Optimal. Leaf size=79 \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )} \]
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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} \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )} \]
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^{10}} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {a b+b^2 x^2}{x^{10}} \, 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^{10}}+\frac {b^2}{x^8}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \[ -\frac {\sqrt {\left (a+b x^2\right )^2} \left (7 a+9 b x^2\right )}{63 x^9 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 15, normalized size = 0.19 \[ -\frac {9 \, b x^{2} + 7 \, a}{63 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 31, normalized size = 0.39 \[ -\frac {9 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 7 \, a \mathrm {sgn}\left (b x^{2} + a\right )}{63 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.46 \[ -\frac {\left (9 b \,x^{2}+7 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{63 \left (b \,x^{2}+a \right ) x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 15, normalized size = 0.19 \[ -\frac {9 \, b x^{2} + 7 \, a}{63 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 35, normalized size = 0.44 \[ -\frac {\left (9\,b\,x^2+7\,a\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{63\,x^9\,\left (b\,x^2+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 15, normalized size = 0.19 \[ \frac {- 7 a - 9 b x^{2}}{63 x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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