3.4.39 \(\int x^2 (a+\frac {b}{c+d x^2})^{3/2} \, dx\) [339]

3.4.39.1 Optimal result

Integrand size = 21, antiderivative size = 331 \[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {(7 b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}+\frac {4 a x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {x \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\sqrt {c} (7 b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {\sqrt {c} (3 b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

1/3*(-a*c+7*b)*x*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d+4/3*a*x*(d*x^2+c)*((a 
*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d-x*(a*d*x^2+a*c+b)*((a*d*x^2+a*c+b)/(d*x^2 
+c))^(1/2)/d-1/3*(-a*c+7*b)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Ellipt 
icE(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)/d^(3/2)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^( 
1/2)+1/3*(-a*c+3*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))*c^(1/2)*((a*d*x^2+a*c+b 
)/(d*x^2+c))^(1/2)/d^(3/2)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)
 

3.4.39.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.42 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.77 \[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {d}{c}} x \left (-3 b^2-2 a b \left (c+d x^2\right )+a^2 \left (c+d x^2\right )^2\right )+i \left (-7 b^2-6 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+4 i b (b+a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 d \sqrt {\frac {d}{c}} \left (b+a \left (c+d x^2\right )\right )} \]

input

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

output

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[d/c]*x*(-3*b^2 - 2*a*b*(c + d 
*x^2) + a^2*(c + d*x^2)^2) + I*(-7*b^2 - 6*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)] + (4*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*d*Sqrt[d/c]*(b + a*(c + d*x^2)))
 

3.4.39.3 Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2057, 2058, 369, 403, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 369

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\frac {\int \frac {d \left (a (7 b-a c) d x^2+(3 b-a c) (b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 d}+\frac {4}{3} a x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{d}-\frac {x \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\frac {1}{3} \int \frac {a (7 b-a c) d x^2+(3 b-a c) (b+a c)}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {4}{3} a x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{d}-\frac {x \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\frac {1}{3} \left ((3 b-a c) (a c+b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+a d (7 b-a c) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx\right )+\frac {4}{3} a x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{d}-\frac {x \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\frac {1}{3} \left (a d (7 b-a c) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {\sqrt {c} (3 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {4}{3} a x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{d}-\frac {x \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\frac {1}{3} \left (a d (7 b-a c) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {\sqrt {c} (3 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {4}{3} a x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{d}-\frac {x \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\frac {1}{3} \left (a d (7 b-a c) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {\sqrt {c} (3 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {4}{3} a x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{d}-\frac {x \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

input

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

output

(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-((x*(b + a*c + a* 
d*x^2)^(3/2))/(d*Sqrt[c + d*x^2])) + ((4*a*x*Sqrt[c + d*x^2]*Sqrt[b + a*c 
+ a*d*x^2])/3 + (a*(7*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))])) + (Sqrt[c]*(3*b - 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)/d))/S 
qrt[b + a*c + a*d*x^2]
 

3.4.39.3.1 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 369

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/(2* 
b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1))   Int[(e*x)^(m - 2)*(a + b*x^2)^(p 
 + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] 
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && 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 403

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

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]
 

3.4.39.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(819\) vs. \(2(371)=742\).

Time = 9.35 (sec) , antiderivative size = 820, normalized size of antiderivative = 2.48

method	result	size
default	\(\frac {\left (\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a^{2} d^{2} x^{5}+2 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a^{2} c d \,x^{3}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a b d \,x^{3}-\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \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}-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}}\, a b d \,x^{3}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} x -5 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \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 +7 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \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 +\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a b c x +3 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \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}-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}}\, a b c x -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}}\, b^{2} x \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}}\, \left (a d \,x^{2}+a c +b \right )}\)	\(820\)
risch	\(\text {Expression too large to display}\)	\(1213\)

input

int(x^2*(a+b/(d*x^2+c))^(3/2),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*d^2*x^5+2* 
((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a^2*c*d*x^3+((a*d*x 
^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*b*d*x^3-((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)*Elli 
pticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a^2*c^2-3*(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*x^3+((a*d*x^2 
+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a^2*c^2*x-5*((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)*Elli 
pticF(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a*b*c+7*((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)*Elli 
pticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a*b*c+((a*d*x^2+a*c+b)*( 
d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*b*c*x+3*((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))*b^2-3*(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*x-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*x)*((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*d*x^2+a*c+b)
 

3.4.39.5 Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.62 \[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {{\left (a^{2} c^{3} - 7 \, a b c^{2}\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} c^{3} - 7 \, a b c^{2} + {\left (a^{2} c^{2} - 2 \, a b c - 3 \, b^{2}\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) + {\left (a^{2} c d^{2} x^{4} + 4 \, a b c d x^{2} - a^{2} c^{3} + 7 \, a b c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, a c d^{2} x} \]

input

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

output

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

3.4.39.6 Sympy [F]

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

input

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

output

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

3.4.39.7 Maxima [F]

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

input

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

output

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

3.4.39.8 Giac [F]

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

input

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

output

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

3.4.39.9 Mupad [F(-1)]

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

input

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

output

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