Optimal. Leaf size=56 \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^2 c}-\frac{2 a \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^2 c} \]
[Out]
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Rubi [A] time = 0.0762577, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^2 c}-\frac{2 a \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^2 c} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[a + b*Sqrt[c*x^2]],x]
[Out]
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Rubi in Sympy [A] time = 9.28637, size = 48, normalized size = 0.86 \[ - \frac{2 a \left (a + b \sqrt{c x^{2}}\right )^{\frac{3}{2}}}{3 b^{2} c} + \frac{2 \left (a + b \sqrt{c x^{2}}\right )^{\frac{5}{2}}}{5 b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a+b*(c*x**2)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0276478, size = 54, normalized size = 0.96 \[ \frac{2 \sqrt{a+b \sqrt{c x^2}} \left (-2 a^2+a b \sqrt{c x^2}+3 b^2 c x^2\right )}{15 b^2 c} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a + b*Sqrt[c*x^2]],x]
[Out]
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Maple [A] time = 0.007, size = 36, normalized size = 0.6 \[{\frac{2}{15\,{b}^{2}c} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( -2\,a+3\,b\sqrt{c{x}^{2}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a+b*(c*x^2)^(1/2))^(1/2),x)
[Out]
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Maxima [A] time = 1.34161, size = 58, normalized size = 1.04 \[ \frac{2 \,{\left (\frac{3 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{15 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20705, size = 62, normalized size = 1.11 \[ \frac{2 \,{\left (3 \, b^{2} c x^{2} + \sqrt{c x^{2}} a b - 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{15 \, b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{a + b \sqrt{c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a+b*(c*x**2)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215624, size = 46, normalized size = 0.82 \[ \frac{2 \,{\left (3 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a\right )}}{15 \, b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*x,x, algorithm="giac")
[Out]