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 = {1102}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 1102
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 = 0.00, size = 36, normalized size = 0.49 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (3 a x+b x^3\right )}{3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 33, normalized size = 0.45
method | result | size |
gosper | \(\frac {x \left (b \,x^{2}+3 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 b \,x^{2}+3 a}\) | \(33\) |
default | \(\frac {x \left (b \,x^{2}+3 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 b \,x^{2}+3 a}\) | \(33\) |
risch | \(\frac {a x \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \,x^{2}+a}+\frac {b \,x^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 b \,x^{2}+3 a}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 10, normalized size = 0.14 \begin {gather*} \frac {1}{3} \, b x^{3} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 10, normalized size = 0.14 \begin {gather*} \frac {1}{3} \, b x^{3} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 8, normalized size = 0.11 \begin {gather*} a x + \frac {b x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.43, size = 20, normalized size = 0.27 \begin {gather*} \frac {1}{3} \, {\left (b x^{3} + 3 \, a x\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}
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 \begin {gather*} \int \sqrt {{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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