\(\int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx\) [137]

Optimal result

Integrand size = 21, antiderivative size = 163 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {\left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{\left (a^2-b^2 c\right ) x}-\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \left (a-b \sqrt {c}\right )^{3/2} \sqrt {c}}+\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \left (a+b \sqrt {c}\right )^{3/2} \sqrt {c}} \] Output:

-(a-b*(d*x+c)^(1/2))*(a+b*(d*x+c)^(1/2))^(1/2)/(-b^2*c+a^2)/x-1/2*b*d*arct 
anh((a+b*(d*x+c)^(1/2))^(1/2)/(a-b*c^(1/2))^(1/2))/(a-b*c^(1/2))^(3/2)/c^( 
1/2)+1/2*b*d*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a+b*c^(1/2))^(1/2))/(a+b*c 
^(1/2))^(3/2)/c^(1/2)
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {1}{2} \left (-\frac {2 \left (a-b \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{\left (a^2-b^2 c\right ) x}+\frac {b d \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a-b \sqrt {c}}}\right )}{\left (-a-b \sqrt {c}\right )^{3/2} \sqrt {c}}-\frac {b d \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a+b \sqrt {c}}}\right )}{\left (-a+b \sqrt {c}\right )^{3/2} \sqrt {c}}\right ) \] Input:

Integrate[1/(x^2*Sqrt[a + b*Sqrt[c + d*x]]),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.68 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {896, 1732, 561, 25, 27, 1492, 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.

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

\(\Big \downarrow \) 896

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

\(\Big \downarrow \) 1732

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1492

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

Int[1/(x^2*Sqrt[a + b*Sqrt[c + d*x]]),x]
 

Output:

(-4*d*((b^2*(2*a - c - d*x)*Sqrt[a + b*Sqrt[c + d*x]])/(4*(a^2 - b^2*c)*(a 
^2/b^2 - c - (2*a*(c + d*x))/b^2 + (c + d*x)^2/b^2)) + (b^2*((b*(b + a/Sqr 
t[c])*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(2*Sqrt[a - 
b*Sqrt[c]]) + (b^2*(1 - a/(b*Sqrt[c]))*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/S 
qrt[a + b*Sqrt[c]]])/(2*Sqrt[a + b*Sqrt[c]])))/(4*(a^2 - b^2*c))))/b^2
 

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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(258\) vs. \(2(125)=250\).

Time = 0.40 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(4 d \,b^{2} \left (-\frac {\sqrt {b^{2} c}\, \left (\frac {2 \sqrt {a +b \sqrt {d x +c}}}{\left (4 \sqrt {b^{2} c}-4 a \right ) \left (b \sqrt {d x +c}+\sqrt {b^{2} c}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{\left (4 \sqrt {b^{2} c}-4 a \right ) \sqrt {\sqrt {b^{2} c}-a}}\right )}{4 b^{2} c}+\frac {\sqrt {b^{2} c}\, \left (-\frac {2 \sqrt {a +b \sqrt {d x +c}}}{\left (-4 \sqrt {b^{2} c}-4 a \right ) \left (-b \sqrt {d x +c}+\sqrt {b^{2} c}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{\left (-4 \sqrt {b^{2} c}-4 a \right ) \sqrt {-\sqrt {b^{2} c}-a}}\right )}{4 b^{2} c}\right )\)	\(259\)
default	\(4 d \,b^{2} \left (-\frac {\sqrt {b^{2} c}\, \left (\frac {2 \sqrt {a +b \sqrt {d x +c}}}{\left (4 \sqrt {b^{2} c}-4 a \right ) \left (b \sqrt {d x +c}+\sqrt {b^{2} c}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{\left (4 \sqrt {b^{2} c}-4 a \right ) \sqrt {\sqrt {b^{2} c}-a}}\right )}{4 b^{2} c}+\frac {\sqrt {b^{2} c}\, \left (-\frac {2 \sqrt {a +b \sqrt {d x +c}}}{\left (-4 \sqrt {b^{2} c}-4 a \right ) \left (-b \sqrt {d x +c}+\sqrt {b^{2} c}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{\left (-4 \sqrt {b^{2} c}-4 a \right ) \sqrt {-\sqrt {b^{2} c}-a}}\right )}{4 b^{2} c}\right )\)	\(259\)

