Optimal. Leaf size=205 \[ \frac {a^{11}}{18 b^{12} \left (a+b x^2\right )^9}-\frac {11 a^{10}}{16 b^{12} \left (a+b x^2\right )^8}+\frac {55 a^9}{14 b^{12} \left (a+b x^2\right )^7}-\frac {55 a^8}{4 b^{12} \left (a+b x^2\right )^6}+\frac {33 a^7}{b^{12} \left (a+b x^2\right )^5}-\frac {231 a^6}{4 b^{12} \left (a+b x^2\right )^4}+\frac {77 a^5}{b^{12} \left (a+b x^2\right )^3}-\frac {165 a^4}{2 b^{12} \left (a+b x^2\right )^2}+\frac {165 a^3}{2 b^{12} \left (a+b x^2\right )}+\frac {55 a^2 \log \left (a+b x^2\right )}{2 b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^4}{4 b^{10}} \]
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Rubi [A] time = 0.21, antiderivative size = 205, 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} \[ \frac {a^{11}}{18 b^{12} \left (a+b x^2\right )^9}-\frac {11 a^{10}}{16 b^{12} \left (a+b x^2\right )^8}+\frac {55 a^9}{14 b^{12} \left (a+b x^2\right )^7}-\frac {55 a^8}{4 b^{12} \left (a+b x^2\right )^6}+\frac {33 a^7}{b^{12} \left (a+b x^2\right )^5}-\frac {231 a^6}{4 b^{12} \left (a+b x^2\right )^4}+\frac {77 a^5}{b^{12} \left (a+b x^2\right )^3}-\frac {165 a^4}{2 b^{12} \left (a+b x^2\right )^2}+\frac {165 a^3}{2 b^{12} \left (a+b x^2\right )}+\frac {55 a^2 \log \left (a+b x^2\right )}{2 b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^4}{4 b^{10}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^{11}}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {10 a}{b^{11}}+\frac {x}{b^{10}}-\frac {a^{11}}{b^{11} (a+b x)^{10}}+\frac {11 a^{10}}{b^{11} (a+b x)^9}-\frac {55 a^9}{b^{11} (a+b x)^8}+\frac {165 a^8}{b^{11} (a+b x)^7}-\frac {330 a^7}{b^{11} (a+b x)^6}+\frac {462 a^6}{b^{11} (a+b x)^5}-\frac {462 a^5}{b^{11} (a+b x)^4}+\frac {330 a^4}{b^{11} (a+b x)^3}-\frac {165 a^3}{b^{11} (a+b x)^2}+\frac {55 a^2}{b^{11} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {5 a x^2}{b^{11}}+\frac {x^4}{4 b^{10}}+\frac {a^{11}}{18 b^{12} \left (a+b x^2\right )^9}-\frac {11 a^{10}}{16 b^{12} \left (a+b x^2\right )^8}+\frac {55 a^9}{14 b^{12} \left (a+b x^2\right )^7}-\frac {55 a^8}{4 b^{12} \left (a+b x^2\right )^6}+\frac {33 a^7}{b^{12} \left (a+b x^2\right )^5}-\frac {231 a^6}{4 b^{12} \left (a+b x^2\right )^4}+\frac {77 a^5}{b^{12} \left (a+b x^2\right )^3}-\frac {165 a^4}{2 b^{12} \left (a+b x^2\right )^2}+\frac {165 a^3}{2 b^{12} \left (a+b x^2\right )}+\frac {55 a^2 \log \left (a+b x^2\right )}{2 b^{12}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 158, normalized size = 0.77 \[ \frac {42131 a^{11}+351459 a^{10} b x^2+1281096 a^9 b^2 x^4+2656584 a^8 b^3 x^6+3402756 a^7 b^4 x^8+2704212 a^6 b^5 x^{10}+1220688 a^5 b^6 x^{12}+190512 a^4 b^7 x^{14}-77112 a^3 b^8 x^{16}-36288 a^2 b^9 x^{18}+27720 a^2 \left (a+b x^2\right )^9 \log \left (a+b x^2\right )-2772 a b^{10} x^{20}+252 b^{11} x^{22}}{1008 b^{12} \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 335, normalized size = 1.63 \[ \frac {252 \, b^{11} x^{22} - 2772 \, a b^{10} x^{20} - 36288 \, a^{2} b^{9} x^{18} - 77112 \, a^{3} b^{8} x^{16} + 190512 \, a^{4} b^{7} x^{14} + 1220688 \, a^{5} b^{6} x^{12} + 2704212 \, a^{6} b^{5} x^{10} + 3402756 \, a^{7} b^{4} x^{8} + 2656584 \, a^{8} b^{3} x^{6} + 1281096 \, a^{9} b^{2} x^{4} + 351459 \, a^{10} b x^{2} + 42131 \, a^{11} + 27720 \, {\left (a^{2} b^{9} x^{18} + 9 \, a^{3} b^{8} x^{16} + 36 \, a^{4} b^{7} x^{14} + 84 \, a^{5} b^{6} x^{12} + 126 \, a^{6} b^{5} x^{10} + 126 \, a^{7} b^{4} x^{8} + 84 \, a^{8} b^{3} x^{6} + 36 \, a^{9} b^{2} x^{4} + 9 \, a^{10} b x^{2} + a^{11}\right )} \log \left (b x^{2} + a\right )}{1008 \, {\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 157, normalized size = 0.77 \[ \frac {55 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{12}} + \frac {b^{10} x^{4} - 20 \, a b^{9} x^{2}}{4 \, b^{20}} - \frac {78419 \, a^{2} b^{9} x^{18} + 622611 \, a^{3} b^{8} x^{16} + 2240964 \, a^{4} b^{7} x^{14} + 4763220 \, a^{5} b^{6} x^{12} + 6562710 \, a^{6} b^{5} x^{10} + 6063750 \, a^{7} b^{4} x^{8} + 3751440 \, a^{8} b^{3} x^{6} + 1496880 \, a^{9} b^{2} x^{4} + 349272 \, a^{10} b x^{2} + 36288 \, a^{11}}{1008 \, {\left (b x^{2} + a\right )}^{9} b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 188, normalized size = 0.92 \[ \frac {a^{11}}{18 \left (b \,x^{2}+a \right )^{9} b^{12}}-\frac {11 a^{10}}{16 \left (b \,x^{2}+a \right )^{8} b^{12}}+\frac {55 a^{9}}{14 \left (b \,x^{2}+a \right )^{7} b^{12}}-\frac {55 a^{8}}{4 \left (b \,x^{2}+a \right )^{6} b^{12}}+\frac {33 a^{7}}{\left (b \,x^{2}+a \right )^{5} b^{12}}-\frac {231 a^{6}}{4 \left (b \,x^{2}+a \right )^{4} b^{12}}+\frac {x^{4}}{4 b^{10}}+\frac {77 a^{5}}{\left (b \,x^{2}+a \right )^{3} b^{12}}-\frac {165 a^{4}}{2 \left (b \,x^{2}+a \right )^{2} b^{12}}-\frac {5 a \,x^{2}}{b^{11}}+\frac {165 a^{3}}{2 \left (b \,x^{2}+a \right ) b^{12}}+\frac {55 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.58, size = 231, normalized size = 1.13 \[ \frac {83160 \, a^{3} b^{8} x^{16} + 582120 \, a^{4} b^{7} x^{14} + 1823976 \, a^{5} b^{6} x^{12} + 3318084 \, a^{6} b^{5} x^{10} + 3817044 \, a^{7} b^{4} x^{8} + 2835756 \, a^{8} b^{3} x^{6} + 1326204 \, a^{9} b^{2} x^{4} + 356499 \, a^{10} b x^{2} + 42131 \, a^{11}}{1008 \, {\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}} + \frac {55 \, a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{12}} + \frac {b x^{4} - 20 \, a x^{2}}{4 \, b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 230, normalized size = 1.12 \[ \frac {\frac {42131\,a^{11}}{1008\,b}+\frac {39611\,a^{10}\,x^2}{112}+\frac {36839\,a^9\,b\,x^4}{28}+\frac {11253\,a^8\,b^2\,x^6}{4}+\frac {15147\,a^7\,b^3\,x^8}{4}+\frac {13167\,a^6\,b^4\,x^{10}}{4}+\frac {3619\,a^5\,b^5\,x^{12}}{2}+\frac {1155\,a^4\,b^6\,x^{14}}{2}+\frac {165\,a^3\,b^7\,x^{16}}{2}}{a^9\,b^{11}+9\,a^8\,b^{12}\,x^2+36\,a^7\,b^{13}\,x^4+84\,a^6\,b^{14}\,x^6+126\,a^5\,b^{15}\,x^8+126\,a^4\,b^{16}\,x^{10}+84\,a^3\,b^{17}\,x^{12}+36\,a^2\,b^{18}\,x^{14}+9\,a\,b^{19}\,x^{16}+b^{20}\,x^{18}}+\frac {x^4}{4\,b^{10}}-\frac {5\,a\,x^2}{b^{11}}+\frac {55\,a^2\,\ln \left (b\,x^2+a\right )}{2\,b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.06, size = 245, normalized size = 1.20 \[ \frac {55 a^{2} \log {\left (a + b x^{2} \right )}}{2 b^{12}} - \frac {5 a x^{2}}{b^{11}} + \frac {42131 a^{11} + 356499 a^{10} b x^{2} + 1326204 a^{9} b^{2} x^{4} + 2835756 a^{8} b^{3} x^{6} + 3817044 a^{7} b^{4} x^{8} + 3318084 a^{6} b^{5} x^{10} + 1823976 a^{5} b^{6} x^{12} + 582120 a^{4} b^{7} x^{14} + 83160 a^{3} b^{8} x^{16}}{1008 a^{9} b^{12} + 9072 a^{8} b^{13} x^{2} + 36288 a^{7} b^{14} x^{4} + 84672 a^{6} b^{15} x^{6} + 127008 a^{5} b^{16} x^{8} + 127008 a^{4} b^{17} x^{10} + 84672 a^{3} b^{18} x^{12} + 36288 a^{2} b^{19} x^{14} + 9072 a b^{20} x^{16} + 1008 b^{21} x^{18}} + \frac {x^{4}}{4 b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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