Optimal. Leaf size=53 \[ \frac{\left (a+b x^2\right )^{3/2} (2 A b-5 a B)}{15 a^2 x^3}-\frac{A \left (a+b x^2\right )^{3/2}}{5 a x^5} \]
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Rubi [A] time = 0.0211285, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {453, 264} \[ \frac{\left (a+b x^2\right )^{3/2} (2 A b-5 a B)}{15 a^2 x^3}-\frac{A \left (a+b x^2\right )^{3/2}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^6} \, dx &=-\frac{A \left (a+b x^2\right )^{3/2}}{5 a x^5}-\frac{(2 A b-5 a B) \int \frac{\sqrt{a+b x^2}}{x^4} \, dx}{5 a}\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{5 a x^5}+\frac{(2 A b-5 a B) \left (a+b x^2\right )^{3/2}}{15 a^2 x^3}\\ \end{align*}
Mathematica [A] time = 0.0131374, size = 40, normalized size = 0.75 \[ -\frac{\left (a+b x^2\right )^{3/2} \left (3 a A+5 a B x^2-2 A b x^2\right )}{15 a^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 0.7 \begin{align*} -{\frac{-2\,Ab{x}^{2}+5\,Ba{x}^{2}+3\,Aa}{15\,{x}^{5}{a}^{2}} \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 = 1.54684, size = 126, normalized size = 2.38 \begin{align*} -\frac{{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} +{\left (5 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, a^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.32608, size = 119, normalized size = 2.25 \begin{align*} - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a x^{2}} + \frac{2 A b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{2}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13378, size = 313, normalized size = 5.91 \begin{align*} \frac{2 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B b^{\frac{3}{2}} - 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a b^{\frac{3}{2}} + 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A b^{\frac{5}{2}} + 20 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{2} b^{\frac{3}{2}} + 10 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a b^{\frac{5}{2}} - 10 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{3} b^{\frac{3}{2}} + 10 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{2} b^{\frac{5}{2}} + 5 \, B a^{4} b^{\frac{3}{2}} - 2 \, A a^{3} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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