Optimal. Leaf size=34 \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0262264, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c*x^2]],x]
[Out]
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Rubi in Sympy [A] time = 2.15646, size = 29, normalized size = 0.85 \[ \frac{2 x \left (a + b \sqrt{c x^{2}}\right )^{\frac{3}{2}}}{3 b \sqrt{c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x**2)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.00634238, size = 34, normalized size = 1. \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*Sqrt[c*x^2]],x]
[Out]
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Maple [A] time = 0.004, size = 27, normalized size = 0.8 \[{\frac{2\,x}{3\,b} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x^2)^(1/2))^(1/2),x)
[Out]
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Maxima [A] time = 1.38011, size = 57, normalized size = 1.68 \[ \frac{{\left ({\left (c^{\frac{3}{2}} + c\right )} b x + a{\left (c + \sqrt{c}\right )}\right )} \sqrt{b \sqrt{c} x + a}}{{\left (c^{2} + 2 \, c^{\frac{3}{2}} + c\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206791, size = 54, normalized size = 1.59 \[ \frac{2 \,{\left (b c x^{2} + \sqrt{c x^{2}} a\right )} \sqrt{\sqrt{c x^{2}} b + a}}{3 \, b c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b \sqrt{c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x**2)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213123, size = 24, normalized size = 0.71 \[ \frac{2 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}}}{3 \, b \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a),x, algorithm="giac")
[Out]