Optimal. Leaf size=99 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac{\left (c+d x^2\right )^{3/2} \left (35 b^2 c^2-4 a d (7 b c-2 a d)\right )}{105 c^3 x^3}-\frac{2 a \left (c+d x^2\right )^{3/2} (7 b c-2 a d)}{35 c^2 x^5} \]
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Rubi [A] time = 0.0747952, antiderivative size = 100, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {462, 453, 264} \[ -\frac{\left (c+d x^2\right )^{3/2} \left (8 a^2 d^2-28 a b c d+35 b^2 c^2\right )}{105 c^3 x^3}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac{2 a \left (c+d x^2\right )^{3/2} (7 b c-2 a d)}{35 c^2 x^5} \]
Antiderivative was successfully verified.
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Rule 462
Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^8} \, dx &=-\frac{a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}+\frac{\int \frac{\left (2 a (7 b c-2 a d)+7 b^2 c x^2\right ) \sqrt{c+d x^2}}{x^6} \, dx}{7 c}\\ &=-\frac{a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac{2 a (7 b c-2 a d) \left (c+d x^2\right )^{3/2}}{35 c^2 x^5}-\frac{1}{35} \left (-35 b^2+\frac{4 a d (7 b c-2 a d)}{c^2}\right ) \int \frac{\sqrt{c+d x^2}}{x^4} \, dx\\ &=-\frac{a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac{2 a (7 b c-2 a d) \left (c+d x^2\right )^{3/2}}{35 c^2 x^5}-\frac{\left (35 b^2-\frac{4 a d (7 b c-2 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{105 c x^3}\\ \end{align*}
Mathematica [A] time = 0.0257616, size = 76, normalized size = 0.77 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (a^2 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+14 a b c x^2 \left (3 c-2 d x^2\right )+35 b^2 c^2 x^4\right )}{105 c^3 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 78, normalized size = 0.8 \begin{align*} -{\frac{8\,{a}^{2}{d}^{2}{x}^{4}-28\,abcd{x}^{4}+35\,{b}^{2}{c}^{2}{x}^{4}-12\,{a}^{2}cd{x}^{2}+42\,a{c}^{2}b{x}^{2}+15\,{a}^{2}{c}^{2}}{105\,{x}^{7}{c}^{3}} \left ( d{x}^{2}+c \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 = 1.8319, size = 238, normalized size = 2.4 \begin{align*} -\frac{{\left ({\left (35 \, b^{2} c^{2} d - 28 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{6} + 15 \, a^{2} c^{3} +{\left (35 \, b^{2} c^{3} + 14 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x^{4} + 3 \,{\left (14 \, a b c^{3} + a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{105 \, c^{3} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.62894, size = 510, normalized size = 5.15 \begin{align*} - \frac{15 a^{2} c^{5} d^{\frac{9}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{33 a^{2} c^{4} d^{\frac{11}{2}} x^{2} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{17 a^{2} c^{3} d^{\frac{13}{2}} x^{4} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{3 a^{2} c^{2} d^{\frac{15}{2}} x^{6} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{12 a^{2} c d^{\frac{17}{2}} x^{8} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{8 a^{2} d^{\frac{19}{2}} x^{10} \sqrt{\frac{c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac{2 a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{5 x^{4}} - \frac{2 a b d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c x^{2}} + \frac{4 a b d^{\frac{5}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c^{2}} - \frac{b^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{b^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15813, size = 662, normalized size = 6.69 \begin{align*} \frac{2 \,{\left (105 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{12} b^{2} d^{\frac{3}{2}} - 420 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{10} b^{2} c d^{\frac{3}{2}} + 420 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{10} a b d^{\frac{5}{2}} + 665 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} c^{2} d^{\frac{3}{2}} - 700 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a b c d^{\frac{5}{2}} + 560 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a^{2} d^{\frac{7}{2}} - 560 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c^{3} d^{\frac{3}{2}} + 280 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac{5}{2}} + 280 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a^{2} c d^{\frac{7}{2}} + 315 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{4} d^{\frac{3}{2}} - 168 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac{5}{2}} + 168 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac{7}{2}} - 140 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{5} d^{\frac{3}{2}} + 196 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac{5}{2}} - 56 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac{7}{2}} + 35 \, b^{2} c^{6} d^{\frac{3}{2}} - 28 \, a b c^{5} d^{\frac{5}{2}} + 8 \, a^{2} c^{4} d^{\frac{7}{2}}\right )}}{105 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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