Optimal. Leaf size=117 \[ \frac{8 b^2 \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^4 x}-\frac{4 b \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} (6 A b-7 a B)}{35 a^2 x^5}-\frac{A \sqrt{a+b x^2}}{7 a x^7} \]
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Rubi [A] time = 0.0473258, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ \frac{8 b^2 \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^4 x}-\frac{4 b \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} (6 A b-7 a B)}{35 a^2 x^5}-\frac{A \sqrt{a+b x^2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 453
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^8 \sqrt{a+b x^2}} \, dx &=-\frac{A \sqrt{a+b x^2}}{7 a x^7}-\frac{(6 A b-7 a B) \int \frac{1}{x^6 \sqrt{a+b x^2}} \, dx}{7 a}\\ &=-\frac{A \sqrt{a+b x^2}}{7 a x^7}+\frac{(6 A b-7 a B) \sqrt{a+b x^2}}{35 a^2 x^5}+\frac{(4 b (6 A b-7 a B)) \int \frac{1}{x^4 \sqrt{a+b x^2}} \, dx}{35 a^2}\\ &=-\frac{A \sqrt{a+b x^2}}{7 a x^7}+\frac{(6 A b-7 a B) \sqrt{a+b x^2}}{35 a^2 x^5}-\frac{4 b (6 A b-7 a B) \sqrt{a+b x^2}}{105 a^3 x^3}-\frac{\left (8 b^2 (6 A b-7 a B)\right ) \int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx}{105 a^3}\\ &=-\frac{A \sqrt{a+b x^2}}{7 a x^7}+\frac{(6 A b-7 a B) \sqrt{a+b x^2}}{35 a^2 x^5}-\frac{4 b (6 A b-7 a B) \sqrt{a+b x^2}}{105 a^3 x^3}+\frac{8 b^2 (6 A b-7 a B) \sqrt{a+b x^2}}{105 a^4 x}\\ \end{align*}
Mathematica [A] time = 0.0289058, size = 84, normalized size = 0.72 \[ \frac{\left (\frac{b x^2}{a}+1\right ) \left (3 a^2-4 a b x^2+8 b^2 x^4\right ) (6 A b-7 a B)}{105 a^3 x^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 83, normalized size = 0.7 \begin{align*} -{\frac{-48\,A{b}^{3}{x}^{6}+56\,Ba{b}^{2}{x}^{6}+24\,Aa{b}^{2}{x}^{4}-28\,B{a}^{2}b{x}^{4}-18\,A{a}^{2}b{x}^{2}+21\,B{a}^{3}{x}^{2}+15\,A{a}^{3}}{105\,{x}^{7}{a}^{4}}\sqrt{b{x}^{2}+a}} \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 = 1.68952, size = 185, normalized size = 1.58 \begin{align*} -\frac{{\left (8 \,{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{6} - 4 \,{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 3 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, a^{4} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.31546, size = 819, normalized size = 7. \begin{align*} - \frac{5 A a^{6} b^{\frac{19}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac{9 A a^{5} b^{\frac{21}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac{5 A a^{4} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac{5 A a^{3} b^{\frac{25}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac{30 A a^{2} b^{\frac{27}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac{40 A a b^{\frac{29}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac{16 A b^{\frac{31}{2}} x^{12} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac{3 B a^{4} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{2 B a^{3} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{3 B a^{2} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{12 B a b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{8 B b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11841, size = 313, normalized size = 2.68 \begin{align*} \frac{16 \,{\left (70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B b^{\frac{5}{2}} - 175 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a b^{\frac{5}{2}} + 210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A b^{\frac{7}{2}} + 147 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{2} b^{\frac{5}{2}} - 126 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a b^{\frac{7}{2}} - 49 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{3} b^{\frac{5}{2}} + 42 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{2} b^{\frac{7}{2}} + 7 \, B a^{4} b^{\frac{5}{2}} - 6 \, A a^{3} b^{\frac{7}{2}}\right )}}{105 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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