Optimal. Leaf size=218 \[ \frac {1616615 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{23/2}}-\frac {1616615 a x}{65536 b^{11}}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}+\frac {1616615 x^3}{196608 b^{10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac {1616615 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{23/2}}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}-\frac {1616615 a x}{65536 b^{11}}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}+\frac {1616615 x^3}{196608 b^{10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}+\frac {7 \int \frac {x^{20}}{\left (a+b x^2\right )^9} \, dx}{6 b}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}+\frac {133 \int \frac {x^{18}}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}+\frac {323 \int \frac {x^{16}}{\left (a+b x^2\right )^7} \, dx}{192 b^3}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}+\frac {1615 \int \frac {x^{14}}{\left (a+b x^2\right )^6} \, dx}{768 b^4}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}+\frac {4199 \int \frac {x^{12}}{\left (a+b x^2\right )^5} \, dx}{1536 b^5}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}+\frac {46189 \int \frac {x^{10}}{\left (a+b x^2\right )^4} \, dx}{12288 b^6}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}+\frac {46189 \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx}{8192 b^7}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}+\frac {323323 \int \frac {x^6}{\left (a+b x^2\right )^2} \, dx}{32768 b^8}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac {1616615 \int \frac {x^4}{a+b x^2} \, dx}{65536 b^9}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac {1616615 \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{65536 b^9}\\ &=-\frac {1616615 a x}{65536 b^{11}}+\frac {1616615 x^3}{196608 b^{10}}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac {\left (1616615 a^2\right ) \int \frac {1}{a+b x^2} \, dx}{65536 b^{11}}\\ &=-\frac {1616615 a x}{65536 b^{11}}+\frac {1616615 x^3}{196608 b^{10}}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac {1616615 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{23/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 155, normalized size = 0.71 \[ \frac {14549535 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\frac {\sqrt {b} x \left (-14549535 a^{10}-126095970 a^9 b x^2-483044562 a^8 b^2 x^4-1071677178 a^7 b^3 x^6-1513521152 a^6 b^4 x^8-1404993798 a^5 b^5 x^{10}-850547502 a^4 b^6 x^{12}-318434718 a^3 b^7 x^{14}-63897057 a^2 b^8 x^{16}-4128768 a b^9 x^{18}+196608 b^{10} x^{20}\right )}{\left (a+b x^2\right )^9}}{589824 b^{23/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 692, normalized size = 3.17 \[ \left [\frac {393216 \, b^{10} x^{21} - 8257536 \, a b^{9} x^{19} - 127794114 \, a^{2} b^{8} x^{17} - 636869436 \, a^{3} b^{7} x^{15} - 1701095004 \, a^{4} b^{6} x^{13} - 2809987596 \, a^{5} b^{5} x^{11} - 3027042304 \, a^{6} b^{4} x^{9} - 2143354356 \, a^{7} b^{3} x^{7} - 966089124 \, a^{8} b^{2} x^{5} - 252191940 \, a^{9} b x^{3} - 29099070 \, a^{10} x + 14549535 \, {\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{1179648 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}}, \frac {196608 \, b^{10} x^{21} - 4128768 \, a b^{9} x^{19} - 63897057 \, a^{2} b^{8} x^{17} - 318434718 \, a^{3} b^{7} x^{15} - 850547502 \, a^{4} b^{6} x^{13} - 1404993798 \, a^{5} b^{5} x^{11} - 1513521152 \, a^{6} b^{4} x^{9} - 1071677178 \, a^{7} b^{3} x^{7} - 483044562 \, a^{8} b^{2} x^{5} - 126095970 \, a^{9} b x^{3} - 14549535 \, a^{10} x + 14549535 \, {\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{589824 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.65, size = 150, normalized size = 0.69 \[ \frac {1616615 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{11}} - \frac {17890785 \, a^{2} b^{8} x^{17} + 122613150 \, a^{3} b^{7} x^{15} + 379867950 \, a^{4} b^{6} x^{13} + 686588166 \, a^{5} b^{5} x^{11} + 786857984 \, a^{6} b^{4} x^{9} + 583302906 \, a^{7} b^{3} x^{7} + 272477394 \, a^{8} b^{2} x^{5} + 73208418 \, a^{9} b x^{3} + 8651295 \, a^{10} x}{589824 \, {\left (b x^{2} + a\right )}^{9} b^{11}} + \frac {b^{20} x^{3} - 30 \, a b^{19} x}{3 \, b^{30}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 217, normalized size = 1.00 \[ -\frac {1987865 a^{2} x^{17}}{65536 \left (b \,x^{2}+a \right )^{9} b^{3}}-\frac {20435525 a^{3} x^{15}}{98304 \left (b \,x^{2}+a \right )^{9} b^{4}}-\frac {21103775 a^{4} x^{13}}{32768 \left (b \,x^{2}+a \right )^{9} b^{5}}-\frac {38143787 a^{5} x^{11}}{32768 \left (b \,x^{2}+a \right )^{9} b^{6}}-\frac {24013 a^{6} x^{9}}{18 \left (b \,x^{2}+a \right )^{9} b^{7}}-\frac {32405717 a^{7} x^{7}}{32768 \left (b \,x^{2}+a \right )^{9} b^{8}}-\frac {15137633 a^{8} x^{5}}{32768 \left (b \,x^{2}+a \right )^{9} b^{9}}-\frac {12201403 a^{9} x^{3}}{98304 \left (b \,x^{2}+a \right )^{9} b^{10}}-\frac {961255 a^{10} x}{65536 \left (b \,x^{2}+a \right )^{9} b^{11}}+\frac {x^{3}}{3 b^{10}}+\frac {1616615 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, b^{11}}-\frac {10 a x}{b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.25, size = 236, normalized size = 1.08 \[ -\frac {17890785 \, a^{2} b^{8} x^{17} + 122613150 \, a^{3} b^{7} x^{15} + 379867950 \, a^{4} b^{6} x^{13} + 686588166 \, a^{5} b^{5} x^{11} + 786857984 \, a^{6} b^{4} x^{9} + 583302906 \, a^{7} b^{3} x^{7} + 272477394 \, a^{8} b^{2} x^{5} + 73208418 \, a^{9} b x^{3} + 8651295 \, a^{10} x}{589824 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} + \frac {1616615 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{11}} + \frac {b x^{3} - 30 \, a x}{3 \, b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.40, size = 231, normalized size = 1.06 \[ \frac {x^3}{3\,b^{10}}-\frac {\frac {961255\,a^{10}\,x}{65536}+\frac {12201403\,a^9\,b\,x^3}{98304}+\frac {15137633\,a^8\,b^2\,x^5}{32768}+\frac {32405717\,a^7\,b^3\,x^7}{32768}+\frac {24013\,a^6\,b^4\,x^9}{18}+\frac {38143787\,a^5\,b^5\,x^{11}}{32768}+\frac {21103775\,a^4\,b^6\,x^{13}}{32768}+\frac {20435525\,a^3\,b^7\,x^{15}}{98304}+\frac {1987865\,a^2\,b^8\,x^{17}}{65536}}{a^9\,b^{11}+9\,a^8\,b^{12}\,x^2+36\,a^7\,b^{13}\,x^4+84\,a^6\,b^{14}\,x^6+126\,a^5\,b^{15}\,x^8+126\,a^4\,b^{16}\,x^{10}+84\,a^3\,b^{17}\,x^{12}+36\,a^2\,b^{18}\,x^{14}+9\,a\,b^{19}\,x^{16}+b^{20}\,x^{18}}+\frac {1616615\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,b^{23/2}}-\frac {10\,a\,x}{b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.92, size = 299, normalized size = 1.37 \[ - \frac {10 a x}{b^{11}} - \frac {1616615 \sqrt {- \frac {a^{3}}{b^{23}}} \log {\left (x - \frac {b^{11} \sqrt {- \frac {a^{3}}{b^{23}}}}{a} \right )}}{131072} + \frac {1616615 \sqrt {- \frac {a^{3}}{b^{23}}} \log {\left (x + \frac {b^{11} \sqrt {- \frac {a^{3}}{b^{23}}}}{a} \right )}}{131072} + \frac {- 8651295 a^{10} x - 73208418 a^{9} b x^{3} - 272477394 a^{8} b^{2} x^{5} - 583302906 a^{7} b^{3} x^{7} - 786857984 a^{6} b^{4} x^{9} - 686588166 a^{5} b^{5} x^{11} - 379867950 a^{4} b^{6} x^{13} - 122613150 a^{3} b^{7} x^{15} - 17890785 a^{2} b^{8} x^{17}}{589824 a^{9} b^{11} + 5308416 a^{8} b^{12} x^{2} + 21233664 a^{7} b^{13} x^{4} + 49545216 a^{6} b^{14} x^{6} + 74317824 a^{5} b^{15} x^{8} + 74317824 a^{4} b^{16} x^{10} + 49545216 a^{3} b^{17} x^{12} + 21233664 a^{2} b^{18} x^{14} + 5308416 a b^{19} x^{16} + 589824 b^{20} x^{18}} + \frac {x^{3}}{3 b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________