\(\int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx\) [131]

Optimal result

Integrand size = 21, antiderivative size = 224 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx=-\frac {\sqrt {a+b \sqrt {c+d x}}}{2 x^2}+\frac {b d \left (b c-a \sqrt {c+d x}\right ) \sqrt {a+b \sqrt {c+d x}}}{8 c \left (a^2-b^2 c\right ) x}-\frac {b \left (2 a-3 b \sqrt {c}\right ) d^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{16 \left (a-b \sqrt {c}\right )^{3/2} c^{3/2}}+\frac {b \left (2 a+3 b \sqrt {c}\right ) d^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{16 \left (a+b \sqrt {c}\right )^{3/2} c^{3/2}} \] Output:

-1/2*(a+b*(d*x+c)^(1/2))^(1/2)/x^2+1/8*b*d*(b*c-a*(d*x+c)^(1/2))*(a+b*(d*x 
+c)^(1/2))^(1/2)/c/(-b^2*c+a^2)/x-1/16*b*(2*a-3*b*c^(1/2))*d^2*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^(3/2)+1/ 
16*b*(2*a+3*b*c^(1/2))*d^2*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^(3/2)
 

Mathematica [A] (verified)

Time = 2.50 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {a+b \sqrt {c+d x}} \left (4 a^2 c+a b d x \sqrt {c+d x}-b^2 c (4 c+d x)\right )}{\left (a^2-b^2 c\right ) x^2}+\frac {b \left (2 a+3 b \sqrt {c}\right ) d^2 \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a-b \sqrt {c}}}\right )}{\left (-a-b \sqrt {c}\right )^{3/2}}+\frac {b \left (-2 a+3 b \sqrt {c}\right ) d^2 \arctan \left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a+b \sqrt {c}}}\right )}{\left (-a+b \sqrt {c}\right )^{3/2}}}{16 c^{3/2}} \] Input:

Integrate[Sqrt[a + b*Sqrt[c + d*x]]/x^3,x]
 

Output:

((-2*Sqrt[c]*Sqrt[a + b*Sqrt[c + d*x]]*(4*a^2*c + a*b*d*x*Sqrt[c + d*x] - 
b^2*c*(4*c + d*x)))/((a^2 - b^2*c)*x^2) + (b*(2*a + 3*b*Sqrt[c])*d^2*ArcTa 
n[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a - b*Sqrt[c]]])/(-a - b*Sqrt[c])^(3/2) 
+ (b*(-2*a + 3*b*Sqrt[c])*d^2*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a + b 
*Sqrt[c]]])/(-a + b*Sqrt[c])^(3/2))/(16*c^(3/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.91 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {896, 25, 1732, 561, 27, 1598, 27, 1405, 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.

\(\displaystyle \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx\)

\(\Big \downarrow \) 896

\(\displaystyle d^2 \int \frac {\sqrt {a+b \sqrt {c+d x}}}{d^3 x^3}d(c+d x)\)

\(\Big \downarrow \) 25

\(\displaystyle -d^2 \int -\frac {\sqrt {a+b \sqrt {c+d x}}}{d^3 x^3}d(c+d x)\)

\(\Big \downarrow \) 1732

\(\displaystyle -2 d^2 \int -\frac {\sqrt {c+d x} \sqrt {a+b \sqrt {c+d x}}}{d^3 x^3}d\sqrt {c+d x}\)

\(\Big \downarrow \) 561

\(\displaystyle -\frac {4 d^2 \int \frac {(a-c-d x) (c+d x)}{b \left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^3}d\sqrt {a+b \sqrt {c+d x}}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {4 d^2 \int \frac {(a-c-d x) (c+d x)}{\left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^3}d\sqrt {a+b \sqrt {c+d x}}}{b^2}\)

\(\Big \downarrow \) 1598

\(\displaystyle -\frac {4 d^2 \left (\frac {b^2 \int -\frac {2 c}{\left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}d\sqrt {a+b \sqrt {c+d x}}}{16 c}+\frac {b^2 \sqrt {a+b \sqrt {c+d x}}}{8 \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}\right )}{b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {4 d^2 \left (\frac {b^2 \sqrt {a+b \sqrt {c+d x}}}{8 \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}-\frac {1}{8} b^2 \int \frac {1}{\left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}d\sqrt {a+b \sqrt {c+d x}}\right )}{b^2}\)

