Optimal. Leaf size=216 \[ -\frac {a^{12}}{18 b^{13} \left (a+b x^2\right )^9}+\frac {3 a^{11}}{4 b^{13} \left (a+b x^2\right )^8}-\frac {33 a^{10}}{7 b^{13} \left (a+b x^2\right )^7}+\frac {55 a^9}{3 b^{13} \left (a+b x^2\right )^6}-\frac {99 a^8}{2 b^{13} \left (a+b x^2\right )^5}+\frac {99 a^7}{b^{13} \left (a+b x^2\right )^4}-\frac {154 a^6}{b^{13} \left (a+b x^2\right )^3}+\frac {198 a^5}{b^{13} \left (a+b x^2\right )^2}-\frac {495 a^4}{2 b^{13} \left (a+b x^2\right )}-\frac {110 a^3 \log \left (a+b x^2\right )}{b^{13}}+\frac {55 a^2 x^2}{2 b^{12}}-\frac {5 a x^4}{2 b^{11}}+\frac {x^6}{6 b^{10}} \]
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Rubi [A] time = 0.26, antiderivative size = 216, 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} \[ -\frac {a^{12}}{18 b^{13} \left (a+b x^2\right )^9}+\frac {3 a^{11}}{4 b^{13} \left (a+b x^2\right )^8}-\frac {33 a^{10}}{7 b^{13} \left (a+b x^2\right )^7}+\frac {55 a^9}{3 b^{13} \left (a+b x^2\right )^6}-\frac {99 a^8}{2 b^{13} \left (a+b x^2\right )^5}+\frac {99 a^7}{b^{13} \left (a+b x^2\right )^4}-\frac {154 a^6}{b^{13} \left (a+b x^2\right )^3}+\frac {198 a^5}{b^{13} \left (a+b x^2\right )^2}-\frac {495 a^4}{2 b^{13} \left (a+b x^2\right )}+\frac {55 a^2 x^2}{2 b^{12}}-\frac {110 a^3 \log \left (a+b x^2\right )}{b^{13}}-\frac {5 a x^4}{2 b^{11}}+\frac {x^6}{6 b^{10}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^{12}}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {55 a^2}{b^{12}}-\frac {10 a x}{b^{11}}+\frac {x^2}{b^{10}}+\frac {a^{12}}{b^{12} (a+b x)^{10}}-\frac {12 a^{11}}{b^{12} (a+b x)^9}+\frac {66 a^{10}}{b^{12} (a+b x)^8}-\frac {220 a^9}{b^{12} (a+b x)^7}+\frac {495 a^8}{b^{12} (a+b x)^6}-\frac {792 a^7}{b^{12} (a+b x)^5}+\frac {924 a^6}{b^{12} (a+b x)^4}-\frac {792 a^5}{b^{12} (a+b x)^3}+\frac {495 a^4}{b^{12} (a+b x)^2}-\frac {220 a^3}{b^{12} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {55 a^2 x^2}{2 b^{12}}-\frac {5 a x^4}{2 b^{11}}+\frac {x^6}{6 b^{10}}-\frac {a^{12}}{18 b^{13} \left (a+b x^2\right )^9}+\frac {3 a^{11}}{4 b^{13} \left (a+b x^2\right )^8}-\frac {33 a^{10}}{7 b^{13} \left (a+b x^2\right )^7}+\frac {55 a^9}{3 b^{13} \left (a+b x^2\right )^6}-\frac {99 a^8}{2 b^{13} \left (a+b x^2\right )^5}+\frac {99 a^7}{b^{13} \left (a+b x^2\right )^4}-\frac {154 a^6}{b^{13} \left (a+b x^2\right )^3}+\frac {198 a^5}{b^{13} \left (a+b x^2\right )^2}-\frac {495 a^4}{2 b^{13} \left (a+b x^2\right )}-\frac {110 a^3 \log \left (a+b x^2\right )}{b^{13}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 