Optimal. Leaf size=79 \[ -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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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 = {1355, 14} \[ -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1355
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^6} \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x^6}+\frac {b^2}{x^3}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (2 a+5 b x^3\right )}{10 x^5 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 15, normalized size = 0.19 \[ -\frac {5 \, b x^{3} + 2 \, a}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 31, normalized size = 0.39 \[ -\frac {5 \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 2 \, a \mathrm {sgn}\left (b x^{3} + a\right )}{10 \, x^{5}} \]
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 \[ -\frac {\left (5 b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{10 \left (b \,x^{3}+a \right ) x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 15, normalized size = 0.19 \[ -\frac {5 \, b x^{3} + 2 \, a}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 35, normalized size = 0.44 \[ -\frac {\left (5\,b\,x^3+2\,a\right )\,\sqrt {{\left (b\,x^3+a\right )}^2}}{10\,x^5\,\left (b\,x^3+a\right )} \]
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 \[ \frac {- 2 a - 5 b x^{3}}{10 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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