Optimal. Leaf size=56 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^2 d}-\frac{4 a \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d} \]
[Out]
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Rubi [A] time = 0.0679516, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^2 d}-\frac{4 a \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c + d*x]],x]
[Out]
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Rubi in Sympy [A] time = 4.01873, size = 48, normalized size = 0.86 \[ - \frac{4 a \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}}}{3 b^{2} d} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{5}{2}}}{5 b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(d*x+c)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0493423, size = 55, normalized size = 0.98 \[ \frac{4 \sqrt{a+b \sqrt{c+d x}} \left (-2 a^2+a b \sqrt{c+d x}+3 b^2 (c+d x)\right )}{15 b^2 d} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*Sqrt[c + d*x]],x]
[Out]
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Maple [A] time = 0.003, size = 41, normalized size = 0.7 \[ 4\,{\frac{1/5\, \left ( a+b\sqrt{dx+c} \right ) ^{5/2}-1/3\, \left ( a+b\sqrt{dx+c} \right ) ^{3/2}a}{{b}^{2}d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(d*x+c)^(1/2))^(1/2),x)
[Out]
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Maxima [A] time = 0.693783, size = 58, normalized size = 1.04 \[ \frac{4 \,{\left (\frac{3 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(d*x + c)*b + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.341986, size = 68, normalized size = 1.21 \[ \frac{4 \,{\left (3 \, b^{2} d x + 3 \, b^{2} c + \sqrt{d x + c} a b - 2 \, a^{2}\right )} \sqrt{\sqrt{d x + c} b + a}}{15 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(d*x + c)*b + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(d*x+c)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.28698, size = 159, normalized size = 2.84 \[ \frac{4 \,{\left (3 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{2} b^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 5 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} a b^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )}{\left | b \right |}}{15 \, b^{6} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(d*x + c)*b + a),x, algorithm="giac")
[Out]