Optimal. Leaf size=188 \[ -\frac {a^{10}}{18 b^{11} \left (a+b x^2\right )^9}+\frac {5 a^9}{8 b^{11} \left (a+b x^2\right )^8}-\frac {45 a^8}{14 b^{11} \left (a+b x^2\right )^7}+\frac {10 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac {21 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac {63 a^5}{2 b^{11} \left (a+b x^2\right )^4}-\frac {35 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac {30 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac {45 a^2}{2 b^{11} \left (a+b x^2\right )}-\frac {5 a \log \left (a+b x^2\right )}{b^{11}}+\frac {x^2}{2 b^{10}} \]
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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.
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Rule 43
Rule 266
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.
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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.
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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.
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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.
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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.
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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.
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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.
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