Optimal. Leaf size=123 \[ -\frac{d^2 (2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{5/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{16 c^2}+\frac{x^4 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{8 c}+\frac{a x^6 \left (c+\frac{d}{x^2}\right )^{3/2}}{6 c} \]
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Rubi [A] time = 0.0942834, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {446, 78, 47, 51, 63, 208} \[ -\frac{d^2 (2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{5/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{16 c^2}+\frac{x^4 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{8 c}+\frac{a x^6 \left (c+\frac{d}{x^2}\right )^{3/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} x^5 \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x) \sqrt{c+d x}}{x^4} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^6}{6 c}-\frac{\left (3 b c-\frac{3 a d}{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x^3} \, dx,x,\frac{1}{x^2}\right )}{6 c}\\ &=\frac{(2 b c-a d) \sqrt{c+\frac{d}{x^2}} x^4}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^6}{6 c}-\frac{(d (2 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )}{16 c}\\ &=\frac{d (2 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{16 c^2}+\frac{(2 b c-a d) \sqrt{c+\frac{d}{x^2}} x^4}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^6}{6 c}+\frac{\left (d^2 (2 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )}{32 c^2}\\ &=\frac{d (2 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{16 c^2}+\frac{(2 b c-a d) \sqrt{c+\frac{d}{x^2}} x^4}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^6}{6 c}+\frac{(d (2 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+\frac{d}{x^2}}\right )}{16 c^2}\\ &=\frac{d (2 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{16 c^2}+\frac{(2 b c-a d) \sqrt{c+\frac{d}{x^2}} x^4}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^6}{6 c}-\frac{d^2 (2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.198508, size = 121, normalized size = 0.98 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c} x \sqrt{\frac{c x^2}{d}+1} \left (a \left (8 c^2 x^4+2 c d x^2-3 d^2\right )+6 b c \left (2 c x^2+d\right )\right )+3 d^{3/2} (a d-2 b c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )\right )}{48 c^{5/2} \sqrt{\frac{c x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 162, normalized size = 1.3 \begin{align*}{\frac{x}{48}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 8\,{c}^{3/2} \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{3}a-6\,\sqrt{c} \left ( c{x}^{2}+d \right ) ^{3/2}xad+12\,{c}^{3/2} \left ( c{x}^{2}+d \right ) ^{3/2}xb+3\,\sqrt{c}\sqrt{c{x}^{2}+d}xa{d}^{2}-6\,{c}^{3/2}\sqrt{c{x}^{2}+d}xbd+3\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) a{d}^{3}-6\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) bc{d}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{c}^{-{\frac{5}{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 = 1.66413, size = 533, normalized size = 4.33 \begin{align*} \left [-\frac{3 \,{\left (2 \, b c d^{2} - a d^{3}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right ) - 2 \,{\left (8 \, a c^{3} x^{6} + 2 \,{\left (6 \, b c^{3} + a c^{2} d\right )} x^{4} + 3 \,{\left (2 \, b c^{2} d - a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{96 \, c^{3}}, \frac{3 \,{\left (2 \, b c d^{2} - a d^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) +{\left (8 \, a c^{3} x^{6} + 2 \,{\left (6 \, b c^{3} + a c^{2} d\right )} x^{4} + 3 \,{\left (2 \, b c^{2} d - a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{48 \, c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 37.8391, size = 226, normalized size = 1.84 \begin{align*} \frac{a c x^{7}}{6 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{5 a \sqrt{d} x^{5}}{24 \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{\frac{3}{2}} x^{3}}{48 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{\frac{5}{2}} x}{16 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{a d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{16 c^{\frac{5}{2}}} + \frac{b c x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 b \sqrt{d} x^{3}}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{b d^{\frac{3}{2}} x}{8 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14942, size = 193, normalized size = 1.57 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (4 \, a x^{2} \mathrm{sgn}\left (x\right ) + \frac{6 \, b c^{4} \mathrm{sgn}\left (x\right ) + a c^{3} d \mathrm{sgn}\left (x\right )}{c^{4}}\right )} x^{2} + \frac{3 \,{\left (2 \, b c^{3} d \mathrm{sgn}\left (x\right ) - a c^{2} d^{2} \mathrm{sgn}\left (x\right )\right )}}{c^{4}}\right )} \sqrt{c x^{2} + d} x + \frac{{\left (2 \, b c d^{2} \mathrm{sgn}\left (x\right ) - a d^{3} \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + d} \right |}\right )}{16 \, c^{\frac{5}{2}}} - \frac{{\left (2 \, b c d^{2} \log \left ({\left | d \right |}\right ) - a d^{3} \log \left ({\left | d \right |}\right )\right )} \mathrm{sgn}\left (x\right )}{32 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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