Integrand size = 17, antiderivative size = 54 \[ \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 a \sqrt {a+b \sqrt {c+d x}}}{b^2 d}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {253, 196, 45} \[ \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d}-\frac {4 a \sqrt {a+b \sqrt {c+d x}}}{b^2 d} \]
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Rule 45
Rule 196
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {x}}} \, dx,x,c+d x\right )}{d} \\ & = \frac {2 \text {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,\sqrt {c+d x}\right )}{d} \\ & = \frac {2 \text {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,\sqrt {c+d x}\right )}{d} \\ & = -\frac {4 a \sqrt {a+b \sqrt {c+d x}}}{b^2 d}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \left (-2 a+b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{3 b^2 d} \]
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Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}-4 a \sqrt {a +b \sqrt {d x +c}}}{b^{2} d}\) | \(41\) |
default | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}-4 a \sqrt {a +b \sqrt {d x +c}}}{b^{2} d}\) | \(41\) |
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none
Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \, \sqrt {\sqrt {d x + c} b + a} {\left (\sqrt {d x + c} b - 2 \, a\right )}}{3 \, b^{2} d} \]
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Time = 0.36 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \frac {2 \left (- a \sqrt {a + b \sqrt {c + d x}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {3}{2}}}{3}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {c + d x}{2 \sqrt {a}} & \text {otherwise} \end {cases}\right )}{d} & \text {for}\: d \neq 0 \\\frac {x}{\sqrt {a + b \sqrt {c}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \, {\left (\frac {{\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}}{b^{2}} - \frac {3 \, \sqrt {\sqrt {d x + c} b + a} a}{b^{2}}\right )}}{3 \, d} \]
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none
Time = 0.41 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {\sqrt {d x + c} b + a} a\right )}}{3 \, b^{2} d} \]
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Time = 17.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4\,{\left (a+b\,\sqrt {c+d\,x}\right )}^{3/2}}{3\,b^2\,d}-\frac {4\,a\,\sqrt {a+b\,\sqrt {c+d\,x}}}{b^2\,d} \]
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