Optimal. Leaf size=53 \[ \frac{x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (7 b c-2 a d)}{35 c^2}+\frac{a x^7 \left (c+\frac{d}{x^2}\right )^{5/2}}{7 c} \]
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Rubi [A] time = 0.0253675, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {453, 264} \[ \frac{x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (7 b c-2 a d)}{35 c^2}+\frac{a x^7 \left (c+\frac{d}{x^2}\right )^{5/2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 453
Rule 264
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2} x^6 \, dx &=\frac{a \left (c+\frac{d}{x^2}\right )^{5/2} x^7}{7 c}+\frac{(7 b c-2 a d) \int \left (c+\frac{d}{x^2}\right )^{3/2} x^4 \, dx}{7 c}\\ &=\frac{(7 b c-2 a d) \left (c+\frac{d}{x^2}\right )^{5/2} x^5}{35 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{5/2} x^7}{7 c}\\ \end{align*}
Mathematica [A] time = 0.0266944, size = 44, normalized size = 0.83 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (5 a c x^2-2 a d+7 b c\right )}{35 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 45, normalized size = 0.9 \begin{align*}{\frac{{x}^{3} \left ( 5\,a{x}^{2}c-2\,ad+7\,bc \right ) \left ( c{x}^{2}+d \right ) }{35\,{c}^{2}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.936792, size = 74, normalized size = 1.4 \begin{align*} \frac{b{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} x^{5}}{5 \, c} + \frac{{\left (5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} x^{7} - 7 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d x^{5}\right )} a}{35 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31448, size = 174, normalized size = 3.28 \begin{align*} \frac{{\left (5 \, a c^{3} x^{7} +{\left (7 \, b c^{3} + 8 \, a c^{2} d\right )} x^{5} +{\left (14 \, b c^{2} d + a c d^{2}\right )} x^{3} +{\left (7 \, b c d^{2} - 2 \, a d^{3}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{35 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.04296, size = 498, normalized size = 9.4 \begin{align*} \frac{15 a c^{6} d^{\frac{9}{2}} x^{10} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{33 a c^{5} d^{\frac{11}{2}} x^{8} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{17 a c^{4} d^{\frac{13}{2}} x^{6} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{3 a c^{3} d^{\frac{15}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{12 a c^{2} d^{\frac{17}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{8 a c d^{\frac{19}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{a d^{\frac{3}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{a d^{\frac{5}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c} - \frac{2 a d^{\frac{7}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{2}} + \frac{b c \sqrt{d} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{2 b d^{\frac{3}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{b d^{\frac{5}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0995, size = 203, normalized size = 3.83 \begin{align*} \frac{35 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b d \mathrm{sgn}\left (x\right ) + 7 \,{\left (3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d\right )} b \mathrm{sgn}\left (x\right ) + \frac{7 \,{\left (3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d\right )} a d \mathrm{sgn}\left (x\right )}{c} + \frac{{\left (15 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d + 35 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{2}\right )} a \mathrm{sgn}\left (x\right )}{c}}{105 \, c} - \frac{{\left (7 \, b c d^{\frac{5}{2}} - 2 \, a d^{\frac{7}{2}}\right )} \mathrm{sgn}\left (x\right )}{35 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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