Optimal. Leaf size=141 \[ -\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}-\frac{2 d x \left (15 b^2 c^2-8 a d (5 b c-3 a d)\right )}{15 c^4 \sqrt{c+d x^2}}-\frac{15 b^2 c^2-8 a d (5 b c-3 a d)}{15 c^3 x \sqrt{c+d x^2}}-\frac{2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.113882, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {462, 453, 271, 191} \[ -\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}-\frac{2 d x \left (15 b^2 c^2-8 a d (5 b c-3 a d)\right )}{15 c^4 \sqrt{c+d x^2}}-\frac{15 b^2-\frac{8 a d (5 b c-3 a d)}{c^2}}{15 c x \sqrt{c+d x^2}}-\frac{2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 453
Rule 271
Rule 191
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx &=-\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}+\frac{\int \frac{2 a (5 b c-3 a d)+5 b^2 c x^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx}{5 c}\\ &=-\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}-\frac{2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt{c+d x^2}}-\frac{1}{15} \left (-15 b^2+\frac{8 a d (5 b c-3 a d)}{c^2}\right ) \int \frac{1}{x^2 \left (c+d x^2\right )^{3/2}} \, dx\\ &=-\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}-\frac{2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt{c+d x^2}}-\frac{15 b^2-\frac{8 a d (5 b c-3 a d)}{c^2}}{15 c x \sqrt{c+d x^2}}-\frac{\left (2 d \left (15 b^2-\frac{8 a d (5 b c-3 a d)}{c^2}\right )\right ) \int \frac{1}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c}\\ &=-\frac{a^2}{5 c x^5 \sqrt{c+d x^2}}-\frac{2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt{c+d x^2}}-\frac{15 b^2-\frac{8 a d (5 b c-3 a d)}{c^2}}{15 c x \sqrt{c+d x^2}}-\frac{2 d \left (15 b^2-\frac{8 a d (5 b c-3 a d)}{c^2}\right ) x}{15 c^2 \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0806853, size = 105, normalized size = 0.74 \[ \sqrt{c+d x^2} \left (\frac{-33 a^2 d^2+50 a b c d-15 b^2 c^2}{15 c^4 x}-\frac{a^2}{5 c^2 x^5}+\frac{a (9 a d-10 b c)}{15 c^3 x^3}-\frac{d x (b c-a d)^2}{c^4 \left (c+d x^2\right )}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 117, normalized size = 0.8 \begin{align*} -{\frac{48\,{x}^{6}{a}^{2}{d}^{3}-80\,{x}^{6}abc{d}^{2}+30\,{x}^{6}{b}^{2}{c}^{2}d+24\,{x}^{4}{a}^{2}c{d}^{2}-40\,{x}^{4}ab{c}^{2}d+15\,{x}^{4}{b}^{2}{c}^{3}-6\,{x}^{2}{a}^{2}{c}^{2}d+10\,{x}^{2}ab{c}^{3}+3\,{a}^{2}{c}^{3}}{15\,{x}^{5}{c}^{4}}{\frac{1}{\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 = 1.50501, size = 258, normalized size = 1.83 \begin{align*} -\frac{{\left (2 \,{\left (15 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 24 \, a^{2} d^{3}\right )} x^{6} + 3 \, a^{2} c^{3} +{\left (15 \, b^{2} c^{3} - 40 \, a b c^{2} d + 24 \, a^{2} c d^{2}\right )} x^{4} + 2 \,{\left (5 \, a b c^{3} - 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \,{\left (c^{4} d x^{7} + c^{5} x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{2}}{x^{6} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17203, size = 610, normalized size = 4.33 \begin{align*} -\frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x}{\sqrt{d x^{2} + c} c^{4}} + \frac{2 \,{\left (15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} c^{2} \sqrt{d} - 30 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a b c d^{\frac{3}{2}} + 15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a^{2} d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c^{3} \sqrt{d} + 180 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac{3}{2}} - 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a^{2} c d^{\frac{5}{2}} + 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{4} \sqrt{d} - 320 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac{3}{2}} + 240 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{5} \sqrt{d} + 220 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac{3}{2}} - 150 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac{5}{2}} + 15 \, b^{2} c^{6} \sqrt{d} - 50 \, a b c^{5} d^{\frac{3}{2}} + 33 \, a^{2} c^{4} d^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{5} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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