3.2925 \(\int x \sqrt{a+b \sqrt{c x^2}} \, dx\)

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]

(-2*a*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^2*c) + (2*(a + b*Sqrt[c*x^2])^(5/2))/(5*b^
2*c)

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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]

(-2*a*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^2*c) + (2*(a + b*Sqrt[c*x^2])^(5/2))/(5*b^
2*c)

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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]

-2*a*(a + b*sqrt(c*x**2))**(3/2)/(3*b**2*c) + 2*(a + b*sqrt(c*x**2))**(5/2)/(5*b
**2*c)

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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]

(2*Sqrt[a + b*Sqrt[c*x^2]]*(-2*a^2 + 3*b^2*c*x^2 + a*b*Sqrt[c*x^2]))/(15*b^2*c)

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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]

2/15*(a+b*(c*x^2)^(1/2))^(3/2)*(-2*a+3*b*(c*x^2)^(1/2))/b^2/c

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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]

2/15*(3*(sqrt(c*x^2)*b + a)^(5/2)/b^2 - 5*(sqrt(c*x^2)*b + a)^(3/2)*a/b^2)/c

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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]

2/15*(3*b^2*c*x^2 + sqrt(c*x^2)*a*b - 2*a^2)*sqrt(sqrt(c*x^2)*b + a)/(b^2*c)

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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]

Integral(x*sqrt(a + b*sqrt(c*x**2)), x)

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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]

2/15*(3*(b*sqrt(c)*x + a)^(5/2) - 5*(b*sqrt(c)*x + a)^(3/2)*a)/(b^2*c)