3.4.51 \(\int \frac {x^2}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx\) [351]

3.4.51.1 Optimal result

Integrand size = 21, antiderivative size = 354 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {x \left (b+a c+a d x^2\right )}{3 a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(2 b+a c) x \left (b+a c+a d x^2\right )}{3 a^2 d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\sqrt {c} (2 b+a c) \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^2 d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 a d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

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

3.4.51.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.62 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )+i \left (2 b^2+3 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-2 i b (b+a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 a^2 d \sqrt {\frac {d}{c}} \left (b+a \left (c+d x^2\right )\right )} \]

input

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

3.4.51.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2057, 2058, 380, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 380

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

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

output

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

3.4.51.3.1 Defintions of rubi rules used

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 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]
 

3.4.51.4 Maple [A] (verified)

Time = 6.00 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {\left (\sqrt {-\frac {a d}{a c +b}}\, a \,d^{2} x^{5}+2 \sqrt {-\frac {a d}{a c +b}}\, a c d \,x^{3}+\sqrt {-\frac {a d}{a c +b}}\, b d \,x^{3}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,c^{2}+\sqrt {-\frac {a d}{a c +b}}\, a \,c^{2} x +\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c -2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c +\sqrt {-\frac {a d}{a c +b}}\, b c x \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, a \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\)	\(409\)
risch	\(\frac {x \left (a d \,x^{2}+a c +b \right )}{3 a d \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {\left (\frac {a \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {b c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (a c d +2 b d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{3 d a \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(531\)

input

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

output

1/3*((-a*d/(a*c+b))^(1/2)*a*d^2*x^5+2*(-a*d/(a*c+b))^(1/2)*a*c*d*x^3+(-a*d 
/(a*c+b))^(1/2)*b*d*x^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*c^2+(-a*d/(a*c+b 
))^(1/2)*a*c^2*x+((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*Ellip 
ticF(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*b*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*c+(-a*d/(a*c+b))^(1/2)*b*c*x)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d 
*x^2+c))^(1/2)/d/(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)
 

3.4.51.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.47 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {{\left (a c^{2} + 2 \, b c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a c^{2} + 2 \, b c + {\left (a c + b\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) + {\left (a d^{2} x^{4} - 2 \, b d x^{2} - a c^{2} - 2 \, b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, a^{2} d^{2} x} \]

input

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

output

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

3.4.51.6 Sympy [F]

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

input

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

output

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

3.4.51.7 Maxima [F]

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

input

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

output

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

3.4.51.8 Giac [F]

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

input

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

output

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

3.4.51.9 Mupad [F(-1)]

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

input

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

output

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