\(\Big \downarrow \) 1405

\(\displaystyle -\frac {4 d^2 \left (\frac {b^2 \sqrt {a+b \sqrt {c+d x}}}{8 \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}-\frac {1}{8} b^2 \left (\frac {\sqrt {a+b \sqrt {c+d x}} \left (a^2-a (c+d x)+b^2 c\right )}{4 c \left (a^2-b^2 c\right ) \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}-\frac {b^4 \int \frac {2 \left (a^2+(c+d x) a-3 b^2 c\right )}{b^4 \left (\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}d\sqrt {a+b \sqrt {c+d x}}}{8 c \left (a^2-b^2 c\right )}\right )\right )}{b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {4 d^2 \left (\frac {b^2 \sqrt {a+b \sqrt {c+d x}}}{8 \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}-\frac {1}{8} b^2 \left (\frac {\sqrt {a+b \sqrt {c+d x}} \left (a^2-a (c+d x)+b^2 c\right )}{4 c \left (a^2-b^2 c\right ) \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}-\frac {\int \frac {a^2+(c+d x) a-3 b^2 c}{\frac {a^2}{b^2}-\frac {2 (c+d x) a}{b^2}+\frac {(c+d x)^2}{b^2}-c}d\sqrt {a+b \sqrt {c+d x}}}{4 c \left (a^2-b^2 c\right )}\right )\right )}{b^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {4 d^2 \left (\frac {b^2 \sqrt {a+b \sqrt {c+d x}}}{8 \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}-\frac {1}{8} b^2 \left (\frac {\sqrt {a+b \sqrt {c+d x}} \left (a^2-a (c+d x)+b^2 c\right )}{4 c \left (a^2-b^2 c\right ) \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}-\frac {\frac {1}{2} \left (-\frac {2 a^2}{b \sqrt {c}}+a+3 b \sqrt {c}\right ) \int \frac {1}{\frac {c+d x}{b^2}-\frac {a-b \sqrt {c}}{b^2}}d\sqrt {a+b \sqrt {c+d x}}+\frac {1}{2} \left (\frac {2 a^2}{b \sqrt {c}}+a-3 b \sqrt {c}\right ) \int \frac {1}{\frac {c+d x}{b^2}-\frac {a+b \sqrt {c}}{b^2}}d\sqrt {a+b \sqrt {c+d x}}}{4 c \left (a^2-b^2 c\right )}\right )\right )}{b^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {4 d^2 \left (\frac {b^2 \sqrt {a+b \sqrt {c+d x}}}{8 \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )^2}-\frac {1}{8} b^2 \left (\frac {\sqrt {a+b \sqrt {c+d x}} \left (a^2-a (c+d x)+b^2 c\right )}{4 c \left (a^2-b^2 c\right ) \left (\frac {a^2}{b^2}-\frac {2 a (c+d x)}{b^2}+\frac {(c+d x)^2}{b^2}-c\right )}-\frac {-\frac {b^2 \left (-\frac {2 a^2}{b \sqrt {c}}+a+3 b \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {a-b \sqrt {c}}}-\frac {b^2 \left (\frac {2 a^2}{b \sqrt {c}}+a-3 b \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {a+b \sqrt {c}}}}{4 c \left (a^2-b^2 c\right )}\right )\right )}{b^2}\)

Input:

Int[Sqrt[a + b*Sqrt[c + d*x]]/x^3,x]
 

Output:

(-4*d^2*((b^2*Sqrt[a + b*Sqrt[c + d*x]])/(8*(a^2/b^2 - c - (2*a*(c + d*x)) 
/b^2 + (c + d*x)^2/b^2)^2) - (b^2*((Sqrt[a + b*Sqrt[c + d*x]]*(a^2 + b^2*c 
 - a*(c + d*x)))/(4*c*(a^2 - b^2*c)*(a^2/b^2 - c - (2*a*(c + d*x))/b^2 + ( 
c + d*x)^2/b^2)) - (-1/2*(b^2*(a - (2*a^2)/(b*Sqrt[c]) + 3*b*Sqrt[c])*ArcT 
anh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/Sqrt[a - b*Sqrt[c]] - 
(b^2*(a + (2*a^2)/(b*Sqrt[c]) - 3*b*Sqrt[c])*ArcTanh[Sqrt[a + b*Sqrt[c + d 
*x]]/Sqrt[a + b*Sqrt[c]]])/(2*Sqrt[a + b*Sqrt[c]]))/(4*c*(a^2 - b^2*c))))/ 
8))/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 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*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[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(174)=348\).

