3.4.29 \(\int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx\) [329]

3.4.29.1 Optimal result

Integrand size = 21, antiderivative size = 362 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx=-\frac {(2 b+a c) d^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^2 (b+a c)}-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c x^3}+\frac {(2 b+a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^2 (b+a c) x}+\frac {(2 b+a c) d^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{3/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {a d^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

-1/3*(a*c+2*b)*d^2*x*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^2/(a*c+b)-1/3*(d* 
x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c/x^3+1/3*(a*c+2*b)*d*(d*x^2+c)*( 
(a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^2/(a*c+b)/x+1/3*(a*c+2*b)*d^(3/2)*(1/(1 
+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c) 
^(1/2),(b/(a*c+b))^(1/2))*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^(3/2)/(a*c+b 
)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)-1/3*a*d^(3/2)*(1/(1+d*x^2/c) 
)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(b 
/(a*c+b))^(1/2))*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*c+b)/c^(1/2)/(c*(a*d 
*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)
 

3.4.29.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.74 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx=-\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (b^2 \left (c-2 d x^2\right )+a^2 c \left (c^2-d^2 x^4\right )+2 a b \left (c^2-c d x^2-d^2 x^4\right )\right )-i \left (2 b^2+3 a b c+a^2 c^2\right ) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+2 i b (b+a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 c^2 (b+a c) \sqrt {\frac {d}{c}} x^3 \left (b+a \left (c+d x^2\right )\right )} \]

input

Integrate[Sqrt[a + b/(c + d*x^2)]/x^4,x]
 

output

-1/3*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[d/c]*(c + d*x^2)*(b^2*(c 
 - 2*d*x^2) + a^2*c*(c^2 - d^2*x^4) + 2*a*b*(c^2 - c*d*x^2 - d^2*x^4)) - I 
*(2*b^2 + 3*a*b*c + a^2*c^2)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*S 
qrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)] + (2 
*I)*b*(b + a*c)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^ 
2)/c]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)]))/(c^2*(b + a*c)* 
Sqrt[d/c]*x^3*(b + a*(c + d*x^2)))
 

3.4.29.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2057, 2058, 377, 25, 27, 445, 25, 27, 406, 320, 388, 313}

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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (\frac {a d \left (d (a c+2 b) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {c^{3/2} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c (a c+b)}-\frac {(a c+2 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (\frac {a d \left (d (a c+2 b) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {c^{3/2} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c (a c+b)}-\frac {(a c+2 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (\frac {a d \left (\frac {c^{3/2} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+d (a c+2 b) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )\right )}{c (a c+b)}-\frac {(a c+2 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{\sqrt {a c+a d x^2+b}}\)

input

Int[Sqrt[a + b/(c + d*x^2)]/x^4,x]
 

output

(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-1/3*(Sqrt[c + d*x 
^2]*Sqrt[b + a*c + a*d*x^2])/(c*x^3) - (d*(-(((2*b + a*c)*Sqrt[c + d*x^2]* 
Sqrt[b + a*c + a*d*x^2])/(c*(b + a*c)*x)) + (a*d*((2*b + a*c)*d*((x*Sqrt[b 
 + a*c + a*d*x^2])/(a*d*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[b + a*c + a*d*x^2 
]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(a*d^(3/2)*Sqrt[c + 
 d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))])) + (c^(3/2) 
*Sqrt[b + a*c + a*d*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c 
)])/(Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + 
d*x^2))])))/(c*(b + a*c))))/(3*c)))/Sqrt[b + a*c + a*d*x^2]
 

3.4.29.3.1 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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 377

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

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 445

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

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

3.4.29.4 Maple [A] (verified)

Time = 7.00 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.57

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (-a c d \,x^{2}-2 b d \,x^{2}+a \,c^{2}+b c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 c^{2} x^{3} \left (a c +b \right )}-\frac {a \,d^{2} \left (\frac {a \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {b c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (a c d +2 b d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{3 c^{2} \left (a c +b \right ) \left (a d \,x^{2}+a c +b \right )}\)	\(569\)
default	\(-\frac {\left (-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{3} x^{6}-2 \sqrt {-\frac {a d}{a c +b}}\, a b \,d^{3} x^{6}+\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} c^{2} d^{2} x^{3}-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d^{2} x^{4}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}+2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}-4 \sqrt {-\frac {a d}{a c +b}}\, a b c \,d^{2} x^{4}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} d \,x^{2}-2 \sqrt {-\frac {a d}{a c +b}}\, b^{2} d^{2} x^{4}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{4}-\sqrt {-\frac {a d}{a c +b}}\, b^{2} c d \,x^{2}+2 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{3}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \left (a c +b \right ) x^{3} c^{2} \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\)	\(571\)

