Optimal. Leaf size=117 \[ \frac{8 b^2 \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{315 a^4 x^3}-\frac{4 b \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{105 a^3 x^5}+\frac{\left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{21 a^2 x^7}-\frac{A \left (a+b x^2\right )^{3/2}}{9 a x^9} \]
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Rubi [A] time = 0.0554634, 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} \[ \frac{8 b^2 \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{315 a^4 x^3}-\frac{4 b \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{105 a^3 x^5}+\frac{\left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{21 a^2 x^7}-\frac{A \left (a+b x^2\right )^{3/2}}{9 a x^9} \]
Antiderivative was successfully verified.
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Rule 453
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx &=-\frac{A \left (a+b x^2\right )^{3/2}}{9 a x^9}-\frac{(6 A b-9 a B) \int \frac{\sqrt{a+b x^2}}{x^8} \, dx}{9 a}\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac{(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}+\frac{(4 b (2 A b-3 a B)) \int \frac{\sqrt{a+b x^2}}{x^6} \, dx}{21 a^2}\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac{(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac{4 b (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}-\frac{\left (8 b^2 (2 A b-3 a B)\right ) \int \frac{\sqrt{a+b x^2}}{x^4} \, dx}{105 a^3}\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac{(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac{4 b (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}+\frac{8 b^2 (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{315 a^4 x^3}\\ \end{align*}
Mathematica [A] time = 0.0382665, size = 81, normalized size = 0.69 \[ \frac{\left (a+b x^2\right )^{3/2} \left (6 a^2 b x^2 \left (5 A+6 B x^2\right )-5 a^3 \left (7 A+9 B x^2\right )-24 a b^2 x^4 \left (A+B x^2\right )+16 A b^3 x^6\right )}{315 a^4 x^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 83, normalized size = 0.7 \begin{align*} -{\frac{-16\,A{b}^{3}{x}^{6}+24\,Ba{b}^{2}{x}^{6}+24\,Aa{b}^{2}{x}^{4}-36\,B{a}^{2}b{x}^{4}-30\,A{a}^{2}b{x}^{2}+45\,B{a}^{3}{x}^{2}+35\,A{a}^{3}}{315\,{x}^{9}{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{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 = 2.17845, size = 231, normalized size = 1.97 \begin{align*} -\frac{{\left (8 \,{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{8} - 4 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + 35 \, A a^{4} + 3 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4} + 5 \,{\left (9 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{315 \, a^{4} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.13412, size = 957, normalized size = 8.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16652, size = 464, normalized size = 3.97 \begin{align*} \frac{16 \,{\left (210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B b^{\frac{7}{2}} - 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a b^{\frac{7}{2}} + 630 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} A b^{\frac{9}{2}} + 63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{2} b^{\frac{7}{2}} + 378 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a b^{\frac{9}{2}} - 42 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{3} b^{\frac{7}{2}} + 168 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a^{2} b^{\frac{9}{2}} + 108 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{4} b^{\frac{7}{2}} - 72 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{3} b^{\frac{9}{2}} - 27 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{5} b^{\frac{7}{2}} + 18 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{4} b^{\frac{9}{2}} + 3 \, B a^{6} b^{\frac{7}{2}} - 2 \, A a^{5} b^{\frac{9}{2}}\right )}}{315 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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