Optimal. Leaf size=53 \[ \frac {\sqrt {a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac {A \sqrt {a+b x^2}}{3 a x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 53, 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, 264} \[ \frac {\sqrt {a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac {A \sqrt {a+b x^2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^4 \sqrt {a+b x^2}} \, dx &=-\frac {A \sqrt {a+b x^2}}{3 a x^3}-\frac {(2 A b-3 a B) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{3 a}\\ &=-\frac {A \sqrt {a+b x^2}}{3 a x^3}+\frac {(2 A b-3 a B) \sqrt {a+b x^2}}{3 a^2 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.74 \[ -\frac {\sqrt {a+b x^2} \left (a \left (A+3 B x^2\right )-2 A b x^2\right )}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 34, normalized size = 0.64 \[ -\frac {{\left ({\left (3 \, B a - 2 \, A b\right )} x^{2} + A a\right )} \sqrt {b x^{2} + a}}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 120, normalized size = 2.26 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B \sqrt {b} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A b^{\frac {3}{2}} + 3 \, B a^{2} \sqrt {b} - 2 \, A a b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.68 \[ -\frac {\sqrt {b \,x^{2}+a}\, \left (-2 A b \,x^{2}+3 B a \,x^{2}+A a \right )}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 56, normalized size = 1.06 \[ -\frac {\sqrt {b x^{2} + a} B}{a x} + \frac {2 \, \sqrt {b x^{2} + a} A b}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a} A}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 35, normalized size = 0.66 \[ -\frac {\sqrt {b\,x^2+a}\,\left (A\,a-2\,A\,b\,x^2+3\,B\,a\,x^2\right )}{3\,a^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.87, size = 70, normalized size = 1.32 \[ - \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {2 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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