Optimal. Leaf size=79 \[ \frac {b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac {a x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.02, 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} \[ \frac {b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac {a x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1112
Rubi steps
\begin {align*} \int x^2 \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 x^2 \left (a b+b^2 x^2\right ) \, dx}{a b+b^2 x^2}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a b x^2+b^2 x^4\right ) \, dx}{a b+b^2 x^2}\\ &=\frac {a x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \left (5 a x^3+3 b x^5\right )}{15 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 13, normalized size = 0.16 \[ \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.37 \[ \frac {1}{5} \, b x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{3} \, a x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.46 \[ \frac {\left (3 b \,x^{2}+5 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x^{3}}{15 b \,x^{2}+15 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 13, normalized size = 0.16 \[ \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]
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 \[ \int x^2\,\sqrt {{\left (b\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 12, normalized size = 0.15 \[ \frac {a x^{3}}{3} + \frac {b x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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