Optimal. Leaf size=123 \[ -\frac{c^2 (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{16 d^{5/2}}+\frac{c \sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{16 d^2 x}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{8 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3} \]
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Rubi [A] time = 0.0805208, 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 = {459, 335, 279, 321, 217, 206} \[ -\frac{c^2 (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{16 d^{5/2}}+\frac{c \sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{16 d^2 x}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{8 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3} \]
Antiderivative was successfully verified.
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Rule 459
Rule 335
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}}}{x^4} \, dx &=-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3}+\frac{(-3 b c+6 a d) \int \frac{\sqrt{c+\frac{d}{x^2}}}{x^4} \, dx}{6 d}\\ &=-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3}-\frac{(-3 b c+6 a d) \operatorname{Subst}\left (\int x^2 \sqrt{c+d x^2} \, dx,x,\frac{1}{x}\right )}{6 d}\\ &=\frac{(b c-2 a d) \sqrt{c+\frac{d}{x^2}}}{8 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3}+\frac{(c (b c-2 a d)) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )}{8 d}\\ &=\frac{(b c-2 a d) \sqrt{c+\frac{d}{x^2}}}{8 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3}+\frac{c (b c-2 a d) \sqrt{c+\frac{d}{x^2}}}{16 d^2 x}-\frac{\left (c^2 (b c-2 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )}{16 d^2}\\ &=\frac{(b c-2 a d) \sqrt{c+\frac{d}{x^2}}}{8 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3}+\frac{c (b c-2 a d) \sqrt{c+\frac{d}{x^2}}}{16 d^2 x}-\frac{\left (c^2 (b c-2 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{1}{\sqrt{c+\frac{d}{x^2}} x}\right )}{16 d^2}\\ &=\frac{(b c-2 a d) \sqrt{c+\frac{d}{x^2}}}{8 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3}+\frac{c (b c-2 a d) \sqrt{c+\frac{d}{x^2}}}{16 d^2 x}-\frac{c^2 (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{\sqrt{c+\frac{d}{x^2}} x}\right )}{16 d^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.025872, size = 68, normalized size = 0.55 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (c^2 x^6 (b c-2 a d) \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{c x^2}{d}+1\right )-b d^3\right )}{6 d^4 x^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 220, normalized size = 1.8 \begin{align*}{\frac{1}{48\,{x}^{5}{d}^{3}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 6\,{d}^{3/2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{6}a{c}^{2}-3\,\sqrt{d}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{6}b{c}^{3}-6\,\sqrt{c{x}^{2}+d}{x}^{6}a{c}^{2}d+3\,\sqrt{c{x}^{2}+d}{x}^{6}b{c}^{3}+6\, \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{4}acd-3\, \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{4}b{c}^{2}-12\, \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}a{d}^{2}+6\, \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}bcd-8\, \left ( c{x}^{2}+d \right ) ^{3/2}b{d}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}} \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 = 1.38809, size = 552, normalized size = 4.49 \begin{align*} \left [-\frac{3 \,{\left (b c^{3} - 2 \, a c^{2} d\right )} \sqrt{d} x^{5} \log \left (-\frac{c x^{2} + 2 \, \sqrt{d} x \sqrt{\frac{c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (b c^{2} d - 2 \, a c d^{2}\right )} x^{4} - 8 \, b d^{3} - 2 \,{\left (b c d^{2} + 6 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{96 \, d^{3} x^{5}}, \frac{3 \,{\left (b c^{3} - 2 \, a c^{2} d\right )} \sqrt{-d} x^{5} \arctan \left (\frac{\sqrt{-d} x \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) +{\left (3 \,{\left (b c^{2} d - 2 \, a c d^{2}\right )} x^{4} - 8 \, b d^{3} - 2 \,{\left (b c d^{2} + 6 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{48 \, d^{3} x^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 10.8581, size = 226, normalized size = 1.84 \begin{align*} - \frac{a c^{\frac{3}{2}}}{8 d x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 a \sqrt{c}}{8 x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{a c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 d^{\frac{3}{2}}} - \frac{a d}{4 \sqrt{c} x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{\frac{5}{2}}}{16 d^{2} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{\frac{3}{2}}}{48 d x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{5 b \sqrt{c}}{24 x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{16 d^{\frac{5}{2}}} - \frac{b d}{6 \sqrt{c} x^{7} \sqrt{1 + \frac{d}{c x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24678, size = 207, normalized size = 1.68 \begin{align*} \frac{\frac{3 \,{\left (b c^{4} \mathrm{sgn}\left (x\right ) - 2 \, a c^{3} d \mathrm{sgn}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d} d^{2}} + \frac{3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} b c^{4} \mathrm{sgn}\left (x\right ) - 6 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} a c^{3} d \mathrm{sgn}\left (x\right ) - 8 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b c^{4} d \mathrm{sgn}\left (x\right ) - 3 \, \sqrt{c x^{2} + d} b c^{4} d^{2} \mathrm{sgn}\left (x\right ) + 6 \, \sqrt{c x^{2} + d} a c^{3} d^{3} \mathrm{sgn}\left (x\right )}{c^{3} d^{2} x^{6}}}{48 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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