Optimal. Leaf size=53 \[ \frac{\left (a+b x^2\right )^{5/2} (2 A b-7 a B)}{35 a^2 x^5}-\frac{A \left (a+b x^2\right )^{5/2}}{7 a x^7} \]
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Rubi [A] time = 0.0221359, 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} \[ \frac{\left (a+b x^2\right )^{5/2} (2 A b-7 a B)}{35 a^2 x^5}-\frac{A \left (a+b x^2\right )^{5/2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^8} \, dx &=-\frac{A \left (a+b x^2\right )^{5/2}}{7 a x^7}-\frac{(2 A b-7 a B) \int \frac{\left (a+b x^2\right )^{3/2}}{x^6} \, dx}{7 a}\\ &=-\frac{A \left (a+b x^2\right )^{5/2}}{7 a x^7}+\frac{(2 A b-7 a B) \left (a+b x^2\right )^{5/2}}{35 a^2 x^5}\\ \end{align*}
Mathematica [A] time = 0.0170152, size = 40, normalized size = 0.75 \[ -\frac{\left (a+b x^2\right )^{5/2} \left (5 a A+7 a B x^2-2 A b x^2\right )}{35 a^2 x^7} \]
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}+7\,Ba{x}^{2}+5\,Aa}{35\,{x}^{7}{a}^{2}} \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 = 1.63191, size = 173, normalized size = 3.26 \begin{align*} -\frac{{\left ({\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} +{\left (14 \, B a^{2} b + A a b^{2}\right )} x^{4} + 5 \, A a^{3} +{\left (7 \, B a^{3} + 8 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{35 \, a^{2} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.67851, size = 518, normalized size = 9.77 \begin{align*} - \frac{15 A a^{6} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{33 A a^{5} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{17 A a^{4} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{3 A a^{3} b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{12 A a^{2} b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{8 A a b^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{A b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a x^{2}} + \frac{2 A b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{2}} - \frac{B a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{2 B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{2}} - \frac{B b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1331, size = 464, normalized size = 8.75 \begin{align*} \frac{2 \,{\left (35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B b^{\frac{5}{2}} - 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a b^{\frac{5}{2}} + 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} A b^{\frac{7}{2}} + 105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{2} b^{\frac{5}{2}} + 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a b^{\frac{7}{2}} - 140 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{3} b^{\frac{5}{2}} + 140 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a^{2} b^{\frac{7}{2}} + 77 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{4} b^{\frac{5}{2}} + 28 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{3} b^{\frac{7}{2}} - 14 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{5} b^{\frac{5}{2}} + 14 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{4} b^{\frac{7}{2}} + 7 \, B a^{6} b^{\frac{5}{2}} - 2 \, A a^{5} b^{\frac{7}{2}}\right )}}{35 \,{\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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