Optimal. Leaf size=66 \[ \frac {2 a x}{\sqrt {b \sqrt {\frac {a^2}{b^2}+c x^2}+a}}+\frac {2 b^2 c x^3}{3 \left (b \sqrt {\frac {a^2}{b^2}+c x^2}+a\right )^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2129} \begin {gather*} \frac {2 b^2 c x^3}{3 \left (b \sqrt {\frac {a^2}{b^2}+c x^2}+a\right )^{3/2}}+\frac {2 a x}{\sqrt {b \sqrt {\frac {a^2}{b^2}+c x^2}+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2129
Rubi steps
\begin {align*} \int \sqrt {a+b \sqrt {\frac {a^2}{b^2}+c x^2}} \, dx &=\frac {2 b^2 c x^3}{3 \left (a+b \sqrt {\frac {a^2}{b^2}+c x^2}\right )^{3/2}}+\frac {2 a x}{\sqrt {a+b \sqrt {\frac {a^2}{b^2}+c x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 55, normalized size = 0.83 \begin {gather*} \frac {2 b x \sqrt {\frac {a^2}{b^2}+c x^2}+4 a x}{3 \sqrt {b \sqrt {\frac {a^2}{b^2}+c x^2}+a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 6.79, size = 109, normalized size = 1.65 \begin {gather*} \frac {2 a \sqrt {\frac {a^2}{b^2}+c x^2} \sqrt {b \sqrt {\frac {a^2}{b^2}+c x^2}+a}}{3 b c x}+\frac {2 \left (b^2 c x^2-a^2\right ) \sqrt {b \sqrt {\frac {a^2}{b^2}+c x^2}+a}}{3 b^2 c x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 70, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (b^{2} c x^{2} + a b \sqrt {\frac {b^{2} c x^{2} + a^{2}}{b^{2}}} - a^{2}\right )} \sqrt {b \sqrt {\frac {b^{2} c x^{2} + a^{2}}{b^{2}}} + a}}{3 \, b^{2} c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sqrt {c x^{2} + \frac {a^{2}}{b^{2}}} b + a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a +\sqrt {c \,x^{2}+\frac {a^{2}}{b^{2}}}\, b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sqrt {c x^{2} + \frac {a^{2}}{b^{2}}} b + a}\,{d x} \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 = -0.02 \begin {gather*} \int \sqrt {a+b\,\sqrt {c\,x^2+\frac {a^2}{b^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \sqrt {\frac {a^{2}}{b^{2}} + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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