Optimal. Leaf size=77 \[ -\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1112, 14} \begin {gather*} -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1112
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {a b+b^2 x^2}{x^4} \, dx}{a b+b^2 x^2}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {a b}{x^4}+\frac {b^2}{x^2}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (a+3 b x^2\right )}{3 x^3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 13.39, size = 39, normalized size = 0.51 \begin {gather*} \frac {\left (-a-3 b x^2\right ) \sqrt {\left (a+b x^2\right )^2}}{3 x^3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 13, normalized size = 0.17 \begin {gather*} -\frac {3 \, b x^{2} + a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 30, normalized size = 0.39 \begin {gather*} -\frac {3 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a \mathrm {sgn}\left (b x^{2} + a\right )}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.44 \begin {gather*} -\frac {\left (3 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 \left (b \,x^{2}+a \right ) x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 13, normalized size = 0.17 \begin {gather*} -\frac {3 \, b x^{2} + a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.24, size = 33, normalized size = 0.43 \begin {gather*} -\frac {\left (3\,b\,x^2+a\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{3\,x^3\,\left (b\,x^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 14, normalized size = 0.18 \begin {gather*} \frac {- a - 3 b x^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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