3.2932 \(\int x \sqrt{a+b \sqrt{c x^2}} \, dx\)

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 = 0.0250762, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {368, 43} \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^2 c}-\frac{2 a \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^2 c} \]

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)

Rule 368

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x \sqrt{a+b \sqrt{c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x \sqrt{a+b x} \, dx,x,\sqrt{c x^2}\right )}{c}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx,x,\sqrt{c x^2}\right )}{c}\\ &=-\frac{2 a \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^2 c}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^2 c}\\ \end{align*}

Mathematica [A]  time = 0.0159931, size = 43, normalized size = 0.77 \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{3/2} \left (3 b \sqrt{c x^2}-2 a\right )}{15 b^2 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

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

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Maple [A]  time = 0.007, size = 36, normalized size = 0.6 \begin{align*}{\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 ) } \end{align*}

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 = 0.931034, size = 58, normalized size = 1.04 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*(c*x^2)^(1/2))^(1/2),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 = 1.15857, size = 105, normalized size = 1.88 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*(c*x^2)^(1/2))^(1/2),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. \begin{align*} \int x \sqrt{a + b \sqrt{c x^{2}}}\, dx \end{align*}

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 [A]  time = 1.12015, size = 46, normalized size = 0.82 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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