Optimal. Leaf size=184 \[ -\frac{9 b}{2 a^{10} \left (a+b x^2\right )}-\frac{2 b}{a^9 \left (a+b x^2\right )^2}-\frac{7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac{3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac{b}{2 a^6 \left (a+b x^2\right )^5}-\frac{b}{3 a^5 \left (a+b x^2\right )^6}-\frac{3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac{b}{8 a^3 \left (a+b x^2\right )^8}-\frac{b}{18 a^2 \left (a+b x^2\right )^9}+\frac{5 b \log \left (a+b x^2\right )}{a^{11}}-\frac{10 b \log (x)}{a^{11}}-\frac{1}{2 a^{10} x^2} \]
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Rubi [A] time = 0.185709, antiderivative size = 184, 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} \[ -\frac{9 b}{2 a^{10} \left (a+b x^2\right )}-\frac{2 b}{a^9 \left (a+b x^2\right )^2}-\frac{7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac{3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac{b}{2 a^6 \left (a+b x^2\right )^5}-\frac{b}{3 a^5 \left (a+b x^2\right )^6}-\frac{3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac{b}{8 a^3 \left (a+b x^2\right )^8}-\frac{b}{18 a^2 \left (a+b x^2\right )^9}+\frac{5 b \log \left (a+b x^2\right )}{a^{11}}-\frac{10 b \log (x)}{a^{11}}-\frac{1}{2 a^{10} x^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^{10} x^2}-\frac{10 b}{a^{11} x}+\frac{b^2}{a^2 (a+b x)^{10}}+\frac{2 b^2}{a^3 (a+b x)^9}+\frac{3 b^2}{a^4 (a+b x)^8}+\frac{4 b^2}{a^5 (a+b x)^7}+\frac{5 b^2}{a^6 (a+b x)^6}+\frac{6 b^2}{a^7 (a+b x)^5}+\frac{7 b^2}{a^8 (a+b x)^4}+\frac{8 b^2}{a^9 (a+b x)^3}+\frac{9 b^2}{a^{10} (a+b x)^2}+\frac{10 b^2}{a^{11} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a^{10} x^2}-\frac{b}{18 a^2 \left (a+b x^2\right )^9}-\frac{b}{8 a^3 \left (a+b x^2\right )^8}-\frac{3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac{b}{3 a^5 \left (a+b x^2\right )^6}-\frac{b}{2 a^6 \left (a+b x^2\right )^5}-\frac{3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac{7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac{2 b}{a^9 \left (a+b x^2\right )^2}-\frac{9 b}{2 a^{10} \left (a+b x^2\right )}-\frac{10 b \log (x)}{a^{11}}+\frac{5 b \log \left (a+b x^2\right )}{a^{11}}\\ \end{align*}
Mathematica [A] time = 0.120819, size = 136, normalized size = 0.74 \[ -\frac{\frac{a \left (80220 a^2 b^7 x^{14}+173250 a^3 b^6 x^{12}+236754 a^4 b^5 x^{10}+210756 a^5 b^4 x^8+120564 a^6 b^3 x^6+41481 a^7 b^2 x^4+7129 a^8 b x^2+252 a^9+21420 a b^8 x^{16}+2520 b^9 x^{18}\right )}{x^2 \left (a+b x^2\right )^9}-2520 b \log \left (a+b x^2\right )+5040 b \log (x)}{504 a^{11}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 167, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{10}{x}^{2}}}-{\frac{b}{18\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{b}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{3\,b}{14\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{7}}}-{\frac{b}{3\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{6}}}-{\frac{b}{2\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{3\,b}{4\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{7\,b}{6\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{3}}}-2\,{\frac{b}{{a}^{9} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,b}{2\,{a}^{10} \left ( b{x}^{2}+a \right ) }}-10\,{\frac{b\ln \left ( x \right ) }{{a}^{11}}}+5\,{\frac{b\ln \left ( b{x}^{2}+a \right ) }{{a}^{11}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26732, size = 312, normalized size = 1.7 \begin{align*} -\frac{2520 \, b^{9} x^{18} + 21420 \, a b^{8} x^{16} + 80220 \, a^{2} b^{7} x^{14} + 173250 \, a^{3} b^{6} x^{12} + 236754 \, a^{4} b^{5} x^{10} + 210756 \, a^{5} b^{4} x^{8} + 120564 \, a^{6} b^{3} x^{6} + 41481 \, a^{7} b^{2} x^{4} + 7129 \, a^{8} b x^{2} + 252 \, a^{9}}{504 \,{\left (a^{10} b^{9} x^{20} + 9 \, a^{11} b^{8} x^{18} + 36 \, a^{12} b^{7} x^{16} + 84 \, a^{13} b^{6} x^{14} + 126 \, a^{14} b^{5} x^{12} + 126 \, a^{15} b^{4} x^{10} + 84 \, a^{16} b^{3} x^{8} + 36 \, a^{17} b^{2} x^{6} + 9 \, a^{18} b x^{4} + a^{19} x^{2}\right )}} + \frac{5 \, b \log \left (b x^{2} + a\right )}{a^{11}} - \frac{5 \, b \log \left (x^{2}\right )}{a^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.38937, size = 996, normalized size = 5.41 \begin{align*} -\frac{2520 \, a b^{9} x^{18} + 21420 \, a^{2} b^{8} x^{16} + 80220 \, a^{3} b^{7} x^{14} + 173250 \, a^{4} b^{6} x^{12} + 236754 \, a^{5} b^{5} x^{10} + 210756 \, a^{6} b^{4} x^{8} + 120564 \, a^{7} b^{3} x^{6} + 41481 \, a^{8} b^{2} x^{4} + 7129 \, a^{9} b x^{2} + 252 \, a^{10} - 2520 \,{\left (b^{10} x^{20} + 9 \, a b^{9} x^{18} + 36 \, a^{2} b^{8} x^{16} + 84 \, a^{3} b^{7} x^{14} + 126 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 84 \, a^{6} b^{4} x^{8} + 36 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{4} + a^{9} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 5040 \,{\left (b^{10} x^{20} + 9 \, a b^{9} x^{18} + 36 \, a^{2} b^{8} x^{16} + 84 \, a^{3} b^{7} x^{14} + 126 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 84 \, a^{6} b^{4} x^{8} + 36 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{4} + a^{9} b x^{2}\right )} \log \left (x\right )}{504 \,{\left (a^{11} b^{9} x^{20} + 9 \, a^{12} b^{8} x^{18} + 36 \, a^{13} b^{7} x^{16} + 84 \, a^{14} b^{6} x^{14} + 126 \, a^{15} b^{5} x^{12} + 126 \, a^{16} b^{4} x^{10} + 84 \, a^{17} b^{3} x^{8} + 36 \, a^{18} b^{2} x^{6} + 9 \, a^{19} b x^{4} + a^{20} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 168.877, size = 243, normalized size = 1.32 \begin{align*} - \frac{252 a^{9} + 7129 a^{8} b x^{2} + 41481 a^{7} b^{2} x^{4} + 120564 a^{6} b^{3} x^{6} + 210756 a^{5} b^{4} x^{8} + 236754 a^{4} b^{5} x^{10} + 173250 a^{3} b^{6} x^{12} + 80220 a^{2} b^{7} x^{14} + 21420 a b^{8} x^{16} + 2520 b^{9} x^{18}}{504 a^{19} x^{2} + 4536 a^{18} b x^{4} + 18144 a^{17} b^{2} x^{6} + 42336 a^{16} b^{3} x^{8} + 63504 a^{15} b^{4} x^{10} + 63504 a^{14} b^{5} x^{12} + 42336 a^{13} b^{6} x^{14} + 18144 a^{12} b^{7} x^{16} + 4536 a^{11} b^{8} x^{18} + 504 a^{10} b^{9} x^{20}} - \frac{10 b \log{\left (x \right )}}{a^{11}} + \frac{5 b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.94949, size = 215, normalized size = 1.17 \begin{align*} -\frac{5 \, b \log \left (x^{2}\right )}{a^{11}} + \frac{5 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{11}} + \frac{10 \, b x^{2} - a}{2 \, a^{11} x^{2}} - \frac{7129 \, b^{10} x^{18} + 66429 \, a b^{9} x^{16} + 275796 \, a^{2} b^{8} x^{14} + 669984 \, a^{3} b^{7} x^{12} + 1050336 \, a^{4} b^{6} x^{10} + 1103256 \, a^{5} b^{5} x^{8} + 777840 \, a^{6} b^{4} x^{6} + 356040 \, a^{7} b^{3} x^{4} + 96570 \, a^{8} b^{2} x^{2} + 11990 \, a^{9} b}{504 \,{\left (b x^{2} + a\right )}^{9} a^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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