Optimal. Leaf size=99 \[ -\frac{a^2 \sqrt{c+d x^2}}{5 c x^5}-\frac{\sqrt{c+d x^2} \left (15 b^2 c^2-4 a d (5 b c-2 a d)\right )}{15 c^3 x}-\frac{2 a \sqrt{c+d x^2} (5 b c-2 a d)}{15 c^2 x^3} \]
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Rubi [A] time = 0.0717884, 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} \[ -\frac{\sqrt{c+d x^2} \left (8 a^2 d^2-20 a b c d+15 b^2 c^2\right )}{15 c^3 x}-\frac{a^2 \sqrt{c+d x^2}}{5 c x^5}-\frac{2 a \sqrt{c+d x^2} (5 b c-2 a d)}{15 c^2 x^3} \]
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}{x^6 \sqrt{c+d x^2}} \, dx &=-\frac{a^2 \sqrt{c+d x^2}}{5 c x^5}+\frac{\int \frac{2 a (5 b c-2 a d)+5 b^2 c x^2}{x^4 \sqrt{c+d x^2}} \, dx}{5 c}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{5 c x^5}-\frac{2 a (5 b c-2 a d) \sqrt{c+d x^2}}{15 c^2 x^3}-\frac{1}{15} \left (-15 b^2+\frac{4 a d (5 b c-2 a d)}{c^2}\right ) \int \frac{1}{x^2 \sqrt{c+d x^2}} \, dx\\ &=-\frac{a^2 \sqrt{c+d x^2}}{5 c x^5}-\frac{2 a (5 b c-2 a d) \sqrt{c+d x^2}}{15 c^2 x^3}-\frac{\left (15 b^2-\frac{4 a d (5 b c-2 a d)}{c^2}\right ) \sqrt{c+d x^2}}{15 c x}\\ \end{align*}
Mathematica [A] time = 0.0225581, size = 74, normalized size = 0.75 \[ -\frac{\sqrt{c+d x^2} \left (a^2 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )+10 a b c x^2 \left (c-2 d x^2\right )+15 b^2 c^2 x^4\right )}{15 c^3 x^5} \]
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}-20\,abcd{x}^{4}+15\,{b}^{2}{c}^{2}{x}^{4}-4\,{a}^{2}cd{x}^{2}+10\,a{c}^{2}b{x}^{2}+3\,{a}^{2}{c}^{2}}{15\,{x}^{5}{c}^{3}}\sqrt{d{x}^{2}+c}} \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 = 1.36814, size = 163, normalized size = 1.65 \begin{align*} -\frac{{\left ({\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + 2 \,{\left (5 \, a b c^{2} - 2 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, c^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.16681, size = 391, normalized size = 3.95 \begin{align*} - \frac{3 a^{2} c^{4} d^{\frac{9}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{6} + 15 c^{3} d^{6} x^{8}} - \frac{2 a^{2} c^{3} d^{\frac{11}{2}} x^{2} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{6} + 15 c^{3} d^{6} x^{8}} - \frac{3 a^{2} c^{2} d^{\frac{13}{2}} x^{4} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{6} + 15 c^{3} d^{6} x^{8}} - \frac{12 a^{2} c d^{\frac{15}{2}} x^{6} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{6} + 15 c^{3} d^{6} x^{8}} - \frac{8 a^{2} d^{\frac{17}{2}} x^{8} \sqrt{\frac{c}{d x^{2}} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{6} + 15 c^{3} d^{6} x^{8}} - \frac{2 a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c x^{2}} + \frac{4 a b d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c^{2}} - \frac{b^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15187, size = 421, normalized size = 4.25 \begin{align*} \frac{2 \,{\left (15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} \sqrt{d} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c \sqrt{d} + 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b d^{\frac{3}{2}} + 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{2} \sqrt{d} - 140 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c d^{\frac{3}{2}} + 80 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{3} \sqrt{d} + 100 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{2} d^{\frac{3}{2}} - 40 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c d^{\frac{5}{2}} + 15 \, b^{2} c^{4} \sqrt{d} - 20 \, a b c^{3} d^{\frac{3}{2}} + 8 \, a^{2} c^{2} d^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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