Integrand size = 13, antiderivative size = 42 \[ \int \sqrt {a+b \sqrt {x}} \, dx=-\frac {4 a \left (a+b \sqrt {x}\right )^{3/2}}{3 b^2}+\frac {4 \left (a+b \sqrt {x}\right )^{5/2}}{5 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {196, 45} \[ \int \sqrt {a+b \sqrt {x}} \, dx=\frac {4 \left (a+b \sqrt {x}\right )^{5/2}}{5 b^2}-\frac {4 a \left (a+b \sqrt {x}\right )^{3/2}}{3 b^2} \]
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Rule 45
Rule 196
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x \sqrt {a+b x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {4 a \left (a+b \sqrt {x}\right )^{3/2}}{3 b^2}+\frac {4 \left (a+b \sqrt {x}\right )^{5/2}}{5 b^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \sqrt {a+b \sqrt {x}} \, dx=\frac {4 \sqrt {a+b \sqrt {x}} \left (-2 a^2+a b \sqrt {x}+3 b^2 x\right )}{15 b^2} \]
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Time = 12.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {5}{2}}}{5}-\frac {4 a \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}}{b^{2}}\) | \(30\) |
default | \(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {5}{2}}}{5}-\frac {4 a \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}}{b^{2}}\) | \(30\) |
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76 \[ \int \sqrt {a+b \sqrt {x}} \, dx=\frac {4 \, {\left (3 \, b^{2} x + a b \sqrt {x} - 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{15 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (37) = 74\).
Time = 0.68 (sec) , antiderivative size = 272, normalized size of antiderivative = 6.48 \[ \int \sqrt {a+b \sqrt {x}} \, dx=- \frac {8 a^{\frac {9}{2}} x^{2} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac {5}{2}}} + \frac {8 a^{\frac {9}{2}} x^{2}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac {5}{2}}} - \frac {4 a^{\frac {7}{2}} b x^{\frac {5}{2}} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac {5}{2}}} + \frac {8 a^{\frac {7}{2}} b x^{\frac {5}{2}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac {5}{2}}} + \frac {16 a^{\frac {5}{2}} b^{2} x^{3} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac {5}{2}}} + \frac {12 a^{\frac {3}{2}} b^{3} x^{\frac {7}{2}} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac {5}{2}}} \]
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Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \sqrt {a+b \sqrt {x}} \, dx=\frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}}}{5 \, b^{2}} - \frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a}{3 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (30) = 60\).
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.81 \[ \int \sqrt {a+b \sqrt {x}} \, dx=\frac {4 \, {\left (\frac {5 \, {\left ({\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b \sqrt {x} + a} a\right )} a}{b} + \frac {3 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} - 10 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b \sqrt {x} + a} a^{2}}{b}\right )}}{15 \, b} \]
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Time = 5.83 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int \sqrt {a+b \sqrt {x}} \, dx=\frac {x\,\sqrt {a+b\,\sqrt {x}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},2;\ 3;\ -\frac {b\,\sqrt {x}}{a}\right )}{\sqrt {\frac {b\,\sqrt {x}}{a}+1}} \]
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