Optimal. Leaf size=44 \[ \frac {2 b \sqrt {a+b x^2}}{3 a^2 x}-\frac {\sqrt {a+b x^2}}{3 a x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {2 b \sqrt {a+b x^2}}{3 a^2 x}-\frac {\sqrt {a+b x^2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {a+b x^2}} \, dx &=-\frac {\sqrt {a+b x^2}}{3 a x^3}-\frac {(2 b) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{3 a}\\ &=-\frac {\sqrt {a+b x^2}}{3 a x^3}+\frac {2 b \sqrt {a+b x^2}}{3 a^2 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.66 \[ -\frac {\left (a-2 b x^2\right ) \sqrt {a+b x^2}}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 27, normalized size = 0.61 \[ \frac {{\left (2 \, b x^{2} - a\right )} \sqrt {b x^{2} + a}}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 55, normalized size = 1.25 \[ \frac {4 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} b^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.59 \[ -\frac {\sqrt {b \,x^{2}+a}\, \left (-2 b \,x^{2}+a \right )}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 36, normalized size = 0.82 \[ \frac {2 \, \sqrt {b x^{2} + a} b}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a}}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 25, normalized size = 0.57 \[ -\frac {\sqrt {b\,x^2+a}\,\left (a-2\,b\,x^2\right )}{3\,a^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 46, normalized size = 1.05 \[ - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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