Optimal. Leaf size=92 \[ \frac{16 b^3 \left (a+b x^2\right )^{5/2}}{1155 a^4 x^5}-\frac{8 b^2 \left (a+b x^2\right )^{5/2}}{231 a^3 x^7}+\frac{2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}-\frac{\left (a+b x^2\right )^{5/2}}{11 a x^{11}} \]
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Rubi [A] time = 0.0287707, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{16 b^3 \left (a+b x^2\right )^{5/2}}{1155 a^4 x^5}-\frac{8 b^2 \left (a+b x^2\right )^{5/2}}{231 a^3 x^7}+\frac{2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}-\frac{\left (a+b x^2\right )^{5/2}}{11 a x^{11}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{x^{12}} \, dx &=-\frac{\left (a+b x^2\right )^{5/2}}{11 a x^{11}}-\frac{(6 b) \int \frac{\left (a+b x^2\right )^{3/2}}{x^{10}} \, dx}{11 a}\\ &=-\frac{\left (a+b x^2\right )^{5/2}}{11 a x^{11}}+\frac{2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}+\frac{\left (8 b^2\right ) \int \frac{\left (a+b x^2\right )^{3/2}}{x^8} \, dx}{33 a^2}\\ &=-\frac{\left (a+b x^2\right )^{5/2}}{11 a x^{11}}+\frac{2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}-\frac{8 b^2 \left (a+b x^2\right )^{5/2}}{231 a^3 x^7}-\frac{\left (16 b^3\right ) \int \frac{\left (a+b x^2\right )^{3/2}}{x^6} \, dx}{231 a^3}\\ &=-\frac{\left (a+b x^2\right )^{5/2}}{11 a x^{11}}+\frac{2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}-\frac{8 b^2 \left (a+b x^2\right )^{5/2}}{231 a^3 x^7}+\frac{16 b^3 \left (a+b x^2\right )^{5/2}}{1155 a^4 x^5}\\ \end{align*}
Mathematica [A] time = 0.0132142, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^2\right )^{5/2} \left (70 a^2 b x^2-105 a^3-40 a b^2 x^4+16 b^3 x^6\right )}{1155 a^4 x^{11}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-16\,{b}^{3}{x}^{6}+40\,a{b}^{2}{x}^{4}-70\,{a}^{2}b{x}^{2}+105\,{a}^{3}}{1155\,{x}^{11}{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63865, size = 163, normalized size = 1.77 \begin{align*} \frac{{\left (16 \, b^{5} x^{10} - 8 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 5 \, a^{3} b^{2} x^{4} - 140 \, a^{4} b x^{2} - 105 \, a^{5}\right )} \sqrt{b x^{2} + a}}{1155 \, a^{4} x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.66182, size = 648, normalized size = 7.04 \begin{align*} - \frac{105 a^{8} b^{\frac{19}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} - \frac{455 a^{7} b^{\frac{21}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} - \frac{740 a^{6} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} - \frac{534 a^{5} b^{\frac{25}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} - \frac{145 a^{4} b^{\frac{27}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} + \frac{5 a^{3} b^{\frac{29}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} + \frac{30 a^{2} b^{\frac{31}{2}} x^{12} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} + \frac{40 a b^{\frac{33}{2}} x^{14} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} + \frac{16 b^{\frac{35}{2}} x^{16} \sqrt{\frac{a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.87118, size = 297, normalized size = 3.23 \begin{align*} \frac{32 \,{\left (1155 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} b^{\frac{11}{2}} + 2079 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a b^{\frac{11}{2}} + 2541 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{2} b^{\frac{11}{2}} + 825 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{3} b^{\frac{11}{2}} + 165 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{4} b^{\frac{11}{2}} - 55 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{5} b^{\frac{11}{2}} + 11 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{6} b^{\frac{11}{2}} - a^{7} b^{\frac{11}{2}}\right )}}{1155 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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