Optimal. Leaf size=79 \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 79, 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}}{5 x^5 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \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^6} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {a b+b^2 x^2}{x^6} \, 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^6}+\frac {b^2}{x^4}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (3 a+5 b x^2\right )}{15 x^5 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 16.71, size = 39, normalized size = 0.49 \begin {gather*} \frac {\left (-3 a-5 b x^2\right ) \sqrt {\left (a+b x^2\right )^2}}{15 x^5 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 15, normalized size = 0.19 \begin {gather*} -\frac {5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 31, normalized size = 0.39 \begin {gather*} -\frac {5 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, a \mathrm {sgn}\left (b x^{2} + a\right )}{15 \, x^{5}} \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 = 0.46 \begin {gather*} -\frac {\left (5 b \,x^{2}+3 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{15 \left (b \,x^{2}+a \right ) x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 15, normalized size = 0.19 \begin {gather*} -\frac {5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.21, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (5\,b\,x^2+3\,a\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{15\,x^5\,\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 = 0.18, size = 15, normalized size = 0.19 \begin {gather*} \frac {- 3 a - 5 b x^{2}}{15 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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