Optimal. Leaf size=77 \[ \frac {b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
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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 = {1355, 14} \[ \frac {b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
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^2} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^2} \, 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^2}+b^2 x\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.49 \[ \frac {\left (b x^3-2 a\right ) \sqrt {\left (a+b x^3\right )^2}}{2 x \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 14, normalized size = 0.18 \[ \frac {b x^{3} - 2 \, a}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 29, normalized size = 0.38 \[ \frac {1}{2} \, b x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {a \mathrm {sgn}\left (b x^{3} + a\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.47 \[ -\frac {\left (-b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 \left (b \,x^{3}+a \right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 14, normalized size = 0.18 \[ \frac {b x^{3} - 2 \, a}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {{\left (b\,x^3+a\right )}^2}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 8, normalized size = 0.10 \[ - \frac {a}{x} + \frac {b x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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