3.194 \(\int \frac {x^{21}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=188 \[ -\frac {a^{10}}{18 b^{11} \left (a+b x^2\right )^9}+\frac {5 a^9}{8 b^{11} \left (a+b x^2\right )^8}-\frac {45 a^8}{14 b^{11} \left (a+b x^2\right )^7}+\frac {10 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac {21 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac {63 a^5}{2 b^{11} \left (a+b x^2\right )^4}-\frac {35 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac {30 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac {45 a^2}{2 b^{11} \left (a+b x^2\right )}-\frac {5 a \log \left (a+b x^2\right )}{b^{11}}+\frac {x^2}{2 b^{10}} \]

[Out]

1/2*x^2/b^10-1/18*a^10/b^11/(b*x^2+a)^9+5/8*a^9/b^11/(b*x^2+a)^8-45/14*a^8/b^11/(b*x^2+a)^7+10*a^7/b^11/(b*x^2
+a)^6-21*a^6/b^11/(b*x^2+a)^5+63/2*a^5/b^11/(b*x^2+a)^4-35*a^4/b^11/(b*x^2+a)^3+30*a^3/b^11/(b*x^2+a)^2-45/2*a
^2/b^11/(b*x^2+a)-5*a*ln(b*x^2+a)/b^11

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Rubi [A]  time = 0.19, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac {a^{10}}{18 b^{11} \left (a+b x^2\right )^9}+\frac {5 a^9}{8 b^{11} \left (a+b x^2\right )^8}-\frac {45 a^8}{14 b^{11} \left (a+b x^2\right )^7}+\frac {10 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac {21 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac {63 a^5}{2 b^{11} \left (a+b x^2\right )^4}-\frac {35 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac {30 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac {45 a^2}{2 b^{11} \left (a+b x^2\right )}-\frac {5 a \log \left (a+b x^2\right )}{b^{11}}+\frac {x^2}{2 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^21/(a + b*x^2)^10,x]

[Out]

x^2/(2*b^10) - a^10/(18*b^11*(a + b*x^2)^9) + (5*a^9)/(8*b^11*(a + b*x^2)^8) - (45*a^8)/(14*b^11*(a + b*x^2)^7
) + (10*a^7)/(b^11*(a + b*x^2)^6) - (21*a^6)/(b^11*(a + b*x^2)^5) + (63*a^5)/(2*b^11*(a + b*x^2)^4) - (35*a^4)
/(b^11*(a + b*x^2)^3) + (30*a^3)/(b^11*(a + b*x^2)^2) - (45*a^2)/(2*b^11*(a + b*x^2)) - (5*a*Log[a + b*x^2])/b
^11

Rule 43

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] && NeQ[b*c - a*d, 0] && 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])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^{10}}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b^{10}}+\frac {a^{10}}{b^{10} (a+b x)^{10}}-\frac {10 a^9}{b^{10} (a+b x)^9}+\frac {45 a^8}{b^{10} (a+b x)^8}-\frac {120 a^7}{b^{10} (a+b x)^7}+\frac {210 a^6}{b^{10} (a+b x)^6}-\frac {252 a^5}{b^{10} (a+b x)^5}+\frac {210 a^4}{b^{10} (a+b x)^4}-\frac {120 a^3}{b^{10} (a+b x)^3}+\frac {45 a^2}{b^{10} (a+b x)^2}-\frac {10 a}{b^{10} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 b^{10}}-\frac {a^{10}}{18 b^{11} \left (a+b x^2\right )^9}+\frac {5 a^9}{8 b^{11} \left (a+b x^2\right )^8}-\frac {45 a^8}{14 b^{11} \left (a+b x^2\right )^7}+\frac {10 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac {21 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac {63 a^5}{2 b^{11} \left (a+b x^2\right )^4}-\frac {35 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac {30 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac {45 a^2}{2 b^{11} \left (a+b x^2\right )}-\frac {5 a \log \left (a+b x^2\right )}{b^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 145, normalized size = 0.77 \[ -\frac {4861 a^{10}+41229 a^9 b x^2+153576 a^8 b^2 x^4+328104 a^7 b^3 x^6+439236 a^6 b^4 x^8+375732 a^5 b^5 x^{10}+197568 a^4 b^6 x^{12}+54432 a^3 b^7 x^{14}+2268 a^2 b^8 x^{16}-2268 a b^9 x^{18}+2520 a \left (a+b x^2\right )^9 \log \left (a+b x^2\right )-252 b^{10} x^{20}}{504 b^{11} \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^21/(a + b*x^2)^10,x]

