3.4.42 \(\int \frac {(a+\frac {b}{c+d x^2})^{3/2}}{x^4} \, dx\) [342]

3.4.42.1 Optimal result

Integrand size = 21, antiderivative size = 388 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\frac {b \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c x^3}-\frac {(8 b+a c) d^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^3}-\frac {(4 b+a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^2 x^3}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^3 x}+\frac {(8 b+a c) d^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {a (4 b+a c) d^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 c^{3/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

b*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c/x^3-1/3*(a*c+8*b)*d^2*x*((a*d*x^2+a* 
c+b)/(d*x^2+c))^(1/2)/c^3-1/3*(a*c+4*b)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+ 
c))^(1/2)/c^2/x^3+1/3*(a*c+8*b)*d*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1 
/2)/c^3/x+1/3*(a*c+8*b)*d^(3/2)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*El 
lipticE(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^(5/2)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2) 
-1/3*a*(a*c+4*b)*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)/c^(3/2)/(a*c+b)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2 
)
 

3.4.42.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.92 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (-b^2 c^2-2 a b c^3-a^2 c^4+4 b^2 c d x^2+3 a b c^2 d x^2-a^2 c^3 d x^2+8 b^2 d^2 x^4+13 a b c d^2 x^4+a^2 c^2 d^2 x^4+8 a b d^3 x^6+a^2 c d^3 x^6+i c \left (8 b^2+9 a b c+a^2 c^2\right ) d \sqrt {\frac {d}{c}} x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-i b c (8 b+5 a c) d \sqrt {\frac {d}{c}} x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 c^3 x^3 \left (b+a \left (c+d x^2\right )\right )} \]

input

Integrate[(a + b/(c + d*x^2))^(3/2)/x^4,x]
 

output

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-(b^2*c^2) - 2*a*b*c^3 - a^2*c^4 + 
 4*b^2*c*d*x^2 + 3*a*b*c^2*d*x^2 - a^2*c^3*d*x^2 + 8*b^2*d^2*x^4 + 13*a*b* 
c*d^2*x^4 + a^2*c^2*d^2*x^4 + 8*a*b*d^3*x^6 + a^2*c*d^3*x^6 + I*c*(8*b^2 + 
 9*a*b*c + a^2*c^2)*d*Sqrt[d/c]*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sq 
rt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)] - I*b 
*c*(8*b + 5*a*c)*d*Sqrt[d/c]*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)]))/(3*c^3 
*x^3*(b + a*(c + d*x^2)))
 

3.4.42.3 Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.13, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2057, 2058, 370, 25, 27, 445, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 370

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {d \left (\frac {a d \left (d (a c+8 b) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {c^{3/2} (a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} (a c+b) \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x}\right )}{3 c}-\frac {(a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {d \left (\frac {a d \left (d (a c+8 b) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {c^{3/2} (a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} (a c+b) \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x}\right )}{3 c}-\frac {(a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {d \left (\frac {a d \left (\frac {c^{3/2} (a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} (a c+b) \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+d (a c+8 b) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )\right )}{c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x}\right )}{3 c}-\frac {(a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

input

Int[(a + b/(c + d*x^2))^(3/2)/x^4,x]
 

output

(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*((b*Sqrt[b + a*c + 
a*d*x^2])/(c*x^3*Sqrt[c + d*x^2]) + (-1/3*((4*b + a*c)*Sqrt[c + d*x^2]*Sqr 
t[b + a*c + a*d*x^2])/(c*x^3) - (d*(-(((8*b + a*c)*Sqrt[c + d*x^2]*Sqrt[b 
+ a*c + a*d*x^2])/(c*x)) + (a*d*((8*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[ArcTa 
n[(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)*(4*b + a*c)*Sqrt 
[b + a*c + a*d*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/( 
(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))/(3*c))/c))/Sqrt[b + a*c + a*d*x^2]
 

3.4.42.3.1 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 370

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

3.4.42.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs. \(2(424)=848\).

Time = 10.72 (sec) , antiderivative size = 1037, normalized size of antiderivative = 2.67

method	result	size
default	\(\text {Expression too large to display}\)	\(1037\)
risch	\(\text {Expression too large to display}\)	\(1122\)

input

int((a+b/(d*x^2+c))^(3/2)/x^4,x,method=_RETURNVERBOSE)
 

output

-1/3*(-((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a^2*c*d^3*x^ 
6-5*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*b*d^3*x^6+((a 
*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2+c)/ 
c)^(1/2)*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a^2*c^2*d^2 
*x^3-3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(-a*d/(a*c+b))^(1/2 
)*a*b*d^3*x^6-((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a^2*c 
^2*d^2*x^4-4*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)/(a*c+b))^( 
1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1 
/2))*a*b*c*d^2*x^3+8*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)/(a 
*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b) 
/a/c)^(1/2))*a*b*c*d^2*x^3-10*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c 
+b))^(1/2)*a*b*c*d^2*x^4-3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2) 
*(-a*d/(a*c+b))^(1/2)*a*b*c*d^2*x^4+((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a* 
d/(a*c+b))^(1/2)*a^2*c^3*d*x^2-5*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/( 
a*c+b))^(1/2)*b^2*d^2*x^4-3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2 
)*(-a*d/(a*c+b))^(1/2)*b^2*d^2*x^4-3*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a 
*d/(a*c+b))^(1/2)*a*b*c^2*d*x^2+((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a 
*c+b))^(1/2)*a^2*c^4-4*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1 
/2)*b^2*c*d*x^2+2*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a 
*b*c^3+((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*b^2*c^2)*...
 

3.4.42.5 Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^4} \, dx=-\frac {{\left (a^{2} c + 8 \, a b\right )} \sqrt {-\frac {a d}{a c + b}} d^{3} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c + 8 \, a b\right )} d^{3} + {\left (a^{2} c^{2} + 5 \, a b c + 4 \, b^{2}\right )} d^{2}\right )} \sqrt {-\frac {a d}{a c + b}} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c^{2} + 9 \, a b c + 8 \, b^{2}\right )} d^{2} x^{4} - a^{2} c^{4} - 2 \, a b c^{3} - b^{2} c^{2} + 4 \, {\left (a b c^{2} + b^{2} c\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, {\left (a c^{4} + b c^{3}\right )} x^{3}} \]

input

integrate((a+b/(d*x^2+c))^(3/2)/x^4,x, algorithm="fricas")
 

output

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

3.4.42.6 Sympy [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]

input

integrate((a+b/(d*x**2+c))**(3/2)/x**4,x)
 

output

Integral(((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2)/x**4, x)
 

3.4.42.7 Maxima [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \]

input

integrate((a+b/(d*x^2+c))^(3/2)/x^4,x, algorithm="maxima")
 

output

integrate((a + b/(d*x^2 + c))^(3/2)/x^4, x)
 

3.4.42.8 Giac [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \]

input

integrate((a+b/(d*x^2+c))^(3/2)/x^4,x, algorithm="giac")
 

output

integrate((a + b/(d*x^2 + c))^(3/2)/x^4, x)
 

3.4.42.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}}{x^4} \,d x \]

input

int((a + b/(c + d*x^2))^(3/2)/x^4,x)
 

output

int((a + b/(c + d*x^2))^(3/2)/x^4, x)