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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 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]]

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]

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 = 0.06, 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]])

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 59, normalized size = 0.86 \[ \frac {2 \, {\left (2 \, a x - b \sqrt {\frac {a^{2} x^{2} + b^{2} c}{b^{2}}}\right )} \sqrt {a x + b \sqrt {\frac {a^{2} x^{2} + b^{2} c}{b^{2}}}}}{3 \, a} \]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x + \sqrt {\frac {a^{2} x^{2}}{b^{2}} + c} b}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \sqrt {a x +\sqrt {\frac {a^{2} x^{2}}{b^{2}}+c}\, b}\, dx \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x + \sqrt {\frac {a^{2} x^{2}}{b^{2}} + c} b}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a\,x+b\,\sqrt {c+\frac {a^2\,x^2}{b^2}}} \,d x \]

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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} + c}}\, dx \]

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)

________________________________________________________________________________________