3.2938 \(\int \sqrt{a+b \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=34 \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0105151, 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, Rules used = {254, 32} \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

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

Rule 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{a+b \sqrt{c x^2}} \, dx &=\frac{x \operatorname{Subst}\left (\int \sqrt{a+b x} \, dx,x,\sqrt{c x^2}\right )}{\sqrt{c x^2}}\\ &=\frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.007202, 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]

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

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Maple [A]  time = 0.001, size = 27, normalized size = 0.8 \begin{align*}{\frac{2\,x}{3\,b} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(c*x^2)^(1/2))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.994868, size = 57, normalized size = 1.68 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.55586, size = 85, normalized size = 2.5 \begin{align*} \frac{2 \,{\left (b c x^{2} + \sqrt{c x^{2}} a\right )} \sqrt{\sqrt{c x^{2}} b + a}}{3 \, b c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sqrt{c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x**2)**(1/2))**(1/2),x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**2)), x)

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Giac [A]  time = 1.1316, size = 24, normalized size = 0.71 \begin{align*} \frac{2 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}}}{3 \, b \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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