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 = {260, 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 260
Rubi steps
\begin {align*} \int \sqrt {a+b \sqrt {c x^2}} \, dx &=\frac {x \text {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 = 0.94, 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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Maple [A]
time = 0.26, size = 27, normalized size = 0.79
method | result | size |
default | \(\frac {2 x \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {3}{2}}}{3 b \sqrt {c \,x^{2}}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, 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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Fricas [A]
time = 0.36, 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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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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Giac [A]
time = 1.32, 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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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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