\(\int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx\) [201]

Optimal result

Integrand size = 21, antiderivative size = 336 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 (b+a c)^2 x \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(b-a c) d^{3/2} \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {a \sqrt {c} d^{3/2} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.94 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {\sqrt {\frac {d}{c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (b^2 \left (c+d x^2\right )+a^2 c \left (c^2-d^2 x^4\right )+a b \left (2 c^2+c d x^2+d^2 x^4\right )\right )+i \left (b^2-a^2 c^2\right ) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-i b (b+a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 (b+a c)^2 d x^3 \left (b+a \left (c+d x^2\right )\right )} \] Input:

Integrate[1/(x^4*Sqrt[a + b/(c + d*x^2)]),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2057, 2058, 377, 27, 445, 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.

\(\displaystyle \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx\)

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

Int[1/(x^4*Sqrt[a + b/(c + d*x^2)]),x]
 

Output:

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

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 10.52 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.69

method	result	size
risch	\(-\frac {\left (a d \,x^{2}+a c +b \right ) \left (-a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c \right )}{3 \left (a c +b \right )^{2} x^{3} c \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {d^{2} a \left (\frac {a \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}}+\frac {b c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 d \left (a c -b \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{3 \left (a c +b \right )^{2} c \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(567\)
default	\(-\frac {\left (-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{3} x^{6}+\sqrt {-\frac {a d}{a c +b}}\, a b \,d^{3} x^{6}+\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} c^{2} d^{2} x^{3}-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d^{2} x^{4}+2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}+2 \sqrt {-\frac {a d}{a c +b}}\, a b c \,d^{2} x^{4}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} d^{2} x^{4}+3 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{2} d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{4}+2 \sqrt {-\frac {a d}{a c +b}}\, b^{2} c d \,x^{2}+2 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{3}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, c \,x^{3} \left (a c +b \right )^{2} \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}\)	\(593\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {{\left (a^{2} c - a b\right )} \sqrt {-\frac {a d}{a c + b}} d^{3} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c - a b\right )} d^{3} + {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} d^{2}\right )} \sqrt {-\frac {a d}{a c + b}} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c^{2} - b^{2}\right )} d^{2} x^{4} - a^{2} c^{4} - 2 \, a b c^{3} - b^{2} c^{2} - 2 \, {\left (a b c^{2} + b^{2} c\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, {\left (a^{3} c^{4} + 3 \, a^{2} b c^{3} + 3 \, a b^{2} c^{2} + b^{3} c\right )} x^{3}} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^{4} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{d x^{2} + c}} x^{4}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{d x^{2} + c}} x^{4}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^4\,\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, d +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, x^{2}}{a^{2} c \,d^{2} x^{4}+a b \,d^{2} x^{4}+2 a^{2} c^{2} d \,x^{2}+3 a b c d \,x^{2}+a^{2} c^{3}+b^{2} d \,x^{2}+2 a b \,c^{2}+b^{2} c}d x \right ) a^{2} c \,d^{3} x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, x^{2}}{a^{2} c \,d^{2} x^{4}+a b \,d^{2} x^{4}+2 a^{2} c^{2} d \,x^{2}+3 a b c d \,x^{2}+a^{2} c^{3}+b^{2} d \,x^{2}+2 a b \,c^{2}+b^{2} c}d x \right ) a b \,d^{3} x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}}{a^{2} c \,d^{2} x^{8}+a b \,d^{2} x^{8}+2 a^{2} c^{2} d \,x^{6}+3 a b c d \,x^{6}+a^{2} c^{3} x^{4}+b^{2} d \,x^{6}+2 a b \,c^{2} x^{4}+b^{2} c \,x^{4}}d x \right ) a^{2} c^{4} x +2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}}{a^{2} c \,d^{2} x^{8}+a b \,d^{2} x^{8}+2 a^{2} c^{2} d \,x^{6}+3 a b c d \,x^{6}+a^{2} c^{3} x^{4}+b^{2} d \,x^{6}+2 a b \,c^{2} x^{4}+b^{2} c \,x^{4}}d x \right ) a b \,c^{3} x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}}{a^{2} c \,d^{2} x^{8}+a b \,d^{2} x^{8}+2 a^{2} c^{2} d \,x^{6}+3 a b c d \,x^{6}+a^{2} c^{3} x^{4}+b^{2} d \,x^{6}+2 a b \,c^{2} x^{4}+b^{2} c \,x^{4}}d x \right ) b^{2} c^{2} x}{c x \left (a c +b \right )} \] Input:

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

Output:

( - sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*d + int((sqrt(c + d*x**2)*sq 
rt(a*c + a*d*x**2 + b)*x**2)/(a**2*c**3 + 2*a**2*c**2*d*x**2 + a**2*c*d**2 
*x**4 + 2*a*b*c**2 + 3*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c + b**2*d*x**2 
),x)*a**2*c*d**3*x + int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*x**2)/ 
(a**2*c**3 + 2*a**2*c**2*d*x**2 + a**2*c*d**2*x**4 + 2*a*b*c**2 + 3*a*b*c* 
d*x**2 + a*b*d**2*x**4 + b**2*c + b**2*d*x**2),x)*a*b*d**3*x + int((sqrt(c 
 + d*x**2)*sqrt(a*c + a*d*x**2 + b))/(a**2*c**3*x**4 + 2*a**2*c**2*d*x**6 
+ a**2*c*d**2*x**8 + 2*a*b*c**2*x**4 + 3*a*b*c*d*x**6 + a*b*d**2*x**8 + b* 
*2*c*x**4 + b**2*d*x**6),x)*a**2*c**4*x + 2*int((sqrt(c + d*x**2)*sqrt(a*c 
 + a*d*x**2 + b))/(a**2*c**3*x**4 + 2*a**2*c**2*d*x**6 + a**2*c*d**2*x**8 
+ 2*a*b*c**2*x**4 + 3*a*b*c*d*x**6 + a*b*d**2*x**8 + b**2*c*x**4 + b**2*d* 
x**6),x)*a*b*c**3*x + int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b))/(a** 
2*c**3*x**4 + 2*a**2*c**2*d*x**6 + a**2*c*d**2*x**8 + 2*a*b*c**2*x**4 + 3* 
a*b*c*d*x**6 + a*b*d**2*x**8 + b**2*c*x**4 + b**2*d*x**6),x)*b**2*c**2*x)/ 
(c*x*(a*c + b))