Optimal. Leaf size=47 \[ -\frac {x (2 A b-a B)}{a^2 \sqrt {a+b x^2}}-\frac {A}{a x \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {453, 191} \begin {gather*} -\frac {x (2 A b-a B)}{a^2 \sqrt {a+b x^2}}-\frac {A}{a x \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac {A}{a x \sqrt {a+b x^2}}-\frac {(2 A b-a B) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{a}\\ &=-\frac {A}{a x \sqrt {a+b x^2}}-\frac {(2 A b-a B) x}{a^2 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.77 \begin {gather*} \frac {-a A+a B x^2-2 A b x^2}{a^2 x \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 36, normalized size = 0.77 \begin {gather*} \frac {-a A+a B x^2-2 A b x^2}{a^2 x \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 43, normalized size = 0.91 \begin {gather*} \frac {{\left ({\left (B a - 2 \, A b\right )} x^{2} - A a\right )} \sqrt {b x^{2} + a}}{a^{2} b x^{3} + a^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 57, normalized size = 1.21 \begin {gather*} \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a} + \frac {{\left (B a - A b\right )} x}{\sqrt {b x^{2} + a} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.77 \begin {gather*} -\frac {2 A b \,x^{2}-B a \,x^{2}+A a}{\sqrt {b \,x^{2}+a}\, a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 51, normalized size = 1.09 \begin {gather*} \frac {B x}{\sqrt {b x^{2} + a} a} - \frac {2 \, A b x}{\sqrt {b x^{2} + a} a^{2}} - \frac {A}{\sqrt {b x^{2} + a} a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 46, normalized size = 0.98 \begin {gather*} -\frac {\sqrt {b\,x^2+a}\,\left (\frac {A}{a}-x^2\,\left (\frac {B}{a}-\frac {2\,A\,b}{a^2}\right )\right )}{b\,x^3+a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.67, size = 68, normalized size = 1.45 \begin {gather*} A \left (- \frac {1}{a \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{2}} + 1}}\right ) + \frac {B x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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