Optimal. Leaf size=45 \[ \frac {a x}{c \sqrt {c+\frac {d}{x^2}}}-\frac {b c-2 a d}{c^2 x \sqrt {c+\frac {d}{x^2}}} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {375, 453, 191} \begin {gather*} \frac {a x}{c \sqrt {c+\frac {d}{x^2}}}-\frac {b c-2 a d}{c^2 x \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 375
Rule 453
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b x^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a x}{c \sqrt {c+\frac {d}{x^2}}}+\frac {(-b c+2 a d) \operatorname {Subst}\left (\int \frac {1}{\left (c+d x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {b c-2 a d}{c^2 \sqrt {c+\frac {d}{x^2}} x}+\frac {a x}{c \sqrt {c+\frac {d}{x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 0.73 \begin {gather*} \frac {a c x^2+2 a d-b c}{c^2 x \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 40, normalized size = 0.89 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (a c x^2+2 a d-b c\right )}{c^2 \left (c x^2+d\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 47, normalized size = 1.04 \begin {gather*} \frac {{\left (a c x^{3} - {\left (b c - 2 \, a d\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c^{3} x^{2} + c^{2} d} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 0.96 \begin {gather*} \frac {\left (a \,x^{2} c +2 a d -b c \right ) \left (c \,x^{2}+d \right )}{\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} c^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 53, normalized size = 1.18 \begin {gather*} a {\left (\frac {\sqrt {c + \frac {d}{x^{2}}} x}{c^{2}} + \frac {d}{\sqrt {c + \frac {d}{x^{2}}} c^{2} x}\right )} - \frac {b}{\sqrt {c + \frac {d}{x^{2}}} c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.90, size = 38, normalized size = 0.84 \begin {gather*} \frac {\left (c\,x^2+d\right )\,\left (a\,c\,x^2+2\,a\,d-b\,c\right )}{c^2\,x^3\,{\left (c+\frac {d}{x^2}\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.51, size = 65, normalized size = 1.44 \begin {gather*} a \left (\frac {x^{2}}{c \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {2 \sqrt {d}}{c^{2} \sqrt {\frac {c x^{2}}{d} + 1}}\right ) - \frac {b}{c \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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