Optimal. Leaf size=40 \[ -\frac {4 a \sqrt {a+b \sqrt {x}}}{b^2}+\frac {4 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {196, 45}
\begin {gather*} \frac {4 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^2}-\frac {4 a \sqrt {a+b \sqrt {x}}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {x}}} \, dx &=2 \text {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 a \sqrt {a+b \sqrt {x}}}{b^2}+\frac {4 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.78 \begin {gather*} \frac {4 \left (-2 a+b \sqrt {x}\right ) \sqrt {a+b \sqrt {x}}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 30, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}-4 a \sqrt {a +b \sqrt {x}}}{b^{2}}\) | \(30\) |
default | \(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}-4 a \sqrt {a +b \sqrt {x}}}{b^{2}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 30, normalized size = 0.75 \begin {gather*} \frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}}}{3 \, b^{2}} - \frac {4 \, \sqrt {b \sqrt {x} + a} a}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 23, normalized size = 0.58 \begin {gather*} \frac {4 \, \sqrt {b \sqrt {x} + a} {\left (b \sqrt {x} - 2 \, a\right )}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs.
\(2 (36) = 72\).
time = 0.53, size = 219, normalized size = 5.48 \begin {gather*} - \frac {8 a^{\frac {7}{2}} x^{2} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} + \frac {8 a^{\frac {7}{2}} x^{2}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} - \frac {4 a^{\frac {5}{2}} b x^{\frac {5}{2}} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} + \frac {8 a^{\frac {5}{2}} b x^{\frac {5}{2}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} + \frac {4 a^{\frac {3}{2}} b^{2} x^{3} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 27, normalized size = 0.68 \begin {gather*} \frac {4 \, {\left ({\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b \sqrt {x} + a} a\right )}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 37, normalized size = 0.92 \begin {gather*} \frac {x\,\sqrt {\frac {b\,\sqrt {x}}{a}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ 3;\ -\frac {b\,\sqrt {x}}{a}\right )}{\sqrt {a+b\,\sqrt {x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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