3.21.93 \(\int \frac {\sqrt {c+d \sqrt {b+a x^2}}}{x} \, dx\) [2093]

3.21.93.1 Optimal result

Integrand size = 23, antiderivative size = 151 \[ \int \frac {\sqrt {c+d \sqrt {b+a x^2}}}{x} \, dx=2 \sqrt {c+d \sqrt {b+a x^2}}+\sqrt {-c+\sqrt {b} d} \arctan \left (\frac {\sqrt {-c+\sqrt {b} d} \sqrt {c+d \sqrt {b+a x^2}}}{c-\sqrt {b} d}\right )+\sqrt {-c-\sqrt {b} d} \arctan \left (\frac {\sqrt {-c-\sqrt {b} d} \sqrt {c+d \sqrt {b+a x^2}}}{c+\sqrt {b} d}\right ) \]

output

2*(c+d*(a*x^2+b)^(1/2))^(1/2)+(-c+d*b^(1/2))^(1/2)*arctan((-c+d*b^(1/2))^( 
1/2)*(c+d*(a*x^2+b)^(1/2))^(1/2)/(c-d*b^(1/2)))+(-c-d*b^(1/2))^(1/2)*arcta 
n((-c-d*b^(1/2))^(1/2)*(c+d*(a*x^2+b)^(1/2))^(1/2)/(c+d*b^(1/2)))
 

3.21.93.2 Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {c+d \sqrt {b+a x^2}}}{x} \, dx=2 \sqrt {c+d \sqrt {b+a x^2}}-\sqrt {-c-\sqrt {b} d} \arctan \left (\frac {\sqrt {c+d \sqrt {b+a x^2}}}{\sqrt {-c-\sqrt {b} d}}\right )-\sqrt {-c+\sqrt {b} d} \arctan \left (\frac {\sqrt {c+d \sqrt {b+a x^2}}}{\sqrt {-c+\sqrt {b} d}}\right ) \]

input

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

output

2*Sqrt[c + d*Sqrt[b + a*x^2]] - Sqrt[-c - Sqrt[b]*d]*ArcTan[Sqrt[c + d*Sqr 
t[b + a*x^2]]/Sqrt[-c - Sqrt[b]*d]] - Sqrt[-c + Sqrt[b]*d]*ArcTan[Sqrt[c + 
 d*Sqrt[b + a*x^2]]/Sqrt[-c + Sqrt[b]*d]]
 

3.21.93.3 Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.92, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {7282, 896, 25, 1732, 561, 25, 27, 1602, 25, 25, 27, 1480, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7282

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {c+d \sqrt {a x^2+b}}}{x^2}dx^2\)

\(\Big \downarrow \) 896

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {c+d \sqrt {a x^2+b}}}{a x^2}d\left (a x^2+b\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{2} \int -\frac {\sqrt {c+d \sqrt {a x^2+b}}}{a x^2}d\left (a x^2+b\right )\)

\(\Big \downarrow \) 1732

\(\displaystyle -\int \frac {\sqrt {a x^2+b} \sqrt {c+d \sqrt {a x^2+b}}}{b-x^4}d\sqrt {a x^2+b}\)

\(\Big \downarrow \) 561

\(\displaystyle -\frac {2 \int -\frac {x^4 \left (c-x^4\right )}{d \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+b-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {a x^2+b}}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {x^4 \left (c-x^4\right )}{d \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+b-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {a x^2+b}}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {x^4 \left (c-x^4\right )}{-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+b-\frac {c^2}{d^2}}d\sqrt {c+d \sqrt {a x^2+b}}}{d^2}\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {2 \left (d^2 \int -\frac {c x^4+\left (b-\frac {c^2}{d^2}\right ) d^2}{d^2 \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+b-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {a x^2+b}}+d^2 \sqrt {d \sqrt {a x^2+b}+c}\right )}{d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (d^2 \sqrt {d \sqrt {a x^2+b}+c}-d^2 \int -\frac {-c x^4+c^2-b d^2}{d^2 \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+b-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {a x^2+b}}\right )}{d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (d^2 \int \frac {-c x^4+c^2-b d^2}{d^2 \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+b-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {a x^2+b}}+d^2 \sqrt {d \sqrt {a x^2+b}+c}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\int \frac {-c x^4+c^2-b d^2}{-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+b-\frac {c^2}{d^2}}d\sqrt {c+d \sqrt {a x^2+b}}+d^2 \sqrt {d \sqrt {a x^2+b}+c}\right )}{d^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {2 \left (-\frac {1}{2} \left (c-\sqrt {b} d\right ) \int \frac {1}{\frac {c-\sqrt {b} d}{d^2}-\frac {x^4}{d^2}}d\sqrt {c+d \sqrt {a x^2+b}}-\frac {1}{2} \left (\sqrt {b} d+c\right ) \int \frac {1}{\frac {c+\sqrt {b} d}{d^2}-\frac {x^4}{d^2}}d\sqrt {c+d \sqrt {a x^2+b}}+d^2 \sqrt {d \sqrt {a x^2+b}+c}\right )}{d^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (-\frac {1}{2} d^2 \sqrt {c-\sqrt {b} d} \text {arctanh}\left (\frac {\sqrt {d \sqrt {a x^2+b}+c}}{\sqrt {c-\sqrt {b} d}}\right )-\frac {1}{2} d^2 \sqrt {\sqrt {b} d+c} \text {arctanh}\left (\frac {\sqrt {d \sqrt {a x^2+b}+c}}{\sqrt {\sqrt {b} d+c}}\right )+d^2 \sqrt {d \sqrt {a x^2+b}+c}\right )}{d^2}\)

