Optimal. Leaf size=59 \[ \frac {b c-a d}{c d x \sqrt {c+\frac {d}{x^2}}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{d^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {452, 335, 217, 206} \[ \frac {b c-a d}{c d x \sqrt {c+\frac {d}{x^2}}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 335
Rule 452
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^2} \, dx &=\frac {b c-a d}{c d \sqrt {c+\frac {d}{x^2}} x}+\frac {b \int \frac {1}{\sqrt {c+\frac {d}{x^2}} x^2} \, dx}{d}\\ &=\frac {b c-a d}{c d \sqrt {c+\frac {d}{x^2}} x}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{d}\\ &=\frac {b c-a d}{c d \sqrt {c+\frac {d}{x^2}} x}-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {1}{\sqrt {c+\frac {d}{x^2}} x}\right )}{d}\\ &=\frac {b c-a d}{c d \sqrt {c+\frac {d}{x^2}} x}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 1.20 \[ \frac {\sqrt {d} (b c-a d)-b c \sqrt {c x^2+d} \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{c d^{3/2} x \sqrt {c+\frac {d}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 195, normalized size = 3.31 \[ \left [\frac {2 \, {\left (b c d - a d^{2}\right )} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + {\left (b c^{2} x^{2} + b c d\right )} \sqrt {d} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right )}{2 \, {\left (c^{2} d^{2} x^{2} + c d^{3}\right )}}, \frac {{\left (b c d - a d^{2}\right )} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + {\left (b c^{2} x^{2} + b c d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right )}{c^{2} d^{2} x^{2} + c d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 79, normalized size = 1.34 \[ -\frac {\left (c \,x^{2}+d \right ) \left (\sqrt {c \,x^{2}+d}\, b c d \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+a \,d^{\frac {5}{2}}-b c \,d^{\frac {3}{2}}\right )}{\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} c \,d^{\frac {5}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 80, normalized size = 1.36 \[ \frac {1}{2} \, b {\left (\frac {\log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2}{\sqrt {c + \frac {d}{x^{2}}} d x}\right )} - \frac {a}{\sqrt {c + \frac {d}{x^{2}}} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.11, size = 60, normalized size = 1.02 \[ \frac {b}{d\,x\,\sqrt {c+\frac {d}{x^2}}}-\frac {a}{c\,x\,\sqrt {c+\frac {d}{x^2}}}-\frac {b\,\ln \left (\sqrt {c+\frac {d}{x^2}}+\frac {\sqrt {d}}{x}\right )}{d^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 11.71, size = 206, normalized size = 3.49 \[ - \frac {a}{c \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + b \left (\frac {c d^{2} x^{2} \log {\left (\frac {c x^{2}}{d} \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} - \frac {2 c d^{2} x^{2} \log {\left (\sqrt {\frac {c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} + \frac {2 d^{3} \sqrt {\frac {c x^{2}}{d} + 1}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} + \frac {d^{3} \log {\left (\frac {c x^{2}}{d} \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} - \frac {2 d^{3} \log {\left (\sqrt {\frac {c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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