Optimal. Leaf size=56 \[ \frac {4 \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^2 d}-\frac {4 a \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 56, 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 = {247, 190, 43} \[ \frac {4 \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^2 d}-\frac {4 a \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 247
Rubi steps
\begin {align*} \int \sqrt {a+b \sqrt {c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+b \sqrt {x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int x \sqrt {a+b x} \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=-\frac {4 a \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.77 \[ \frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (3 b \sqrt {c+d x}-2 a\right )}{15 b^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 50, normalized size = 0.89 \[ \frac {4 \, {\left (3 \, b^{2} d x + 3 \, b^{2} c + \sqrt {d x + c} a b - 2 \, a^{2}\right )} \sqrt {\sqrt {d x + c} b + a}}{15 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 99, normalized size = 1.77 \[ \frac {4 \, {\left (\frac {5 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {\sqrt {d x + c} b + a} a\right )} a}{b} + \frac {3 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {\sqrt {d x + c} b + a} a^{2}}{b}\right )}}{15 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.73 \[ \frac {-\frac {4 \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} a}{3}+\frac {4 \left (a +\sqrt {d x +c}\, b \right )^{\frac {5}{2}}}{5}}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 43, normalized size = 0.77 \[ \frac {4 \, {\left (\frac {3 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}}}{b^{2}} - \frac {5 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a}{b^{2}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.39, size = 44, normalized size = 0.79 \[ \frac {4\,{\left (a+b\,\sqrt {c+d\,x}\right )}^{5/2}}{5\,b^2\,d}-\frac {4\,a\,{\left (a+b\,\sqrt {c+d\,x}\right )}^{3/2}}{3\,b^2\,d} \]
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 \[ \int \sqrt {a + b \sqrt {c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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