input

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

output

(2*(d^2*Sqrt[c + d*Sqrt[b + a*x^2]] - (d^2*Sqrt[c - Sqrt[b]*d]*ArcTanh[Sqr 
t[c + d*Sqrt[b + a*x^2]]/Sqrt[c - Sqrt[b]*d]])/2 - (d^2*Sqrt[c + Sqrt[b]*d 
]*ArcTanh[Sqrt[c + d*Sqrt[b + a*x^2]]/Sqrt[c + Sqrt[b]*d]])/2))/d^2
 

3.21.93.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

rule 1732

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb 
ol] :> With[{g = Denominator[n]}, Simp[g   Subst[Int[x^(g - 1)*(d + e*x^(g* 
n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} 
, x] && EqQ[n2, 2*n] && FractionQ[n]
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

3.21.93.4 Maple [F]

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

input

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

output

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

3.21.93.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.21.93.6 Sympy [F]

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

input

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

output

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

3.21.93.7 Maxima [F]

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

input

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

output

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

3.21.93.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (110) = 220\).

Time = 0.30 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {c+d \sqrt {b+a x^2}}}{x} \, dx=\frac {2 \, \sqrt {\sqrt {a x^{2} + b} d + c} d + \frac {{\left (\sqrt {b} c d^{3} \mathrm {sgn}\left ({\left (\sqrt {a x^{2} + b} d + c\right )} d - c d\right ) - b d^{3} {\left | d \right |} + c^{2} d {\left | d \right |} \mathrm {sgn}\left ({\left (\sqrt {a x^{2} + b} d + c\right )} d - c d\right ) - \sqrt {b} c d^{3}\right )} \arctan \left (\frac {\sqrt {\sqrt {a x^{2} + b} d + c}}{\sqrt {-c + \sqrt {b d^{2}}}}\right )}{{\left (\sqrt {b} d + c\right )} \sqrt {\sqrt {b} d - c} {\left | d \right |}} + \frac {{\left (\sqrt {b} c d^{3} \mathrm {sgn}\left ({\left (\sqrt {a x^{2} + b} d + c\right )} d - c d\right ) + b d^{3} {\left | d \right |} - c^{2} d {\left | d \right |} \mathrm {sgn}\left ({\left (\sqrt {a x^{2} + b} d + c\right )} d - c d\right ) - \sqrt {b} c d^{3}\right )} \arctan \left (\frac {\sqrt {\sqrt {a x^{2} + b} d + c}}{\sqrt {-c - \sqrt {b d^{2}}}}\right )}{{\left (\sqrt {b} d - c\right )} \sqrt {-\sqrt {b} d - c} {\left | d \right |}}}{d} \]

input

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

output

(2*sqrt(sqrt(a*x^2 + b)*d + c)*d + (sqrt(b)*c*d^3*sgn((sqrt(a*x^2 + b)*d + 
 c)*d - c*d) - b*d^3*abs(d) + c^2*d*abs(d)*sgn((sqrt(a*x^2 + b)*d + c)*d - 
 c*d) - sqrt(b)*c*d^3)*arctan(sqrt(sqrt(a*x^2 + b)*d + c)/sqrt(-c + sqrt(b 
*d^2)))/((sqrt(b)*d + c)*sqrt(sqrt(b)*d - c)*abs(d)) + (sqrt(b)*c*d^3*sgn( 
(sqrt(a*x^2 + b)*d + c)*d - c*d) + b*d^3*abs(d) - c^2*d*abs(d)*sgn((sqrt(a 
*x^2 + b)*d + c)*d - c*d) - sqrt(b)*c*d^3)*arctan(sqrt(sqrt(a*x^2 + b)*d + 
 c)/sqrt(-c - sqrt(b*d^2)))/((sqrt(b)*d - c)*sqrt(-sqrt(b)*d - c)*abs(d))) 
/d
 

3.21.93.9 Mupad [F(-1)]

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

input

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

output

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