Optimal. Leaf size=54 \[ -\frac {4 a \sqrt {a+b \sqrt {c+d x}}}{b^2 d}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {253, 196, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {c+d x}}} \, dx &=\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*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.78 \begin {gather*} \frac {4 \left (-2 a+b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{3 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 41, normalized size = 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 | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}-4 a \sqrt {a +b \sqrt {d x +c}}}{b^{2} d}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 42, normalized size = 0.78 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 34, normalized size = 0.63 \begin {gather*} \frac {4 \, \sqrt {\sqrt {d x + c} b + a} {\left (\sqrt {d x + c} b - 2 \, a\right )}}{3 \, b^{2} d} \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 \frac {1}{\sqrt {a + b \sqrt {c + d x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.50, size = 38, normalized size = 0.70 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.26, size = 44, normalized size = 0.81 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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