\(\int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [1939]

Optimal result

Integrand size = 35, antiderivative size = 135 \[ \int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2 \sqrt {b^2+a^2 x^2} \left (7 b^4 x+19 a^2 b^2 x^3+4 a^4 x^5\right )}{5 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}+\frac {2 \left (9 b^6+112 a^2 b^4 x^2+147 a^4 b^2 x^4+28 a^6 x^6\right )}{35 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}} \]

[Out]

2/5*(a^2*x^2+b^2)^(1/2)*(4*a^4*x^5+19*a^2*b^2*x^3+7*b^4*x)/(a*x+(a^2*x^2+b^2)^(1/2))^(7/2)+2/35*(28*a^6*x^6+14
7*a^4*b^2*x^4+112*a^2*b^4*x^2+9*b^6)/a/(a*x+(a^2*x^2+b^2)^(1/2))^(7/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2147, 276} \[ \int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {3 b^2 \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{4 a}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}{20 a}-\frac {b^6}{28 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}-\frac {b^4}{4 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}} \]

[In]

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

[Out]

-1/28*b^6/(a*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2)) - b^4/(4*a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + (3*b^2*Sqrt[a*
x + Sqrt[b^2 + a^2*x^2]])/(4*a) + (a*x + Sqrt[b^2 + a^2*x^2])^(5/2)/(20*a)

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2147

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^m, Subst[Int[x^n*((d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1)/(-d + x)^(2*(m + 1
))), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && E
qQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (b^2+x^2\right )^3}{x^{9/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{8 a} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b^6}{x^{9/2}}+\frac {3 b^4}{x^{5/2}}+\frac {3 b^2}{\sqrt {x}}+x^{3/2}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{8 a} \\ & = -\frac {b^6}{28 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}-\frac {b^4}{4 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {3 b^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}{20 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.93 \[ \int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2 \left (9 b^6+28 a^5 x^5 \left (a x+\sqrt {b^2+a^2 x^2}\right )+7 a b^4 x \left (16 a x+7 \sqrt {b^2+a^2 x^2}\right )+7 a^3 b^2 x^3 \left (21 a x+19 \sqrt {b^2+a^2 x^2}\right )\right )}{35 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}} \]

[In]

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

[Out]

(2*(9*b^6 + 28*a^5*x^5*(a*x + Sqrt[b^2 + a^2*x^2]) + 7*a*b^4*x*(16*a*x + 7*Sqrt[b^2 + a^2*x^2]) + 7*a^3*b^2*x^
3*(21*a*x + 19*Sqrt[b^2 + a^2*x^2])))/(35*a*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2))

Maple [F]

\[\int \frac {a^{2} x^{2}+b^{2}}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.61 \[ \int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {2 \, {\left (5 \, a^{4} x^{4} + 12 \, a^{2} b^{2} x^{2} - 9 \, b^{4} - {\left (5 \, a^{3} x^{3} + 13 \, a b^{2} x\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{35 \, a b^{2}} \]

[In]

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

[Out]

-2/35*(5*a^4*x^4 + 12*a^2*b^2*x^2 - 9*b^4 - (5*a^3*x^3 + 13*a*b^2*x)*sqrt(a^2*x^2 + b^2))*sqrt(a*x + sqrt(a^2*
x^2 + b^2))/(a*b^2)

Sympy [F]

\[ \int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {a^{2} x^{2} + b^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {a^{2} x^{2} + b^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {a^{2} x^{2} + b^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {b^2+a^2 x^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {a^2\,x^2+b^2}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]

[In]

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

[Out]

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