3.4.25 \(\int x^4 \sqrt {a+\frac {b}{c+d x^2}} \, dx\) [325]

3.4.25.1 Optimal result

Integrand size = 21, antiderivative size = 368 \[ \int x^4 \sqrt {a+\frac {b}{c+d x^2}} \, dx=-\frac {\left (2 b^2+7 a b c-3 a^2 c^2\right ) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{15 a^2 d^2}+\frac {(b-3 a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{15 a d^2}+\frac {x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}+\frac {\sqrt {c} \left (2 b^2+7 a b c-3 a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 a^2 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} (b-3 a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{15 a d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

-1/15*(-3*a^2*c^2+7*a*b*c+2*b^2)*x*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a^2/d 
^2+1/15*(-3*a*c+b)*x*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a/d^2+1/5 
*x^3*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d-1/15*c^(3/2)*(-3*a*c+b) 
*(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/d^(5/2 
)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)+1/15*(-3*a^2*c^2+7*a*b*c+2*b 
^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))*c^(1/2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/ 
2)/a^2/d^(5/2)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)
 

3.4.25.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.06 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.80 \[ \int x^4 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (b^2-2 a b \left (c-2 d x^2\right )-3 a^2 \left (c^2-d^2 x^4\right )\right )+i \left (2 b^3+9 a b^2 c+4 a^2 b c^2-3 a^3 c^3\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-2 i b \left (b^2+4 a b c+3 a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{15 a^2 c^2 \left (\frac {d}{c}\right )^{5/2} \left (b+a \left (c+d x^2\right )\right )} \]

input

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

output

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(a*Sqrt[d/c]*x*(c + d*x^2)*(b^2 - 2 
*a*b*(c - 2*d*x^2) - 3*a^2*(c^2 - d^2*x^4)) + I*(2*b^3 + 9*a*b^2*c + 4*a^2 
*b*c^2 - 3*a^3*c^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)] - (2*I)*b*(b^2 + 4*a* 
b*c + 3*a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*E 
llipticF[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)]))/(15*a^2*c^2*(d/c)^(5/2 
)*(b + a*(c + d*x^2)))
 

3.4.25.3 Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2057, 2058, 380, 444, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 380

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 d}-\frac {\int \frac {x^2 \left (3 c (b+a c)-(b-3 a c) d x^2\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 d}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 d}-\frac {-\frac {\int -\frac {d \left (\left (2 b^2+7 a c b-3 a^2 c^2\right ) d x^2+c (b-3 a c) (b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d^2}-\frac {x (b-3 a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}}{5 d}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 d}-\frac {\frac {\int \frac {d \left (\left (2 b^2+7 a c b-3 a^2 c^2\right ) d x^2+c (b-3 a c) (b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d^2}-\frac {x (b-3 a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}}{5 d}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 d}-\frac {\frac {\int \frac {\left (2 b^2+7 a c b-3 a^2 c^2\right ) d x^2+c (b-3 a c) (b+a c)}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d}-\frac {x (b-3 a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}}{5 d}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 d}-\frac {\frac {d \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+c (b-3 a c) (a c+b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d}-\frac {x (b-3 a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}}{5 d}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 d}-\frac {\frac {d \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {c^{3/2} (b-3 a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}}{3 a d}-\frac {x (b-3 a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}}{5 d}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 d}-\frac {\frac {d \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {c^{3/2} (b-3 a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}}{3 a d}-\frac {x (b-3 a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}}{5 d}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 d}-\frac {\frac {d \left (-3 a^2 c^2+7 a b c+2 b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} (b-3 a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}}{3 a d}-\frac {x (b-3 a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}}{5 d}\right )}{\sqrt {a c+a d x^2+b}}\)

input

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

output

(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*((x^3*Sqrt[c + d*x^ 
2]*Sqrt[b + a*c + a*d*x^2])/(5*d) - (-1/3*((b - 3*a*c)*x*Sqrt[c + d*x^2]*S 
qrt[b + a*c + a*d*x^2])/(a*d) + ((2*b^2 + 7*a*b*c - 3*a^2*c^2)*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 
)*(b - 3*a*c)*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))]))/(3*a*d))/(5*d)))/Sqrt[b + a*c + a*d*x^2]
 

