Optimal. Leaf size=74 \[ \frac {a x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1088} \[ \frac {b x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {a x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
Antiderivative was successfully verified.
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Rule 1088
Rubi steps
\begin {align*} \int \sqrt {a^2+2 a b x^2+b^2 x^4} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (2 a b+2 b^2 x^2\right ) \, dx}{2 a b+2 b^2 x^2}\\ &=\frac {a x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.49 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \left (3 a x+b x^3\right )}{3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 10, normalized size = 0.14 \[ \frac {1}{3} \, b x^{3} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 20, normalized size = 0.27 \[ \frac {1}{3} \, {\left (b x^{3} + 3 \, a x\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.45 \[ \frac {\left (b \,x^{2}+3 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x}{3 b \,x^{2}+3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 10, normalized size = 0.14 \[ \frac {1}{3} \, b x^{3} + a x \]
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 \[ \int \sqrt {{\left (b\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 8, normalized size = 0.11 \[ a x + \frac {b x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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