Optimal. Leaf size=42 \[ -\frac {b c-a d}{d^2 \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d^2} \]
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Rubi [A] time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 = {444, 43} \begin {gather*} -\frac {b c-a d}{d^2 \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^3} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {a+b x}{(c+d x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {-b c+a d}{d (c+d x)^{3/2}}+\frac {b}{d \sqrt {c+d x}}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {b c-a d}{d^2 \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.86 \begin {gather*} \frac {a d x^2-b \left (2 c x^2+d\right )}{d^2 x^2 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 46, normalized size = 1.10 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (a d x^2-2 b c x^2-b d\right )}{d^2 \left (c x^2+d\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 46, normalized size = 1.10 \begin {gather*} -\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c d^{2} x^{2} + d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 37, normalized size = 0.88 \begin {gather*} -\frac {\frac {{\left (2 \, b c - a d\right )} x^{2}}{d^{2}} + \frac {b}{d}}{\sqrt {c x^{4} + d x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 46, normalized size = 1.10 \begin {gather*} \frac {\left (a d \,x^{2}-2 b c \,x^{2}-b d \right ) \left (c \,x^{2}+d \right )}{\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 46, normalized size = 1.10 \begin {gather*} -b {\left (\frac {\sqrt {c + \frac {d}{x^{2}}}}{d^{2}} + \frac {c}{\sqrt {c + \frac {d}{x^{2}}} d^{2}}\right )} + \frac {a}{\sqrt {c + \frac {d}{x^{2}}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.52, size = 46, normalized size = 1.10 \begin {gather*} \frac {x\,\sqrt {c+\frac {d}{x^2}}\,\left (x^2\,\left (\frac {a}{d}-\frac {2\,b\,c}{d^2}\right )-\frac {b}{d}\right )}{c\,x^3+d\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.42, size = 68, normalized size = 1.62 \begin {gather*} \begin {cases} \frac {a}{d \sqrt {c + \frac {d}{x^{2}}}} - \frac {2 b c}{d^{2} \sqrt {c + \frac {d}{x^{2}}}} - \frac {b}{d x^{2} \sqrt {c + \frac {d}{x^{2}}}} & \text {for}\: d \neq 0 \\\frac {- \frac {a}{2 x^{2}} - \frac {b}{4 x^{4}}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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