Optimal. Leaf size=166 \[ \frac{1}{2 a^9 \left (a+b x^2\right )}+\frac{1}{4 a^8 \left (a+b x^2\right )^2}+\frac{1}{6 a^7 \left (a+b x^2\right )^3}+\frac{1}{8 a^6 \left (a+b x^2\right )^4}+\frac{1}{10 a^5 \left (a+b x^2\right )^5}+\frac{1}{12 a^4 \left (a+b x^2\right )^6}+\frac{1}{14 a^3 \left (a+b x^2\right )^7}+\frac{1}{16 a^2 \left (a+b x^2\right )^8}-\frac{\log \left (a+b x^2\right )}{2 a^{10}}+\frac{\log (x)}{a^{10}}+\frac{1}{18 a \left (a+b x^2\right )^9} \]
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Rubi [A] time = 0.128463, antiderivative size = 166, normalized size of antiderivative = 1., 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, 44} \[ \frac{1}{2 a^9 \left (a+b x^2\right )}+\frac{1}{4 a^8 \left (a+b x^2\right )^2}+\frac{1}{6 a^7 \left (a+b x^2\right )^3}+\frac{1}{8 a^6 \left (a+b x^2\right )^4}+\frac{1}{10 a^5 \left (a+b x^2\right )^5}+\frac{1}{12 a^4 \left (a+b x^2\right )^6}+\frac{1}{14 a^3 \left (a+b x^2\right )^7}+\frac{1}{16 a^2 \left (a+b x^2\right )^8}-\frac{\log \left (a+b x^2\right )}{2 a^{10}}+\frac{\log (x)}{a^{10}}+\frac{1}{18 a \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^{10} x}-\frac{b}{a (a+b x)^{10}}-\frac{b}{a^2 (a+b x)^9}-\frac{b}{a^3 (a+b x)^8}-\frac{b}{a^4 (a+b x)^7}-\frac{b}{a^5 (a+b x)^6}-\frac{b}{a^6 (a+b x)^5}-\frac{b}{a^7 (a+b x)^4}-\frac{b}{a^8 (a+b x)^3}-\frac{b}{a^9 (a+b x)^2}-\frac{b}{a^{10} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{18 a \left (a+b x^2\right )^9}+\frac{1}{16 a^2 \left (a+b x^2\right )^8}+\frac{1}{14 a^3 \left (a+b x^2\right )^7}+\frac{1}{12 a^4 \left (a+b x^2\right )^6}+\frac{1}{10 a^5 \left (a+b x^2\right )^5}+\frac{1}{8 a^6 \left (a+b x^2\right )^4}+\frac{1}{6 a^7 \left (a+b x^2\right )^3}+\frac{1}{4 a^8 \left (a+b x^2\right )^2}+\frac{1}{2 a^9 \left (a+b x^2\right )}+\frac{\log (x)}{a^{10}}-\frac{\log \left (a+b x^2\right )}{2 a^{10}}\\ \end{align*}
Mathematica [A] time = 0.0816384, size = 120, normalized size = 0.72 \[ \frac{\frac{a \left (80220 a^2 b^6 x^{12}+173250 a^3 b^5 x^{10}+236754 a^4 b^4 x^8+210756 a^5 b^3 x^6+120564 a^6 b^2 x^4+41481 a^7 b x^2+7129 a^8+21420 a b^7 x^{14}+2520 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}-2520 \log \left (a+b x^2\right )+5040 \log (x)}{5040 a^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 147, normalized size = 0.9 \begin{align*}{\frac{1}{18\,a \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{1}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{8}}}+{\frac{1}{14\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{7}}}+{\frac{1}{12\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{1}{10\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{1}{8\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{1}{6\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{1}{4\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{1}{2\,{a}^{9} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{10}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.35835, size = 289, normalized size = 1.74 \begin{align*} \frac{2520 \, b^{8} x^{16} + 21420 \, a b^{7} x^{14} + 80220 \, a^{2} b^{6} x^{12} + 173250 \, a^{3} b^{5} x^{10} + 236754 \, a^{4} b^{4} x^{8} + 210756 \, a^{5} b^{3} x^{6} + 120564 \, a^{6} b^{2} x^{4} + 41481 \, a^{7} b x^{2} + 7129 \, a^{8}}{5040 \,{\left (a^{9} b^{9} x^{18} + 9 \, a^{10} b^{8} x^{16} + 36 \, a^{11} b^{7} x^{14} + 84 \, a^{12} b^{6} x^{12} + 126 \, a^{13} b^{5} x^{10} + 126 \, a^{14} b^{4} x^{8} + 84 \, a^{15} b^{3} x^{6} + 36 \, a^{16} b^{2} x^{4} + 9 \, a^{17} b x^{2} + a^{18}\right )}} - \frac{\log \left (b x^{2} + a\right )}{2 \, a^{10}} + \frac{\log \left (x^{2}\right )}{2 \, a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.34803, size = 936, normalized size = 5.64 \begin{align*} \frac{2520 \, a b^{8} x^{16} + 21420 \, a^{2} b^{7} x^{14} + 80220 \, a^{3} b^{6} x^{12} + 173250 \, a^{4} b^{5} x^{10} + 236754 \, a^{5} b^{4} x^{8} + 210756 \, a^{6} b^{3} x^{6} + 120564 \, a^{7} b^{2} x^{4} + 41481 \, a^{8} b x^{2} + 7129 \, a^{9} - 2520 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (b x^{2} + a\right ) + 5040 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (x\right )}{5040 \,{\left (a^{10} b^{9} x^{18} + 9 \, a^{11} b^{8} x^{16} + 36 \, a^{12} b^{7} x^{14} + 84 \, a^{13} b^{6} x^{12} + 126 \, a^{14} b^{5} x^{10} + 126 \, a^{15} b^{4} x^{8} + 84 \, a^{16} b^{3} x^{6} + 36 \, a^{17} b^{2} x^{4} + 9 \, a^{18} b x^{2} + a^{19}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 77.2405, size = 223, normalized size = 1.34 \begin{align*} \frac{7129 a^{8} + 41481 a^{7} b x^{2} + 120564 a^{6} b^{2} x^{4} + 210756 a^{5} b^{3} x^{6} + 236754 a^{4} b^{4} x^{8} + 173250 a^{3} b^{5} x^{10} + 80220 a^{2} b^{6} x^{12} + 21420 a b^{7} x^{14} + 2520 b^{8} x^{16}}{5040 a^{18} + 45360 a^{17} b x^{2} + 181440 a^{16} b^{2} x^{4} + 423360 a^{15} b^{3} x^{6} + 635040 a^{14} b^{4} x^{8} + 635040 a^{13} b^{5} x^{10} + 423360 a^{12} b^{6} x^{12} + 181440 a^{11} b^{7} x^{14} + 45360 a^{10} b^{8} x^{16} + 5040 a^{9} b^{9} x^{18}} + \frac{\log{\left (x \right )}}{a^{10}} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.27445, size = 184, normalized size = 1.11 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, a^{10}} - \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{10}} + \frac{7129 \, b^{9} x^{18} + 66681 \, a b^{8} x^{16} + 278064 \, a^{2} b^{7} x^{14} + 679056 \, a^{3} b^{6} x^{12} + 1071504 \, a^{4} b^{5} x^{10} + 1135008 \, a^{5} b^{4} x^{8} + 809592 \, a^{6} b^{3} x^{6} + 377208 \, a^{7} b^{2} x^{4} + 105642 \, a^{8} b x^{2} + 14258 \, a^{9}}{5040 \,{\left (b x^{2} + a\right )}^{9} a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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