[Out]

-1/504*(4861*a^10 + 41229*a^9*b*x^2 + 153576*a^8*b^2*x^4 + 328104*a^7*b^3*x^6 + 439236*a^6*b^4*x^8 + 375732*a^
5*b^5*x^10 + 197568*a^4*b^6*x^12 + 54432*a^3*b^7*x^14 + 2268*a^2*b^8*x^16 - 2268*a*b^9*x^18 - 252*b^10*x^20 +
2520*a*(a + b*x^2)^9*Log[a + b*x^2])/(b^11*(a + b*x^2)^9)

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fricas [A]  time = 0.77, size = 322, normalized size = 1.71 \[ \frac {252 \, b^{10} x^{20} + 2268 \, a b^{9} x^{18} - 2268 \, a^{2} b^{8} x^{16} - 54432 \, a^{3} b^{7} x^{14} - 197568 \, a^{4} b^{6} x^{12} - 375732 \, a^{5} b^{5} x^{10} - 439236 \, a^{6} b^{4} x^{8} - 328104 \, a^{7} b^{3} x^{6} - 153576 \, a^{8} b^{2} x^{4} - 41229 \, a^{9} b x^{2} - 4861 \, a^{10} - 2520 \, {\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \log \left (b x^{2} + a\right )}{504 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^21/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

1/504*(252*b^10*x^20 + 2268*a*b^9*x^18 - 2268*a^2*b^8*x^16 - 54432*a^3*b^7*x^14 - 197568*a^4*b^6*x^12 - 375732
*a^5*b^5*x^10 - 439236*a^6*b^4*x^8 - 328104*a^7*b^3*x^6 - 153576*a^8*b^2*x^4 - 41229*a^9*b*x^2 - 4861*a^10 - 2
520*(a*b^9*x^18 + 9*a^2*b^8*x^16 + 36*a^3*b^7*x^14 + 84*a^4*b^6*x^12 + 126*a^5*b^5*x^10 + 126*a^6*b^4*x^8 + 84
*a^7*b^3*x^6 + 36*a^8*b^2*x^4 + 9*a^9*b*x^2 + a^10)*log(b*x^2 + a))/(b^20*x^18 + 9*a*b^19*x^16 + 36*a^2*b^18*x
^14 + 84*a^3*b^17*x^12 + 126*a^4*b^16*x^10 + 126*a^5*b^15*x^8 + 84*a^6*b^14*x^6 + 36*a^7*b^13*x^4 + 9*a^8*b^12
*x^2 + a^9*b^11)

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giac [A]  time = 0.63, size = 139, normalized size = 0.74 \[ \frac {x^{2}}{2 \, b^{10}} - \frac {5 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{b^{11}} + \frac {7129 \, a b^{9} x^{18} + 52821 \, a^{2} b^{8} x^{16} + 181044 \, a^{3} b^{7} x^{14} + 369516 \, a^{4} b^{6} x^{12} + 490770 \, a^{5} b^{5} x^{10} + 437850 \, a^{6} b^{4} x^{8} + 261660 \, a^{7} b^{3} x^{6} + 100800 \, a^{8} b^{2} x^{4} + 22680 \, a^{9} b x^{2} + 2268 \, a^{10}}{504 \, {\left (b x^{2} + a\right )}^{9} b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^21/(b*x^2+a)^10,x, algorithm="giac")

[Out]