Time = 0.41 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.66

method	result	size
derivativedivides	\(-4 d^{2} b^{4} \left (\frac {\frac {a \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{32 b^{2} c \left (-b^{2} c +a^{2}\right )}-\frac {\left (b^{2} c +3 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{32 b^{2} c \left (-b^{2} c +a^{2}\right )}+\frac {a \left (b^{2} c +3 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{32 b^{2} c \left (-b^{2} c +a^{2}\right )}-\frac {\left (-3 b^{2} c +a^{2}\right ) \sqrt {a +b \sqrt {d x +c}}}{32 b^{2} c}}{\left (\left (a +b \sqrt {d x +c}\right )^{2}-2 a \left (a +b \sqrt {d x +c}\right )-b^{2} c +a^{2}\right )^{2}}+\frac {\frac {\left (3 b^{2} c +a \sqrt {b^{2} c}-2 a^{2}\right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{2 \sqrt {b^{2} c}\, \sqrt {\sqrt {b^{2} c}-a}}+\frac {\left (-3 b^{2} c +a \sqrt {b^{2} c}+2 a^{2}\right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{2 \sqrt {b^{2} c}\, \sqrt {-\sqrt {b^{2} c}-a}}}{32 b^{2} c \left (-b^{2} c +a^{2}\right )}\right )\)	\(372\)
default	\(-4 d^{2} b^{4} \left (\frac {\frac {a \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{32 b^{2} c \left (-b^{2} c +a^{2}\right )}-\frac {\left (b^{2} c +3 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{32 b^{2} c \left (-b^{2} c +a^{2}\right )}+\frac {a \left (b^{2} c +3 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{32 b^{2} c \left (-b^{2} c +a^{2}\right )}-\frac {\left (-3 b^{2} c +a^{2}\right ) \sqrt {a +b \sqrt {d x +c}}}{32 b^{2} c}}{\left (\left (a +b \sqrt {d x +c}\right )^{2}-2 a \left (a +b \sqrt {d x +c}\right )-b^{2} c +a^{2}\right )^{2}}+\frac {\frac {\left (3 b^{2} c +a \sqrt {b^{2} c}-2 a^{2}\right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{2 \sqrt {b^{2} c}\, \sqrt {\sqrt {b^{2} c}-a}}+\frac {\left (-3 b^{2} c +a \sqrt {b^{2} c}+2 a^{2}\right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{2 \sqrt {b^{2} c}\, \sqrt {-\sqrt {b^{2} c}-a}}}{32 b^{2} c \left (-b^{2} c +a^{2}\right )}\right )\)	\(372\)

Input:

int((a+b*(d*x+c)^(1/2))^(1/2)/x^3,x,method=_RETURNVERBOSE)
 

Output:

-4*d^2*b^4*((1/32*a/b^2/c/(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(7/2)-1/32*(b^2 
*c+3*a^2)/b^2/c/(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(5/2)+1/32*a*(b^2*c+3*a^2 
)/b^2/c/(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(3/2)-1/32*(-3*b^2*c+a^2)/b^2/c*( 
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)^2+1/32/b^2/c/(-b^2*c+a^2)*(1/2*(3*b^2*c+a*(b^2*c)^(1/2)-2*a^2)/( 
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))+1/2*(-3*b^2*c+a*(b^2*c)^(1/2)+2*a^2)/(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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2856 vs. \(2 (175) = 350\).

