 63504 a^{4} b^{12} x^{10} + 42336 a^{3} b^{13} x^{12} + 18144 a^{2} b^{14} x^{14} + 4536 a b^{15} x^{16} + 504 b^{16} x^{18}} \] Input:
integrate(x**13/(b*x**2+a)**10,x)
Output:
(-a**6 - 9*a**5*b*x**2 - 36*a**4*b**2*x**4 - 84*a**3*b**3*x**6 - 126*a**2* b**4*x**8 - 126*a*b**5*x**10 - 84*b**6*x**12)/(504*a**9*b**7 + 4536*a**8*b **8*x**2 + 18144*a**7*b**9*x**4 + 42336*a**6*b**10*x**6 + 63504*a**5*b**11 *x**8 + 63504*a**4*b**12*x**10 + 42336*a**3*b**13*x**12 + 18144*a**2*b**14 *x**14 + 4536*a*b**15*x**16 + 504*b**16*x**18)
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (52) = 104\).
Time = 0.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.90 \[ \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \, {\left (b^{16} x^{18} + 9 \, a b^{15} x^{16} + 36 \, a^{2} b^{14} x^{14} + 84 \, a^{3} b^{13} x^{12} + 126 \, a^{4} b^{12} x^{10} + 126 \, a^{5} b^{11} x^{8} + 84 \, a^{6} b^{10} x^{6} + 36 \, a^{7} b^{9} x^{4} + 9 \, a^{8} b^{8} x^{2} + a^{9} b^{7}\right )}} \] Input:
integrate(x^13/(b*x^2+a)^10,x, algorithm="maxima")
Output:
-1/504*(84*b^6*x^12 + 126*a*b^5*x^10 + 126*a^2*b^4*x^8 + 84*a^3*b^3*x^6 + 36*a^4*b^2*x^4 + 9*a^5*b*x^2 + a^6)/(b^16*x^18 + 9*a*b^15*x^16 + 36*a^2*b^ 14*x^14 + 84*a^3*b^13*x^12 + 126*a^4*b^12*x^10 + 126*a^5*b^11*x^8 + 84*a^6 *b^10*x^6 + 36*a^7*b^9*x^4 + 9*a^8*b^8*x^2 + a^9*b^7)
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.33 \[ \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \, {\left (b x^{2} + a\right )}^{9} b^{7}} \] Input:
integrate(x^13/(b*x^2+a)^10,x, algorithm="giac")
Output:
-1/504*(84*b^6*x^12 + 126*a*b^5*x^10 + 126*a^2*b^4*x^8 + 84*a^3*b^3*x^6 + 36*a^4*b^2*x^4 + 9*a^5*b*x^2 + a^6)/((b*x^2 + a)^9*b^7)
Time = 0.18 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.93 \[ \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {a^6+9\,a^5\,b\,x^2+36\,a^4\,b^2\,x^4+84\,a^3\,b^3\,x^6+126\,a^2\,b^4\,x^8+126\,a\,b^5\,x^{10}+84\,b^6\,x^{12}}{504\,a^9\,b^7+4536\,a^8\,b^8\,x^2+18144\,a^7\,b^9\,x^4+42336\,a^6\,b^{10}\,x^6+63504\,a^5\,b^{11}\,x^8+63504\,a^4\,b^{12}\,x^{10}+42336\,a^3\,b^{13}\,x^{12}+18144\,a^2\,b^{14}\,x^{14}+4536\,a\,b^{15}\,x^{16}+504\,b^{16}\,x^{18}} \] Input:
int(x^13/(a + b*x^2)^10,x)
Output:
-(a^6 + 84*b^6*x^12 + 9*a^5*b*x^2 + 126*a*b^5*x^10 + 36*a^4*b^2*x^4 + 84*a ^3*b^3*x^6 + 126*a^2*b^4*x^8)/(504*a^9*b^7 + 504*b^16*x^18 + 4536*a*b^15*x ^16 + 4536*a^8*b^8*x^2 + 18144*a^7*b^9*x^4 + 42336*a^6*b^10*x^6 + 63504*a^ 5*b^11*x^8 + 63504*a^4*b^12*x^10 + 42336*a^3*b^13*x^12 + 18144*a^2*b^14*x^ 14)
Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.88 \[ \int \frac {x^{13}}{\left (a+b x^2\right )^{10}} \, dx=\frac {-84 b^{6} x^{12}-126 a \,b^{5} x^{10}-126 a^{2} b^{4} x^{8}-84 a^{3} b^{3} x^{6}-36 a^{4} b^{2} x^{4}-9 a^{5} b \,x^{2}-a^{6}}{504 b^{7} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:
int(x^13/(b*x^2+a)^10,x)
Output:
( - a**6 - 9*a**5*b*x**2 - 36*a**4*b**2*x**4 - 84*a**3*b**3*x**6 - 126*a** 2*b**4*x**8 - 126*a*b**5*x**10 - 84*b**6*x**12)/(504*b**7*(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))