Optimal. Leaf size=43 \[ \frac {\sqrt {c+\frac {d}{x^2}} (b c-a d)}{d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^2} \]
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Rubi [A] time = 0.04, antiderivative size = 43, 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} \[ \frac {\sqrt {c+\frac {d}{x^2}} (b c-a d)}{d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^3} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {a+b x}{\sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {-b c+a d}{d \sqrt {c+d x}}+\frac {b \sqrt {c+d x}}{d}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {(b c-a d) \sqrt {c+\frac {d}{x^2}}}{d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 0.91 \[ -\frac {\sqrt {c+\frac {d}{x^2}} \left (3 a d x^2+b \left (d-2 c x^2\right )\right )}{3 d^2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 39, normalized size = 0.91 \[ \frac {{\left ({\left (2 \, b c - 3 \, a d\right )} x^{2} - b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3 \, d^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 88, normalized size = 2.05 \[ \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{2} a + 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )} b \sqrt {c} + b d}{3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 47, normalized size = 1.09 \[ -\frac {\left (3 a d \,x^{2}-2 b c \,x^{2}+b d \right ) \left (c \,x^{2}+d \right )}{3 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, d^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 48, normalized size = 1.12 \[ -\frac {1}{3} \, b {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}}{d^{2}} - \frac {3 \, \sqrt {c + \frac {d}{x^{2}}} c}{d^{2}}\right )} - \frac {a \sqrt {c + \frac {d}{x^{2}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.56, size = 35, normalized size = 0.81 \[ -\frac {\sqrt {c+\frac {d}{x^2}}\,\left (b\,d+3\,a\,d\,x^2-2\,b\,c\,x^2\right )}{3\,d^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.05, size = 138, normalized size = 3.21 \[ \frac {\begin {cases} \frac {- \frac {a}{x^{2}} - \frac {b}{2 x^{4}}}{\sqrt {c}} & \text {for}\: d = 0 \\\frac {\frac {2 a c}{\sqrt {c + \frac {d}{x^{2}}}} + 2 a \left (- \frac {c}{\sqrt {c + \frac {d}{x^{2}}}} - \sqrt {c + \frac {d}{x^{2}}}\right ) + \frac {2 b c \left (- \frac {c}{\sqrt {c + \frac {d}{x^{2}}}} - \sqrt {c + \frac {d}{x^{2}}}\right )}{d} + \frac {2 b \left (\frac {c^{2}}{\sqrt {c + \frac {d}{x^{2}}}} + 2 c \sqrt {c + \frac {d}{x^{2}}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3}\right )}{d}}{d} & \text {otherwise} \end {cases}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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