Optimal. Leaf size=34 \[ \frac {2 x \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {254, 32} \begin {gather*} \frac {2 x \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 254
Rubi steps
\begin {align*} \int \sqrt {a+b \sqrt {c x^2}} \, dx &=\frac {x \operatorname {Subst}\left (\int \sqrt {a+b x} \, dx,x,\sqrt {c x^2}\right )}{\sqrt {c x^2}}\\ &=\frac {2 x \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {2 x \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.58, size = 63, normalized size = 1.85 \begin {gather*} \frac {2 a \sqrt {c x^2} \sqrt {a+b \sqrt {c x^2}}}{3 b c x}+\frac {2}{3} x \sqrt {a+b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 40, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (b c x^{2} + \sqrt {c x^{2}} a\right )} \sqrt {\sqrt {c x^{2}} b + a}}{3 \, b c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 18, normalized size = 0.53 \begin {gather*} \frac {2 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}}}{3 \, b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.79 \begin {gather*} \frac {2 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {3}{2}} x}{3 \sqrt {c \,x^{2}}\, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 42, normalized size = 1.24 \begin {gather*} \frac {{\left ({\left (c^{\frac {3}{2}} + c\right )} b x + a {\left (c + \sqrt {c}\right )}\right )} \sqrt {b \sqrt {c} x + a}}{{\left (c^{2} + 2 \, c^{\frac {3}{2}} + c\right )} b} \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 = -0.03 \begin {gather*} \int \sqrt {a+b\,\sqrt {c\,x^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 {c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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