Time = 0.32 (sec) , antiderivative size = 2856, normalized size of antiderivative = 12.75 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx=\text {Too large to display} \] Input:

integrate((a+b*(d*x+c)^(1/2))^(1/2)/x^3,x, algorithm="fricas")
 

Output:

1/32*((b^2*c^2 - a^2*c)*x^2*sqrt(-((15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^ 
2)*d^4 + (b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3)*sqrt((81*b^14 
*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^ 
4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3))) 
/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3))*log((81*b^10*c^2 - 8 
1*a^2*b^8*c + 20*a^4*b^6)*sqrt(sqrt(d*x + c)*b + a)*d^6 + ((27*b^10*c^4 - 
24*a^2*b^8*c^3 + 5*a^4*b^6*c^2)*d^4 - 2*(2*a*b^8*c^7 - 7*a^3*b^6*c^6 + 9*a 
^5*b^4*c^5 - 5*a^7*b^2*c^4 + a^9*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b^12*c + 
25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^6* 
c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))*sqrt(-((15*a*b^6*c^2 - 
 15*a^3*b^4*c + 4*a^5*b^2)*d^4 + (b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 
- a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 
- 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^ 
10*b^2*c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^ 
3))) - (b^2*c^2 - a^2*c)*x^2*sqrt(-((15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b 
^2)*d^4 + (b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3)*sqrt((81*b^1 
4*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a 
^4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)) 
)/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3))*log((81*b^10*c^2 - 
81*a^2*b^8*c + 20*a^4*b^6)*sqrt(sqrt(d*x + c)*b + a)*d^6 - ((27*b^10*c^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx=\text {Timed out} \] Input:

integrate((a+b*(d*x+c)**(1/2))**(1/2)/x**3,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx=\int { \frac {\sqrt {\sqrt {d x + c} b + a}}{x^{3}} \,d x } \] Input:

integrate((a+b*(d*x+c)^(1/2))^(1/2)/x^3,x, algorithm="maxima")
 

Output:

integrate(sqrt(sqrt(d*x + c)*b + a)/x^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (175) = 350\).

Time = 0.25 (sec) , antiderivative size = 895, normalized size of antiderivative = 4.00 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx =\text {Too large to display} \] Input:

integrate((a+b*(d*x+c)^(1/2))^(1/2)/x^3,x, algorithm="giac")
 

Output:

1/16*(((b^3*c^2 - a^2*b*c)^2*a*b^3*sqrt(c)*d^3 - (3*b^7*c^3 - 4*a^2*b^5*c^ 
2 + a^4*b^3*c)*d^3*abs(b^3*c^2 - a^2*b*c) + (3*a*b^9*c^(9/2) - 8*a^3*b^7*c 
^(7/2) + 7*a^5*b^5*c^(5/2) - 2*a^7*b^3*c^(3/2))*d^3)*arctan(sqrt(sqrt(d*x 
+ c)*b + a)/sqrt(-(a*b^2*c^2 - a^3*c + sqrt((a*b^2*c^2 - a^3*c)^2 + (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)))/((b^5* 
c^(9/2) - a*b^4*c^4 - 2*a^2*b^3*c^(7/2) + 2*a^3*b^2*c^3 + a^4*b*c^(5/2) - 
a^5*c^2)*sqrt(-b*sqrt(c) - a)*abs(b^3*c^2 - a^2*b*c)) + ((b^3*c^2 - a^2*b* 
c)^2*a*b^3*d^3 + (3*b^7*c^(5/2) - 4*a^2*b^5*c^(3/2) + a^4*b^3*sqrt(c))*d^3 
*abs(b^3*c^2 - a^2*b*c) + (3*a*b^9*c^4 - 8*a^3*b^7*c^3 + 7*a^5*b^5*c^2 - 2 
*a^7*b^3*c)*d^3)*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-(a*b^2*c^2 - a^3*c 
 - sqrt((a*b^2*c^2 - a^3*c)^2 + (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)))/((b^5*c^4 + a*b^4*c^(7/2) - 2*a^2*b^3*c^3 
- 2*a^3*b^2*c^(5/2) + a^4*b*c^2 + a^5*c^(3/2))*sqrt(b*sqrt(c) - a)*abs(b^3 
*c^2 - a^2*b*c)) - 2*(3*sqrt(sqrt(d*x + c)*b + a)*b^7*c^2*d^3 + (sqrt(d*x 
+ c)*b + a)^(5/2)*b^5*c*d^3 - (sqrt(d*x + c)*b + a)^(3/2)*a*b^5*c*d^3 - 4* 
sqrt(sqrt(d*x + c)*b + a)*a^2*b^5*c*d^3 - (sqrt(d*x + c)*b + a)^(7/2)*a*b^ 
3*d^3 + 3*(sqrt(d*x + c)*b + a)^(5/2)*a^2*b^3*d^3 - 3*(sqrt(d*x + c)*b + a 
)^(3/2)*a^3*b^3*d^3 + sqrt(sqrt(d*x + c)*b + a)*a^4*b^3*d^3)/((b^2*c^2 - a 
^2*c)*(b^2*c - (sqrt(d*x + c)*b + a)^2 + 2*(sqrt(d*x + c)*b + a)*a - a^2)^ 
2))/(b*d)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c+d\,x}}}{x^3} \,d x \] Input:

int((a + b*(c + d*x)^(1/2))^(1/2)/x^3,x)
 

Output:

int((a + b*(c + d*x)^(1/2))^(1/2)/x^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 719, normalized size of antiderivative = 3.21 \[ \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^3} \, dx =\text {Too large to display} \] Input:

int((a+b*(d*x+c)^(1/2))^(1/2)/x^3,x)
 

Output:

( - 4*sqrt(c)*sqrt(sqrt(c)*b - a)*atan(sqrt(sqrt(c + d*x)*b + a)/sqrt(sqrt 
(c)*b - a))*a**3*b*d**2*x**2 + 8*sqrt(c)*sqrt(sqrt(c)*b - a)*atan(sqrt(sqr 
t(c + d*x)*b + a)/sqrt(sqrt(c)*b - a))*a*b**3*c*d**2*x**2 - 2*sqrt(sqrt(c) 
*b - a)*atan(sqrt(sqrt(c + d*x)*b + a)/sqrt(sqrt(c)*b - a))*a**2*b**2*c*d* 
*2*x**2 + 6*sqrt(sqrt(c)*b - a)*atan(sqrt(sqrt(c + d*x)*b + a)/sqrt(sqrt(c 
)*b - a))*b**4*c**2*d**2*x**2 - 4*sqrt(c + d*x)*sqrt(sqrt(c + d*x)*b + a)* 
a**3*b*c*d*x + 4*sqrt(c + d*x)*sqrt(sqrt(c + d*x)*b + a)*a*b**3*c**2*d*x - 
 16*sqrt(sqrt(c + d*x)*b + a)*a**4*c**2 + 32*sqrt(sqrt(c + d*x)*b + a)*a** 
2*b**2*c**3 + 4*sqrt(sqrt(c + d*x)*b + a)*a**2*b**2*c**2*d*x - 16*sqrt(sqr 
t(c + d*x)*b + a)*b**4*c**4 - 4*sqrt(sqrt(c + d*x)*b + a)*b**4*c**3*d*x - 
2*sqrt(c)*sqrt(sqrt(c)*b + a)*log(sqrt(sqrt(c + d*x)*b + a) - sqrt(sqrt(c) 
*b + a))*a**3*b*d**2*x**2 + 4*sqrt(c)*sqrt(sqrt(c)*b + a)*log(sqrt(sqrt(c 
+ d*x)*b + a) - sqrt(sqrt(c)*b + a))*a*b**3*c*d**2*x**2 + 2*sqrt(c)*sqrt(s 
qrt(c)*b + a)*log(sqrt(sqrt(c + d*x)*b + a) + sqrt(sqrt(c)*b + a))*a**3*b* 
d**2*x**2 - 4*sqrt(c)*sqrt(sqrt(c)*b + a)*log(sqrt(sqrt(c + d*x)*b + a) + 
sqrt(sqrt(c)*b + a))*a*b**3*c*d**2*x**2 + sqrt(sqrt(c)*b + a)*log(sqrt(sqr 
t(c + d*x)*b + a) - sqrt(sqrt(c)*b + a))*a**2*b**2*c*d**2*x**2 - 3*sqrt(sq 
rt(c)*b + a)*log(sqrt(sqrt(c + d*x)*b + a) - sqrt(sqrt(c)*b + a))*b**4*c** 
2*d**2*x**2 - sqrt(sqrt(c)*b + a)*log(sqrt(sqrt(c + d*x)*b + a) + sqrt(sqr 
t(c)*b + a))*a**2*b**2*c*d**2*x**2 + 3*sqrt(sqrt(c)*b + a)*log(sqrt(sqr...