3.4.25.3.1 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 380

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b* 
(m + 2*(p + q) + 1))), x] - Simp[e^2/(b*(m + 2*(p + q) + 1))   Int[(e*x)^(m 
 - 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[a*c*(m - 1) + (a*d*(m - 1) - 2 
*q*(b*c - a*d))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c 
- a*d, 0] && GtQ[q, 0] && GtQ[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 444

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

3.4.25.4 Maple [A] (verified)

Time = 6.52 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.81

method	result	size
default	\(-\frac {\left (-3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} d^{3} x^{7}-3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{2} x^{5}-4 \sqrt {-\frac {a d}{a c +b}}\, a b \,d^{2} x^{5}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d \,x^{3}-2 \sqrt {-\frac {a d}{a c +b}}\, a b c d \,x^{3}-3 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} c^{3}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} x -\sqrt {-\frac {a d}{a c +b}}\, b^{2} d \,x^{3}-9 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b \,c^{2}+7 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b \,c^{2}+2 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{2} x -\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b^{2} c +2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b^{2} c -\sqrt {-\frac {a d}{a c +b}}\, b^{2} c x \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 d^{2} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, a \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\)	\(665\)
risch	\(-\frac {x \left (-3 a d \,x^{2}+3 a c -b \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 d^{2} a}+\frac {\left (\frac {3 a^{2} c^{3} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {b^{2} c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {2 a b \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (3 a^{2} c^{2} d -7 a b c d -2 b^{2} d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{15 a \,d^{2} \left (a d \,x^{2}+a c +b \right )}\)	\(686\)

input

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

output

-1/15*(-3*(-a*d/(a*c+b))^(1/2)*a^2*d^3*x^7-3*(-a*d/(a*c+b))^(1/2)*a^2*c*d^ 
2*x^5-4*(-a*d/(a*c+b))^(1/2)*a*b*d^2*x^5+3*(-a*d/(a*c+b))^(1/2)*a^2*c^2*d* 
x^3-2*(-a*d/(a*c+b))^(1/2)*a*b*c*d*x^3-3*((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^3+3*(-a*d/(a*c+b))^(1/2)*a^2*c^3*x-(-a*d/(a*c+b))^(1/2)*b^2*d*x^3-9*( 
(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^2+7*((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^2+2*(-a*d/(a*c+b))^(1/2)*a*b*c^2*x-((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))*b^ 
2*c+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))*b^2*c-(-a*d/(a*c+b))^(1/2)*b^2*c*x)* 
(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d^2/(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/((a*d*x^2+a*c+b)*(d*x^2+c))^( 
1/2)
 

3.4.25.5 Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.65 \[ \int x^4 \sqrt {a+\frac {b}{c+d x^2}} \, dx=-\frac {{\left (3 \, a^{2} c^{3} - 7 \, a b c^{2} - 2 \, b^{2} c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (3 \, a^{2} c^{3} - 7 \, a b c^{2} - 2 \, b^{2} c + {\left (3 \, a^{2} c^{2} + 2 \, a b c - b^{2}\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (3 \, a^{2} d^{3} x^{6} + a b d^{2} x^{4} + 3 \, a^{2} c^{3} - 7 \, a b c^{2} - 2 \, {\left (3 \, a b c + b^{2}\right )} d x^{2} - 2 \, b^{2} c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{15 \, a^{2} d^{3} x} \]

input

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

output

-1/15*((3*a^2*c^3 - 7*a*b*c^2 - 2*b^2*c)*sqrt(a)*x*sqrt(-c/d)*elliptic_e(a 
rcsin(sqrt(-c/d)/x), (a*c + b)/(a*c)) - (3*a^2*c^3 - 7*a*b*c^2 - 2*b^2*c + 
 (3*a^2*c^2 + 2*a*b*c - b^2)*d)*sqrt(a)*x*sqrt(-c/d)*elliptic_f(arcsin(sqr 
t(-c/d)/x), (a*c + b)/(a*c)) - (3*a^2*d^3*x^6 + a*b*d^2*x^4 + 3*a^2*c^3 - 
7*a*b*c^2 - 2*(3*a*b*c + b^2)*d*x^2 - 2*b^2*c)*sqrt((a*d*x^2 + a*c + b)/(d 
*x^2 + c)))/(a^2*d^3*x)
 

3.4.25.6 Sympy [F]

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

input

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

output

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

3.4.25.7 Maxima [F]

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

input

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

output

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

3.4.25.8 Giac [F]

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

input

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

output

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

3.4.25.9 Mupad [F(-1)]

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

input

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

output

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