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

Optimal. Leaf size=69 \[ \frac{\left (b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x\right )^{3/2}}{3 a}-\frac{b^2 c}{a \sqrt{b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x}} \]

[Out]

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

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Rubi [A]  time = 0.0570967, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2117, 14} \[ \frac{\left (b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x\right )^{3/2}}{3 a}-\frac{b^2 c}{a \sqrt{b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0574548, size = 67, normalized size = 0.97 \[ \frac{2 \left (a b x \sqrt{\frac{a^2 x^2}{b^2}+c}+a^2 x^2+b^2 (-c)\right )}{3 a \sqrt{b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int \sqrt{ax+b\sqrt{c+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.99495, size = 120, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (2 \, a x - b \sqrt{\frac{a^{2} x^{2} + b^{2} c}{b^{2}}}\right )} \sqrt{a x + b \sqrt{\frac{a^{2} x^{2} + b^{2} c}{b^{2}}}}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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