Optimal. Leaf size=134 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-3 a d)}{5 d^5}-\frac{c^3 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{9/2} (4 b c-a d)}{9 d^5}-\frac{3 c \left (c+\frac{d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^5} \]
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Rubi [A] time = 0.101759, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-3 a d)}{5 d^5}-\frac{c^3 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{9/2} (4 b c-a d)}{9 d^5}-\frac{3 c \left (c+\frac{d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^5} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}}}{x^9} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x) \sqrt{c+d x} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{c^3 (b c-a d) \sqrt{c+d x}}{d^4}-\frac{c^2 (4 b c-3 a d) (c+d x)^{3/2}}{d^4}+\frac{3 c (2 b c-a d) (c+d x)^{5/2}}{d^4}+\frac{(-4 b c+a d) (c+d x)^{7/2}}{d^4}+\frac{b (c+d x)^{9/2}}{d^4}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{c^3 (b c-a d) \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^5}+\frac{c^2 (4 b c-3 a d) \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^5}-\frac{3 c (2 b c-a d) \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^5}+\frac{(4 b c-a d) \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^5}\\ \end{align*}
Mathematica [A] time = 0.0633148, size = 90, normalized size = 0.67 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (x^2 \left (\frac{c x^2}{d}+1\right ) \left (24 c^2 d x^4-16 c^3 x^6-30 c d^2 x^2+35 d^3\right ) (8 b c-11 a d)-315 b d^3 \left (c x^2+d\right )\right )}{3465 d^4 x^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 118, normalized size = 0.9 \begin{align*}{\frac{ \left ( 176\,a{c}^{3}d{x}^{8}-128\,b{c}^{4}{x}^{8}-264\,a{c}^{2}{d}^{2}{x}^{6}+192\,b{c}^{3}d{x}^{6}+330\,ac{d}^{3}{x}^{4}-240\,b{c}^{2}{d}^{2}{x}^{4}-385\,a{d}^{4}{x}^{2}+280\,bc{d}^{3}{x}^{2}-315\,b{d}^{4} \right ) \left ( c{x}^{2}+d \right ) }{3465\,{d}^{5}{x}^{10}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957736, size = 205, normalized size = 1.53 \begin{align*} -\frac{1}{3465} \, b{\left (\frac{315 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}}}{d^{5}} - \frac{1540 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} c}{d^{5}} + \frac{2970 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c^{2}}{d^{5}} - \frac{2772 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{3}}{d^{5}} + \frac{1155 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{4}}{d^{5}}\right )} - \frac{1}{315} \, a{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{4}} - \frac{135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{4}} + \frac{189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{4}} - \frac{105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{3}}{d^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79989, size = 302, normalized size = 2.25 \begin{align*} -\frac{{\left (16 \,{\left (8 \, b c^{5} - 11 \, a c^{4} d\right )} x^{10} - 8 \,{\left (8 \, b c^{4} d - 11 \, a c^{3} d^{2}\right )} x^{8} + 6 \,{\left (8 \, b c^{3} d^{2} - 11 \, a c^{2} d^{3}\right )} x^{6} + 315 \, b d^{5} - 5 \,{\left (8 \, b c^{2} d^{3} - 11 \, a c d^{4}\right )} x^{4} + 35 \,{\left (b c d^{4} + 11 \, a d^{5}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3465 \, d^{5} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.68921, size = 146, normalized size = 1.09 \begin{align*} - \frac{a \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} - \frac{b \left (\frac{c^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{4 c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{6 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} - \frac{4 c \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11}\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 7.0966, size = 581, normalized size = 4.34 \begin{align*} \frac{32 \,{\left (3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{14} a c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} b c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 4851 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{9}{2}} d \mathrm{sgn}\left (x\right ) + 7392 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{11}{2}} d \mathrm{sgn}\left (x\right ) + 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{9}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 2640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{11}{2}} d^{2} \mathrm{sgn}\left (x\right ) - 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{9}{2}} d^{3} \mathrm{sgn}\left (x\right ) - 1320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{11}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 1815 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{9}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{11}{2}} d^{4} \mathrm{sgn}\left (x\right ) - 605 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{9}{2}} d^{5} \mathrm{sgn}\left (x\right ) - 88 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{11}{2}} d^{5} \mathrm{sgn}\left (x\right ) + 121 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{9}{2}} d^{6} \mathrm{sgn}\left (x\right ) + 8 \, b c^{\frac{11}{2}} d^{6} \mathrm{sgn}\left (x\right ) - 11 \, a c^{\frac{9}{2}} d^{7} \mathrm{sgn}\left (x\right )\right )}}{3465 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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