Optimal. Leaf size=91 \[ -\frac{a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac{a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac{3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac{a}{3 b^5 \left (a+b x^2\right )^6}-\frac{1}{10 b^5 \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.0676645, antiderivative size = 91, 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, 43} \[ -\frac{a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac{a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac{3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac{a}{3 b^5 \left (a+b x^2\right )^6}-\frac{1}{10 b^5 \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^4}{b^4 (a+b x)^{10}}-\frac{4 a^3}{b^4 (a+b x)^9}+\frac{6 a^2}{b^4 (a+b x)^8}-\frac{4 a}{b^4 (a+b x)^7}+\frac{1}{b^4 (a+b x)^6}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac{a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac{3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac{a}{3 b^5 \left (a+b x^2\right )^6}-\frac{1}{10 b^5 \left (a+b x^2\right )^5}\\ \end{align*}
Mathematica [A] time = 0.0150951, size = 57, normalized size = 0.63 \[ -\frac{36 a^2 b^2 x^4+9 a^3 b x^2+a^4+84 a b^3 x^6+126 b^4 x^8}{1260 b^5 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 82, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{18\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{{a}^{3}}{4\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{3\,{a}^{2}}{7\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{7}}}+{\frac{a}{3\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{6}}}-{\frac{1}{10\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.25224, size = 197, normalized size = 2.16 \begin{align*} -\frac{126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \,{\left (b^{14} x^{18} + 9 \, a b^{13} x^{16} + 36 \, a^{2} b^{12} x^{14} + 84 \, a^{3} b^{11} x^{12} + 126 \, a^{4} b^{10} x^{10} + 126 \, a^{5} b^{9} x^{8} + 84 \, a^{6} b^{8} x^{6} + 36 \, a^{7} b^{7} x^{4} + 9 \, a^{8} b^{6} x^{2} + a^{9} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21416, size = 323, normalized size = 3.55 \begin{align*} -\frac{126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \,{\left (b^{14} x^{18} + 9 \, a b^{13} x^{16} + 36 \, a^{2} b^{12} x^{14} + 84 \, a^{3} b^{11} x^{12} + 126 \, a^{4} b^{10} x^{10} + 126 \, a^{5} b^{9} x^{8} + 84 \, a^{6} b^{8} x^{6} + 36 \, a^{7} b^{7} x^{4} + 9 \, a^{8} b^{6} x^{2} + a^{9} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.24991, size = 155, normalized size = 1.7 \begin{align*} - \frac{a^{4} + 9 a^{3} b x^{2} + 36 a^{2} b^{2} x^{4} + 84 a b^{3} x^{6} + 126 b^{4} x^{8}}{1260 a^{9} b^{5} + 11340 a^{8} b^{6} x^{2} + 45360 a^{7} b^{7} x^{4} + 105840 a^{6} b^{8} x^{6} + 158760 a^{5} b^{9} x^{8} + 158760 a^{4} b^{10} x^{10} + 105840 a^{3} b^{11} x^{12} + 45360 a^{2} b^{12} x^{14} + 11340 a b^{13} x^{16} + 1260 b^{14} x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.84398, size = 74, normalized size = 0.81 \begin{align*} -\frac{126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \,{\left (b x^{2} + a\right )}^{9} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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