1/2*x^2/b^10 - 5*a*log(abs(b*x^2 + a))/b^11 + 1/504*(7129*a*b^9*x^18 + 52821*a^2*b^8*x^16 + 181044*a^3*b^7*x^1
4 + 369516*a^4*b^6*x^12 + 490770*a^5*b^5*x^10 + 437850*a^6*b^4*x^8 + 261660*a^7*b^3*x^6 + 100800*a^8*b^2*x^4 +
22680*a^9*b*x^2 + 2268*a^10)/((b*x^2 + a)^9*b^11)

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maple [A]  time = 0.02, size = 177, normalized size = 0.94 \[ -\frac {a^{10}}{18 \left (b \,x^{2}+a \right )^{9} b^{11}}+\frac {5 a^{9}}{8 \left (b \,x^{2}+a \right )^{8} b^{11}}-\frac {45 a^{8}}{14 \left (b \,x^{2}+a \right )^{7} b^{11}}+\frac {10 a^{7}}{\left (b \,x^{2}+a \right )^{6} b^{11}}-\frac {21 a^{6}}{\left (b \,x^{2}+a \right )^{5} b^{11}}+\frac {63 a^{5}}{2 \left (b \,x^{2}+a \right )^{4} b^{11}}-\frac {35 a^{4}}{\left (b \,x^{2}+a \right )^{3} b^{11}}+\frac {30 a^{3}}{\left (b \,x^{2}+a \right )^{2} b^{11}}+\frac {x^{2}}{2 b^{10}}-\frac {45 a^{2}}{2 \left (b \,x^{2}+a \right ) b^{11}}-\frac {5 a \ln \left (b \,x^{2}+a \right )}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^21/(b*x^2+a)^10,x)

[Out]

1/2*x^2/b^10-1/18*a^10/b^11/(b*x^2+a)^9+5/8*a^9/b^11/(b*x^2+a)^8-45/14*a^8/b^11/(b*x^2+a)^7+10*a^7/b^11/(b*x^2
+a)^6-21*a^6/b^11/(b*x^2+a)^5+63/2*a^5/b^11/(b*x^2+a)^4-35*a^4/b^11/(b*x^2+a)^3+30*a^3/b^11/(b*x^2+a)^2-45/2*a
^2/b^11/(b*x^2+a)-5*a*ln(b*x^2+a)/b^11

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maxima [A]  time = 1.60, size = 220, normalized size = 1.17 \[ -\frac {11340 \, a^{2} b^{8} x^{16} + 75600 \, a^{3} b^{7} x^{14} + 229320 \, a^{4} b^{6} x^{12} + 407484 \, a^{5} b^{5} x^{10} + 460404 \, a^{6} b^{4} x^{8} + 337176 \, a^{7} b^{3} x^{6} + 155844 \, a^{8} b^{2} x^{4} + 41481 \, a^{9} b x^{2} + 4861 \, a^{10}}{504 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} + \frac {x^{2}}{2 \, b^{10}} - \frac {5 \, a \log \left (b x^{2} + a\right )}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^21/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/504*(11340*a^2*b^8*x^16 + 75600*a^3*b^7*x^14 + 229320*a^4*b^6*x^12 + 407484*a^5*b^5*x^10 + 460404*a^6*b^4*x
^8 + 337176*a^7*b^3*x^6 + 155844*a^8*b^2*x^4 + 41481*a^9*b*x^2 + 4861*a^10)/(b^20*x^18 + 9*a*b^19*x^16 + 36*a^
2*b^18*x^14 + 84*a^3*b^17*x^12 + 126*a^4*b^16*x^10 + 126*a^5*b^15*x^8 + 84*a^6*b^14*x^6 + 36*a^7*b^13*x^4 + 9*
a^8*b^12*x^2 + a^9*b^11) + 1/2*x^2/b^10 - 5*a*log(b*x^2 + a)/b^11