input

int((a+b/(d*x^2+c))^(1/2)/x^4,x,method=_RETURNVERBOSE)
 

output

-1/3*(d*x^2+c)*(-a*c*d*x^2-2*b*d*x^2+a*c^2+b*c)/c^2/x^3/(a*c+b)*((a*d*x^2+ 
a*c+b)/(d*x^2+c))^(1/2)-1/3*a*d^2/c^2/(a*c+b)*(a*c^2/(-a*d/(a*c+b))^(1/2)* 
(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x 
^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c 
/a)^(1/2))+b*c/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2 
)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a*d/ 
(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-2*(a*c*d+2*b*d)*(a*c^2+b*c) 
/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2 
*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(2*a*c*d+2*b*d)*(EllipticF(x*(-a 
*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-EllipticE(x*(-a*d/(a*c+b 
))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))))*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/ 
2)*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)/(a*d*x^2+a*c+b)
 

3.4.29.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx=-\frac {{\left (a^{2} c + 2 \, a b\right )} \sqrt {-\frac {a d}{a c + b}} d^{3} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c + 2 \, a b\right )} d^{3} + {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} d^{2}\right )} \sqrt {-\frac {a d}{a c + b}} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c^{2} + 3 \, a b c + 2 \, b^{2}\right )} d^{2} x^{4} - a^{2} c^{4} - 2 \, a b c^{3} - b^{2} c^{2} + {\left (a b c^{2} + b^{2} c\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, {\left (a^{2} c^{4} + 2 \, a b c^{3} + b^{2} c^{2}\right )} x^{3}} \]

input

integrate((a+b/(d*x^2+c))^(1/2)/x^4,x, algorithm="fricas")
 

output

-1/3*((a^2*c + 2*a*b)*sqrt(-a*d/(a*c + b))*d^3*x^3*sqrt((a*c^2 + b*c)/d^2) 
*elliptic_e(arcsin(sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a*c)) - ((a^2*c + 2 
*a*b)*d^3 + (a^2*c^2 + 2*a*b*c + b^2)*d^2)*sqrt(-a*d/(a*c + b))*x^3*sqrt(( 
a*c^2 + b*c)/d^2)*elliptic_f(arcsin(sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a* 
c)) - ((a^2*c^2 + 3*a*b*c + 2*b^2)*d^2*x^4 - a^2*c^4 - 2*a*b*c^3 - b^2*c^2 
 + (a*b*c^2 + b^2*c)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^2*c 
^4 + 2*a*b*c^3 + b^2*c^2)*x^3)
 

3.4.29.6 Sympy [F]

\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx=\int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{4}}\, dx \]

input

integrate((a+b/(d*x**2+c))**(1/2)/x**4,x)
 

output

Integral(sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))/x**4, x)
 

3.4.29.7 Maxima [F]

\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx=\int { \frac {\sqrt {a + \frac {b}{d x^{2} + c}}}{x^{4}} \,d x } \]

input

integrate((a+b/(d*x^2+c))^(1/2)/x^4,x, algorithm="maxima")
 

output

integrate(sqrt(a + b/(d*x^2 + c))/x^4, x)
 

3.4.29.8 Giac [F]

\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx=\int { \frac {\sqrt {a + \frac {b}{d x^{2} + c}}}{x^{4}} \,d x } \]

input

integrate((a+b/(d*x^2+c))^(1/2)/x^4,x, algorithm="giac")
 

output

integrate(sqrt(a + b/(d*x^2 + c))/x^4, x)
 

3.4.29.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^4} \, dx=\int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^4} \,d x \]

input

int((a + b/(c + d*x^2))^(1/2)/x^4,x)
 

output

int((a + b/(c + d*x^2))^(1/2)/x^4, x)