Input:

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

Output:

4*d*b^2*(-1/4*(b^2*c)^(1/2)/b^2/c*(2*(a+b*(d*x+c)^(1/2))^(1/2)/(4*(b^2*c)^ 
(1/2)-4*a)/(b*(d*x+c)^(1/2)+(b^2*c)^(1/2))+2/(4*(b^2*c)^(1/2)-4*a)/((b^2*c 
)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/((b^2*c)^(1/2)-a)^(1/2)) 
)+1/4*(b^2*c)^(1/2)/b^2/c*(-2*(a+b*(d*x+c)^(1/2))^(1/2)/(-4*(b^2*c)^(1/2)- 
4*a)/(-b*(d*x+c)^(1/2)+(b^2*c)^(1/2))+2/(-4*(b^2*c)^(1/2)-4*a)/(-(b^2*c)^( 
1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/(-(b^2*c)^(1/2)-a)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2493 vs. \(2 (127) = 254\).

Time = 0.20 (sec) , antiderivative size = 2493, normalized size of antiderivative = 15.29 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*((b^2*c - a^2)*x*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (b^6*c^4 - 3*a^2*b 
^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)* 
d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8* 
b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2* 
c^2 - a^6*c))*log((b^6*c + 3*a^2*b^4)*sqrt(sqrt(d*x + c)*b + a)*d^3 + (2*( 
a*b^6*c^2 + 3*a^3*b^4*c)*d^2 - (b^8*c^5 - 2*a^2*b^6*c^4 + 2*a^6*b^2*c^2 - 
a^8*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^1 
0*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 
+ a^12*c)))*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (b^6*c^4 - 3*a^2*b^4*c^3 + 
3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12 
*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 
 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6 
*c))) - (b^2*c - a^2)*x*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (b^6*c^4 - 3*a^ 
2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^ 
6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a 
^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b 
^2*c^2 - a^6*c))*log((b^6*c + 3*a^2*b^4)*sqrt(sqrt(d*x + c)*b + a)*d^3 - ( 
2*(a*b^6*c^2 + 3*a^3*b^4*c)*d^2 - (b^8*c^5 - 2*a^2*b^6*c^4 + 2*a^6*b^2*c^2 
 - a^8*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2* 
b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^...
 

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b \sqrt {c + d x}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\int { \frac {1}{\sqrt {\sqrt {d x + c} b + a} x^{2}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 637 vs. \(2 (127) = 254\).

Time = 0.24 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.91 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {d {\left (\frac {{\left ({\left (b^{3} c - a^{2} b\right )}^{2} \sqrt {b \sqrt {c} - a} b^{4} c - 2 \, {\left (a b^{6} c^{\frac {3}{2}} - a^{3} b^{4} \sqrt {c}\right )} \sqrt {b \sqrt {c} - a} {\left | -b^{3} c + a^{2} b \right |} + {\left (a^{2} b^{8} c^{2} - 2 \, a^{4} b^{6} c + a^{6} b^{4}\right )} \sqrt {b \sqrt {c} - a}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{2} c - a^{3} + \sqrt {{\left (a b^{2} c - a^{3}\right )}^{2} + {\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} {\left (b^{2} c - a^{2}\right )}}}{b^{2} c - a^{2}}}}\right )}{{\left (b^{6} c^{\frac {7}{2}} - 3 \, a^{2} b^{4} c^{\frac {5}{2}} + 3 \, a^{4} b^{2} c^{\frac {3}{2}} - a^{6} \sqrt {c}\right )} {\left | -b^{3} c + a^{2} b \right |}} - \frac {{\left ({\left (b^{3} c - a^{2} b\right )}^{2} \sqrt {-b \sqrt {c} - a} b^{4} c + 2 \, {\left (a b^{6} c^{\frac {3}{2}} - a^{3} b^{4} \sqrt {c}\right )} \sqrt {-b \sqrt {c} - a} {\left | -b^{3} c + a^{2} b \right |} + {\left (a^{2} b^{8} c^{2} - 2 \, a^{4} b^{6} c + a^{6} b^{4}\right )} \sqrt {-b \sqrt {c} - a}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-\frac {a b^{2} c - a^{3} - \sqrt {{\left (a b^{2} c - a^{3}\right )}^{2} + {\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} {\left (b^{2} c - a^{2}\right )}}}{b^{2} c - a^{2}}}}\right )}{{\left (b^{6} c^{\frac {7}{2}} - 3 \, a^{2} b^{4} c^{\frac {5}{2}} + 3 \, a^{4} b^{2} c^{\frac {3}{2}} - a^{6} \sqrt {c}\right )} {\left | -b^{3} c + a^{2} b \right |}} + \frac {2 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{4} - 2 \, \sqrt {\sqrt {d x + c} b + a} a b^{4}\right )}}{{\left (b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{2} + 2 \, {\left (\sqrt {d x + c} b + a\right )} a - a^{2}\right )} {\left (b^{2} c - a^{2}\right )}}\right )}}{2 \, b^{2}} \] Input:

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

Output:

1/2*d*(((b^3*c - a^2*b)^2*sqrt(b*sqrt(c) - a)*b^4*c - 2*(a*b^6*c^(3/2) - a 
^3*b^4*sqrt(c))*sqrt(b*sqrt(c) - a)*abs(-b^3*c + a^2*b) + (a^2*b^8*c^2 - 2 
*a^4*b^6*c + a^6*b^4)*sqrt(b*sqrt(c) - a))*arctan(sqrt(sqrt(d*x + c)*b + a 
)/sqrt(-(a*b^2*c - a^3 + sqrt((a*b^2*c - a^3)^2 + (b^4*c^2 - 2*a^2*b^2*c + 
 a^4)*(b^2*c - a^2)))/(b^2*c - a^2)))/((b^6*c^(7/2) - 3*a^2*b^4*c^(5/2) + 
3*a^4*b^2*c^(3/2) - a^6*sqrt(c))*abs(-b^3*c + a^2*b)) - ((b^3*c - a^2*b)^2 
*sqrt(-b*sqrt(c) - a)*b^4*c + 2*(a*b^6*c^(3/2) - a^3*b^4*sqrt(c))*sqrt(-b* 
sqrt(c) - a)*abs(-b^3*c + a^2*b) + (a^2*b^8*c^2 - 2*a^4*b^6*c + a^6*b^4)*s 
qrt(-b*sqrt(c) - a))*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-(a*b^2*c - a^3 
 - sqrt((a*b^2*c - a^3)^2 + (b^4*c^2 - 2*a^2*b^2*c + a^4)*(b^2*c - a^2)))/ 
(b^2*c - a^2)))/((b^6*c^(7/2) - 3*a^2*b^4*c^(5/2) + 3*a^4*b^2*c^(3/2) - a^ 
6*sqrt(c))*abs(-b^3*c + a^2*b)) + 2*((sqrt(d*x + c)*b + a)^(3/2)*b^4 - 2*s 
qrt(sqrt(d*x + c)*b + a)*a*b^4)/((b^2*c - (sqrt(d*x + c)*b + a)^2 + 2*(sqr 
t(d*x + c)*b + a)*a - a^2)*(b^2*c - a^2)))/b^2
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {1}{x^2\,\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.88 \[ \int \frac {1}{x^2 \sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {-2 \sqrt {c}\, \sqrt {\sqrt {c}\, b -a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {d x +c}\, b +a}}{\sqrt {\sqrt {c}\, b -a}}\right ) a^{2} b d x -2 \sqrt {c}\, \sqrt {\sqrt {c}\, b -a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {d x +c}\, b +a}}{\sqrt {\sqrt {c}\, b -a}}\right ) b^{3} c d x -4 \sqrt {\sqrt {c}\, b -a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {d x +c}\, b +a}}{\sqrt {\sqrt {c}\, b -a}}\right ) a \,b^{2} c d x +4 \sqrt {d x +c}\, \sqrt {\sqrt {d x +c}\, b +a}\, a^{2} b c -4 \sqrt {d x +c}\, \sqrt {\sqrt {d x +c}\, b +a}\, b^{3} c^{2}-4 \sqrt {\sqrt {d x +c}\, b +a}\, a^{3} c +4 \sqrt {\sqrt {d x +c}\, b +a}\, a \,b^{2} c^{2}-\sqrt {c}\, \sqrt {\sqrt {c}\, b +a}\, \mathrm {log}\left (\sqrt {\sqrt {d x +c}\, b +a}-\sqrt {\sqrt {c}\, b +a}\right ) a^{2} b d x -\sqrt {c}\, \sqrt {\sqrt {c}\, b +a}\, \mathrm {log}\left (\sqrt {\sqrt {d x +c}\, b +a}-\sqrt {\sqrt {c}\, b +a}\right ) b^{3} c d x +\sqrt {c}\, \sqrt {\sqrt {c}\, b +a}\, \mathrm {log}\left (\sqrt {\sqrt {d x +c}\, b +a}+\sqrt {\sqrt {c}\, b +a}\right ) a^{2} b d x +\sqrt {c}\, \sqrt {\sqrt {c}\, b +a}\, \mathrm {log}\left (\sqrt {\sqrt {d x +c}\, b +a}+\sqrt {\sqrt {c}\, b +a}\right ) b^{3} c d x +2 \sqrt {\sqrt {c}\, b +a}\, \mathrm {log}\left (\sqrt {\sqrt {d x +c}\, b +a}-\sqrt {\sqrt {c}\, b +a}\right ) a \,b^{2} c d x -2 \sqrt {\sqrt {c}\, b +a}\, \mathrm {log}\left (\sqrt {\sqrt {d x +c}\, b +a}+\sqrt {\sqrt {c}\, b +a}\right ) a \,b^{2} c d x}{4 c x \left (b^{4} c^{2}-2 a^{2} b^{2} c +a^{4}\right )} \] Input:

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

Output:

( - 2*sqrt(c)*sqrt(sqrt(c)*b - a)*atan(sqrt(sqrt(c + d*x)*b + a)/sqrt(sqrt 
(c)*b - a))*a**2*b*d*x - 2*sqrt(c)*sqrt(sqrt(c)*b - a)*atan(sqrt(sqrt(c + 
d*x)*b + a)/sqrt(sqrt(c)*b - a))*b**3*c*d*x - 4*sqrt(sqrt(c)*b - a)*atan(s 
qrt(sqrt(c + d*x)*b + a)/sqrt(sqrt(c)*b - a))*a*b**2*c*d*x + 4*sqrt(c + d* 
x)*sqrt(sqrt(c + d*x)*b + a)*a**2*b*c - 4*sqrt(c + d*x)*sqrt(sqrt(c + d*x) 
*b + a)*b**3*c**2 - 4*sqrt(sqrt(c + d*x)*b + a)*a**3*c + 4*sqrt(sqrt(c + d 
*x)*b + a)*a*b**2*c**2 - sqrt(c)*sqrt(sqrt(c)*b + a)*log(sqrt(sqrt(c + d*x 
)*b + a) - sqrt(sqrt(c)*b + a))*a**2*b*d*x - sqrt(c)*sqrt(sqrt(c)*b + a)*l 
og(sqrt(sqrt(c + d*x)*b + a) - sqrt(sqrt(c)*b + a))*b**3*c*d*x + sqrt(c)*s 
qrt(sqrt(c)*b + a)*log(sqrt(sqrt(c + d*x)*b + a) + sqrt(sqrt(c)*b + a))*a* 
*2*b*d*x + sqrt(c)*sqrt(sqrt(c)*b + a)*log(sqrt(sqrt(c + d*x)*b + a) + sqr 
t(sqrt(c)*b + a))*b**3*c*d*x + 2*sqrt(sqrt(c)*b + a)*log(sqrt(sqrt(c + d*x 
)*b + a) - sqrt(sqrt(c)*b + a))*a*b**2*c*d*x - 2*sqrt(sqrt(c)*b + a)*log(s 
qrt(sqrt(c + d*x)*b + a) + sqrt(sqrt(c)*b + a))*a*b**2*c*d*x)/(4*c*x*(a**4 
 - 2*a**2*b**2*c + b**4*c**2))