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mupad [B]  time = 0.43, size = 220, normalized size = 1.17 \[ \frac {x^2}{2\,b^{10}}-\frac {\frac {4861\,a^{10}}{504\,b}+\frac {4609\,a^9\,x^2}{56}+\frac {4329\,a^8\,b\,x^4}{14}+669\,a^7\,b^2\,x^6+\frac {1827\,a^6\,b^3\,x^8}{2}+\frac {1617\,a^5\,b^4\,x^{10}}{2}+455\,a^4\,b^5\,x^{12}+150\,a^3\,b^6\,x^{14}+\frac {45\,a^2\,b^7\,x^{16}}{2}}{a^9\,b^{10}+9\,a^8\,b^{11}\,x^2+36\,a^7\,b^{12}\,x^4+84\,a^6\,b^{13}\,x^6+126\,a^5\,b^{14}\,x^8+126\,a^4\,b^{15}\,x^{10}+84\,a^3\,b^{16}\,x^{12}+36\,a^2\,b^{17}\,x^{14}+9\,a\,b^{18}\,x^{16}+b^{19}\,x^{18}}-\frac {5\,a\,\ln \left (b\,x^2+a\right )}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^21/(a + b*x^2)^10,x)

[Out]

x^2/(2*b^10) - ((4861*a^10)/(504*b) + (4609*a^9*x^2)/56 + (4329*a^8*b*x^4)/14 + 669*a^7*b^2*x^6 + (1827*a^6*b^
3*x^8)/2 + (1617*a^5*b^4*x^10)/2 + 455*a^4*b^5*x^12 + 150*a^3*b^6*x^14 + (45*a^2*b^7*x^16)/2)/(a^9*b^10 + b^19
*x^18 + 9*a*b^18*x^16 + 9*a^8*b^11*x^2 + 36*a^7*b^12*x^4 + 84*a^6*b^13*x^6 + 126*a^5*b^14*x^8 + 126*a^4*b^15*x
^10 + 84*a^3*b^16*x^12 + 36*a^2*b^17*x^14) - (5*a*log(a + b*x^2))/b^11

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sympy [A]  time = 2.03, size = 233, normalized size = 1.24 \[ - \frac {5 a \log {\left (a + b x^{2} \right )}}{b^{11}} + \frac {- 4861 a^{10} - 41481 a^{9} b x^{2} - 155844 a^{8} b^{2} x^{4} - 337176 a^{7} b^{3} x^{6} - 460404 a^{6} b^{4} x^{8} - 407484 a^{5} b^{5} x^{10} - 229320 a^{4} b^{6} x^{12} - 75600 a^{3} b^{7} x^{14} - 11340 a^{2} b^{8} x^{16}}{504 a^{9} b^{11} + 4536 a^{8} b^{12} x^{2} + 18144 a^{7} b^{13} x^{4} + 42336 a^{6} b^{14} x^{6} + 63504 a^{5} b^{15} x^{8} + 63504 a^{4} b^{16} x^{10} + 42336 a^{3} b^{17} x^{12} + 18144 a^{2} b^{18} x^{14} + 4536 a b^{19} x^{16} + 504 b^{20} x^{18}} + \frac {x^{2}}{2 b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**21/(b*x**2+a)**10,x)

[Out]

-5*a*log(a + b*x**2)/b**11 + (-4861*a**10 - 41481*a**9*b*x**2 - 155844*a**8*b**2*x**4 - 337176*a**7*b**3*x**6
- 460404*a**6*b**4*x**8 - 407484*a**5*b**5*x**10 - 229320*a**4*b**6*x**12 - 75600*a**3*b**7*x**14 - 11340*a**2
*b**8*x**16)/(504*a**9*b**11 + 4536*a**8*b**12*x**2 + 18144*a**7*b**13*x**4 + 42336*a**6*b**14*x**6 + 63504*a*
*5*b**15*x**8 + 63504*a**4*b**16*x**10 + 42336*a**3*b**17*x**12 + 18144*a**2*b**18*x**14 + 4536*a*b**19*x**16
+ 504*b**20*x**18) + x**2/(2*b**10)

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