169, normalized size = 0.78 \[ -\frac {35201 a^{12}+289089 a^{11} b x^2+1031616 a^{10} b^2 x^4+2074464 a^9 b^3 x^6+2529576 a^8 b^4 x^8+1831032 a^7 b^5 x^{10}+638568 a^6 b^6 x^{12}-58968 a^5 b^7 x^{14}-139482 a^4 b^8 x^{16}-43218 a^3 b^9 x^{18}+27720 a^3 \left (a+b x^2\right )^9 \log \left (a+b x^2\right )-2772 a^2 b^{10} x^{20}+252 a b^{11} x^{22}-42 b^{12} x^{24}}{252 b^{13} \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 346, normalized size = 1.60 \[ \frac {42 \, b^{12} x^{24} - 252 \, a b^{11} x^{22} + 2772 \, a^{2} b^{10} x^{20} + 43218 \, a^{3} b^{9} x^{18} + 139482 \, a^{4} b^{8} x^{16} + 58968 \, a^{5} b^{7} x^{14} - 638568 \, a^{6} b^{6} x^{12} - 1831032 \, a^{7} b^{5} x^{10} - 2529576 \, a^{8} b^{4} x^{8} - 2074464 \, a^{9} b^{3} x^{6} - 1031616 \, a^{10} b^{2} x^{4} - 289089 \, a^{11} b x^{2} - 35201 \, a^{12} - 27720 \, {\left (a^{3} b^{9} x^{18} + 9 \, a^{4} b^{8} x^{16} + 36 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 126 \, a^{7} b^{5} x^{10} + 126 \, a^{8} b^{4} x^{8} + 84 \, a^{9} b^{3} x^{6} + 36 \, a^{10} b^{2} x^{4} + 9 \, a^{11} b x^{2} + a^{12}\right )} \log \left (b x^{2} + a\right )}{252 \, {\left (b^{22} x^{18} + 9 \, a b^{21} x^{16} + 36 \, a^{2} b^{20} x^{14} + 84 \, a^{3} b^{19} x^{12} + 126 \, a^{4} b^{18} x^{10} + 126 \, a^{5} b^{17} x^{8} + 84 \, a^{6} b^{16} x^{6} + 36 \, a^{7} b^{15} x^{4} + 9 \, a^{8} b^{14} x^{2} + a^{9} b^{13}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 168, normalized size = 0.78 \[ -\frac {110 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{13}} + \frac {78419 \, a^{3} b^{9} x^{18} + 643401 \, a^{4} b^{8} x^{16} + 2374020 \, a^{5} b^{7} x^{14} + 5151300 \, a^{6} b^{6} x^{12} + 7227990 \, a^{7} b^{5} x^{10} + 6791400 \, a^{8} b^{4} x^{8} + 4268880 \, a^{9} b^{3} x^{6} + 1729728 \, a^{10} b^{2} x^{4} + 409752 \, a^{11} b x^{2} + 43218 \, a^{12}}{252 \, {\left (b x^{2} + a\right )}^{9} b^{13}} + \frac {b^{20} x^{6} - 15 \, a b^{19} x^{4} + 165 \, a^{2} b^{18} x^{2}}{6 \, b^{30}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 199, normalized size = 0.92 \[ -\frac {a^{12}}{18 \left (b \,x^{2}+a \right )^{9} b^{13}}+\frac {3 a^{11}}{4 \left (b \,x^{2}+a \right )^{8} b^{13}}-\frac {33 a^{10}}{7 \left (b \,x^{2}+a \right )^{7} b^{13}}+\frac {55 a^{9}}{3 \left (b \,x^{2}+a \right )^{6} b^{13}}+\frac {x^{6}}{6 b^{10}}-\frac {99 a^{8}}{2 \left (b \,x^{2}+a \right )^{5} b^{13}}+\frac {99 a^{7}}{\left (b \,x^{2}+a \right )^{4} b^{13}}-\frac {5 a \,x^{4}}{2 b^{11}}-\frac {154 a^{6}}{\left (b \,x^{2}+a \right )^{3} b^{13}}+\frac {198 a^{5}}{\left (b \,x^{2}+a \right )^{2} b^{13}}+\frac {55 a^{2} x^{2}}{2 b^{12}}-\frac {495 a^{4}}{2 \left (b \,x^{2}+a \right ) b^{13}}-\frac {110 a^{3} \ln \left (b \,x^{2}+a \right )}{b^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.62, size = 242, normalized size = 1.12 \[ -\frac {62370 \, a^{4} b^{8} x^{16} + 449064 \, a^{5} b^{7} x^{14} + 1435896 \, a^{6} b^{6} x^{12} + 2652804 \, a^{7} b^{5} x^{10} + 3089394 \, a^{8} b^{4} x^{8} + 2318316 \, a^{9} b^{3} x^{6} + 1093356 \, a^{10} b^{2} x^{4} + 296019 \, a^{11} b x^{2} + 35201 \, a^{12}}{252 \, {\left (b^{22} x^{18} + 9 \, a b^{21} x^{16} + 36 \, a^{2} b^{20} x^{14} + 84 \, a^{3} b^{19} x^{12} + 126 \, a^{4} b^{18} x^{10} + 126 \, a^{5} b^{17} x^{8} + 84 \, a^{6} b^{16} x^{6} + 36 \, a^{7} b^{15} x^{4} + 9 \, a^{8} b^{14} x^{2} + a^{9} b^{13}\right )}} - \frac {110 \, a^{3} \log \left (b x^{2} + a\right )}{b^{13}} + \frac {b^{2} x^{6} - 15 \, a b x^{4} + 165 \, a^{2} x^{2}}{6 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.29, size = 242, normalized size = 1.12 \[ \frac {x^6}{6\,b^{10}}-\frac {\frac {35201\,a^{12}}{252\,b}+\frac {32891\,a^{11}\,x^2}{28}+\frac {30371\,a^{10}\,b\,x^4}{7}+\frac {27599\,a^9\,b^2\,x^6}{3}+\frac {24519\,a^8\,b^3\,x^8}{2}+10527\,a^7\,b^4\,x^{10}+5698\,a^6\,b^5\,x^{12}+1782\,a^5\,b^6\,x^{14}+\frac {495\,a^4\,b^7\,x^{16}}{2}}{a^9\,b^{12}+9\,a^8\,b^{13}\,x^2+36\,a^7\,b^{14}\,x^4+84\,a^6\,b^{15}\,x^6+126\,a^5\,b^{16}\,x^8+126\,a^4\,b^{17}\,x^{10}+84\,a^3\,b^{18}\,x^{12}+36\,a^2\,b^{19}\,x^{14}+9\,a\,b^{20}\,x^{16}+b^{21}\,x^{18}}-\frac {5\,a\,x^4}{2\,b^{11}}-\frac {110\,a^3\,\ln \left (b\,x^2+a\right )}{b^{13}}+\frac {55\,a^2\,x^2}{2\,b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.15, size = 260, normalized size = 1.20 \[ - \frac {110 a^{3} \log {\left (a + b x^{2} \right )}}{b^{13}} + \frac {55 a^{2} x^{2}}{2 b^{12}} - \frac {5 a x^{4}}{2 b^{11}} + \frac {- 35201 a^{12} - 296019 a^{11} b x^{2} - 1093356 a^{10} b^{2} x^{4} - 2318316 a^{9} b^{3} x^{6} - 3089394 a^{8} b^{4} x^{8} - 2652804 a^{7} b^{5} x^{10} - 1435896 a^{6} b^{6} x^{12} - 449064 a^{5} b^{7} x^{14} - 62370 a^{4} b^{8} x^{16}}{252 a^{9} b^{13} + 2268 a^{8} b^{14} x^{2} + 9072 a^{7} b^{15} x^{4} + 21168 a^{6} b^{16} x^{6} + 31752 a^{5} b^{17} x^{8} + 31752 a^{4} b^{18} x^{10} + 21168 a^{3} b^{19} x^{12} + 9072 a^{2} b^{20} x^{14} + 2268 a b^{21} x^{16} + 252 b^{22} x^{18}} + \frac {x^{6}}{6 b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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