Optimal. Leaf size=21 \[ -\frac {\left (a+b x^2\right )^{3/2}}{3 a x^3} \]
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Rubi [A] time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {264} \[ -\frac {\left (a+b x^2\right )^{3/2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{x^4} \, dx &=-\frac {\left (a+b x^2\right )^{3/2}}{3 a x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ -\frac {\left (a+b x^2\right )^{3/2}}{3 a x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 17, normalized size = 0.81 \[ -\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.68, size = 59, normalized size = 2.81 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {3}{2}} + a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.86 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 17, normalized size = 0.81 \[ -\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.57, size = 17, normalized size = 0.81 \[ -\frac {{\left (b\,x^2+a\right )}^{3/2}}{3\,a\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.71, size = 42, normalized size = 2.00 \[ - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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