3.7.30 \(\int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^2} \, dx\) [630]

3.7.30.1 Optimal result

Integrand size = 21, antiderivative size = 137 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^2} \, dx=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {a-b \sqrt {c}} \sqrt {c}}-\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {a+b \sqrt {c}} \sqrt {c}} \]

output

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

3.7.30.2 Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^2} \, dx=\frac {1}{2} \left (-\frac {2 \sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a-b \sqrt {c}}}\right )}{\sqrt {-a-b \sqrt {c}} \sqrt {c}}-\frac {b d \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a+b \sqrt {c}}}\right )}{\sqrt {-a+b \sqrt {c}} \sqrt {c}}\right ) \]

input

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

output

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

3.7.30.3 Rubi [A] (warning: unable to verify)

Time = 0.34 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.36, 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, 1598, 27, 1406, 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.

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

\(\Big \downarrow \) 896

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

\(\Big \downarrow \) 1732

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1598

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1406

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

\(\Big \downarrow \) 221

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

input

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

output

(-4*d*((b^2*Sqrt[a + b*Sqrt[c + d*x]])/(4*(a^2/b^2 - c - (2*a*(c + d*x))/b 
^2 + (c + d*x)^2/b^2)) - (b^2*((b*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a 
 - b*Sqrt[c]]])/(2*Sqrt[a - b*Sqrt[c]]*Sqrt[c]) - (b*ArcTanh[Sqrt[a + b*Sq 
rt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/(2*Sqrt[a + b*Sqrt[c]]*Sqrt[c])))/4))/b 
^2
 

3.7.30.3.1 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 1406

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

rule 1598

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

3.7.30.4 Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(4 d \,b^{2} \left (-\frac {\sqrt {a +b \sqrt {d x +c}}}{4 \left (\left (a +b \sqrt {d x +c}\right )^{2}-2 a \left (a +b \sqrt {d x +c}\right )-b^{2} c +a^{2}\right )}-\frac {\arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{8 \sqrt {b^{2} c}\, \sqrt {\sqrt {b^{2} c}-a}}+\frac {\arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{8 \sqrt {b^{2} c}\, \sqrt {-\sqrt {b^{2} c}-a}}\right )\)	\(166\)
default	\(4 d \,b^{2} \left (-\frac {\sqrt {a +b \sqrt {d x +c}}}{4 \left (\left (a +b \sqrt {d x +c}\right )^{2}-2 a \left (a +b \sqrt {d x +c}\right )-b^{2} c +a^{2}\right )}-\frac {\arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{8 \sqrt {b^{2} c}\, \sqrt {\sqrt {b^{2} c}-a}}+\frac {\arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{8 \sqrt {b^{2} c}\, \sqrt {-\sqrt {b^{2} c}-a}}\right )\)	\(166\)

input

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

output

4*d*b^2*(-1/4*(a+b*(d*x+c)^(1/2))^(1/2)/((a+b*(d*x+c)^(1/2))^2-2*a*(a+b*(d 
*x+c)^(1/2))-b^2*c+a^2)-1/8/(b^2*c)^(1/2)/((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))+1/8/(b^2*c)^(1/2)/(-(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)))
 

3.7.30.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (101) = 202\).

Time = 0.35 (sec) , antiderivative size = 1003, normalized size of antiderivative = 7.32 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^2} \, dx=-\frac {x \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} + {\left (b^{4} c d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) - x \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} - {\left (b^{4} c d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) + x \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} + {\left (b^{4} c d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) - x \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} - {\left (b^{4} c d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) + 4 \, \sqrt {\sqrt {d x + c} b + a}}{4 \, x} \]

input

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

output

-1/4*(x*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)) 
*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d 
^3 + (b^4*c*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(a*b^2*c 
^2 - a^3*c))*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^ 
4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))) - x*sqrt(-(a*b^2*d^2 + sqrt(b 
^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^ 
2*c))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d^3 - (b^4*c*d^2 - sqrt(b^6*d^4/(b 
^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(a*b^2*c^2 - a^3*c))*sqrt(-(a*b^2*d^2 + s 
qrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 
 - a^2*c))) + x*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + 
 a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))*log(sqrt(sqrt(d*x + c)*b + 
a)*b^4*d^3 + (b^4*c*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))* 
(a*b^2*c^2 - a^3*c))*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2* 
c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))) - x*sqrt(-(a*b^2*d^2 
- sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2* 
c^2 - a^2*c))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d^3 - (b^4*c*d^2 + sqrt(b^ 
6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(a*b^2*c^2 - a^3*c))*sqrt(-(a*b^2 
*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/ 
(b^2*c^2 - a^2*c))) + 4*sqrt(sqrt(d*x + c)*b + a))/x
 

3.7.30.6 Sympy [F]

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

input

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

output

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

3.7.30.7 Maxima [F]

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

input

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

output

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

3.7.30.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (101) = 202\).

Time = 0.41 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^2} \, dx=\frac {\frac {2 \, \sqrt {\sqrt {d x + c} b + a} b^{3} d^{2}}{b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{2} + 2 \, {\left (\sqrt {d x + c} b + a\right )} a - a^{2}} - \frac {{\left (b^{3} c d^{2} {\left | b \right |} + a b^{3} \sqrt {c} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a + \sqrt {b^{2} c}}}\right )}{{\left (b c^{\frac {3}{2}} + a c\right )} \sqrt {b \sqrt {c} - a} {\left | b \right |}} + \frac {{\left (b^{3} c d^{2} {\left | b \right |} - a b^{3} \sqrt {c} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a - \sqrt {b^{2} c}}}\right )}{{\left (b c^{\frac {3}{2}} - a c\right )} \sqrt {-b \sqrt {c} - a} {\left | b \right |}}}{2 \, b d} \]

input

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

output

1/2*(2*sqrt(sqrt(d*x + c)*b + a)*b^3*d^2/(b^2*c - (sqrt(d*x + c)*b + a)^2 
+ 2*(sqrt(d*x + c)*b + a)*a - a^2) - (b^3*c*d^2*abs(b) + a*b^3*sqrt(c)*d^2 
)*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-a + sqrt(b^2*c)))/((b*c^(3/2) + a 
*c)*sqrt(b*sqrt(c) - a)*abs(b)) + (b^3*c*d^2*abs(b) - a*b^3*sqrt(c)*d^2)*a 
rctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-a - sqrt(b^2*c)))/((b*c^(3/2) - a*c) 
*sqrt(-b*sqrt(c) - a)*abs(b)))/(b*d)
 

3.7.30.9 Mupad [F(-1)]

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

input

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

output

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