Optimal. Leaf size=21 \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]
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Rubi [A] time = 0.0046399, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx &=-\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7}\\ \end{align*}
Mathematica [A] time = 0.0080252, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,a{x}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{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 [B] time = 1.58271, size = 100, normalized size = 4.76 \begin{align*} -\frac{{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a}}{7 \, a x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.695, size = 95, normalized size = 4.52 \begin{align*} - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{6}} - \frac{3 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{4}} - \frac{3 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{2}} - \frac{b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.08011, size = 153, normalized size = 7.29 \begin{align*} \frac{2 \,{\left (7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} b^{\frac{7}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{2} b^{\frac{7}{2}} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{4} b^{\frac{7}{2}} + a^{6} b^{\frac{7}{2}}\right )}}{7 \,{\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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