3.4.61 \(\int \frac {x^4}{(a+\frac {b}{c+d x^2})^{3/2}} \, dx\) [361]

3.4.61.1 Optimal result

Integrand size = 21, antiderivative size = 482 \[ \int \frac {x^4}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {x^3 \left (c+d x^2\right )}{a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(8 b+a c) x \left (b+a c+a d x^2\right )}{5 a^3 d^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {6 x^3 \left (b+a c+a d x^2\right )}{5 a^2 d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) x \left (b+a c+a d x^2\right )}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} \left (16 b^2+16 a b c+a^2 c^2\right ) \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a^4 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {c^{3/2} (8 b+a c) \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{5 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

-x^3*(d*x^2+c)/a/d/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/5*(a*c+8*b)*x*(a*d* 
x^2+a*c+b)/a^3/d^2/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+6/5*x^3*(a*d*x^2+a*c+ 
b)/a^2/d/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/5*(a^2*c^2+16*a*b*c+16*b^2)*x 
*(a*d*x^2+a*c+b)/a^4/d^2/(d*x^2+c)/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/5*c 
^(3/2)*(a*c+8*b)*(a*d*x^2+a*c+b)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*E 
llipticF(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(b/(a*c+b))^(1/2))/a^3/d^(5/2 
)/(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)-1/5*(a^2*c^2+16*a*b*c+16*b^2)*(a*d*x^2+a*c+b)*(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^4/d^(5/2)/(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)
 

3.4.61.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.61 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.61 \[ \int \frac {x^4}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (8 b^2+a b \left (9 c+2 d x^2\right )+a^2 \left (c^2-d^2 x^4\right )\right )+i \left (16 b^3+32 a b^2 c+17 a^2 b c^2+a^3 c^3\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-8 i b \left (2 b^2+3 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{5 a^4 c^2 \left (\frac {d}{c}\right )^{5/2} \left (b+a \left (c+d x^2\right )\right )} \]

input

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

output

-1/5*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(a*Sqrt[d/c]*x*(c + d*x^2)*(8* 
b^2 + a*b*(9*c + 2*d*x^2) + a^2*(c^2 - d^2*x^4)) + I*(16*b^3 + 32*a*b^2*c 
+ 17*a^2*b*c^2 + 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)] - (8*I)*b*(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]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)]))/(a^4*c^2*(d/c)^( 
5/2)*(b + a*(c + d*x^2)))
 

3.4.61.3 Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {2057, 2058, 369, 27, 443, 25, 27, 444, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 369

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 443

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {\int -\frac {d x^2 \left ((8 b+a c) d x^2+c (6 b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 a d}+\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}-\frac {\int \frac {d x^2 \left ((8 b+a c) d x^2+c (6 b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 a d}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}-\frac {\int \frac {x^2 \left ((8 b+a c) d x^2+c (6 b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 a}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}-\frac {\frac {x (a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}-\frac {\int \frac {d \left (\left (16 b^2+16 a c b+a^2 c^2\right ) d x^2+c (b+a c) (8 b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d^2}}{5 a}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}-\frac {\frac {x (a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}-\frac {\int \frac {\left (16 b^2+16 a c b+a^2 c^2\right ) d x^2+c (b+a c) (8 b+a c)}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d}}{5 a}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}-\frac {\frac {x (a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}-\frac {d \left (a^2 c^2+16 a b c+16 b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+c (a c+b) (a c+8 b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d}}{5 a}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}-\frac {\frac {x (a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}-\frac {d \left (a^2 c^2+16 a b c+16 b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {c^{3/2} (a c+8 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}}{3 a d}}{5 a}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}-\frac {\frac {x (a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}-\frac {d \left (a^2 c^2+16 a b c+16 b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {c^{3/2} (a c+8 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}}{3 a d}}{5 a}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {3 \left (\frac {2 x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 a}-\frac {\frac {x (a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a d}-\frac {d \left (a^2 c^2+16 a b c+16 b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} (a c+8 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}}{3 a d}}{5 a}\right )}{a d}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\)

input

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

output

(Sqrt[b + a*c + a*d*x^2]*(-((x^3*(c + d*x^2)^(3/2))/(a*d*Sqrt[b + a*c + a* 
d*x^2])) + (3*((2*x^3*Sqrt[c + d*x^2]*Sqrt[b + a*c + a*d*x^2])/(5*a) - ((( 
8*b + a*c)*x*Sqrt[c + d*x^2]*Sqrt[b + a*c + a*d*x^2])/(3*a*d) - ((16*b^2 + 
 16*a*b*c + 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)*(8*b + a*c)*Sqrt[b + a*c + a*d*x^2]*Elli 
pticF[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*a)))/( 
a*d)))/(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])
 

3.4.61.3.1 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 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 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 443

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

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]
 

3.4.61.4 Maple [A] (verified)

Time = 11.98 (sec) , antiderivative size = 879, normalized size of antiderivative = 1.82

method	result	size
risch	\(-\frac {x \left (-a d \,x^{2}+a c +3 b \right ) \left (a d \,x^{2}+a c +b \right )}{5 d^{2} a^{3} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {\left (-\frac {2 d \left (a^{2} c^{2}+11 a b c +11 b^{2}\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 )}+\frac {\left (c^{3} a^{3}+4 a^{2} b \,c^{2}-2 a \,b^{2} c -5 b^{3}\right ) \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 )}{a \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 {5 b^{2} \left (a^{2} c^{2}+2 a b c +b^{2}\right ) \left (-\frac {\left (a \,d^{2} x^{2}+a c d \right ) x}{\left (a c +b \right ) b d \sqrt {\left (x^{2}+\frac {a c +b}{a d}\right ) \left (a \,d^{2} x^{2}+a c d \right )}}+\frac {\left (\frac {1}{a c +b}+\frac {a c}{\left (a c +b \right ) b}\right ) \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 d \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 )}{b \left (a c +b \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 )}{a}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{5 a^{3} d^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(879\)
default	\(\text {Expression too large to display}\)	\(1159\)

input

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

output

-1/5/d^2*x*(-a*d*x^2+a*c+3*b)*(a*d*x^2+a*c+b)/a^3/((a*d*x^2+a*c+b)/(d*x^2+ 
c))^(1/2)+1/5/a^3/d^2*(-2*d*(a^2*c^2+11*a*b*c+11*b^2)*(a*c^2+b*c)/(-a*d/(a 
*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(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^3*c^3+4*a^2*b*c^2-2*a*b^2*c-5*b^3)/a/( 
-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(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))+5*b^2*(a^2*c^2+2*a*b*c+b^2)/a*(-(a*d^2*x^2+ 
a*c*d)/(a*c+b)/b*x/d/((x^2+(a*c+b)/a/d)*(a*d^2*x^2+a*c*d))^(1/2)+(1/(a*c+b 
)+a*c/(a*c+b)/b)/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x 
^2)^(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/b*a*d/(a*c+b)*(a*c^2+b* 
c)/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(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)*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)/(d*x^2+c)
 

3.4.61.5 Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.90 \[ \int \frac {x^4}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {{\left ({\left (a^{3} c^{3} + 16 \, a^{2} b c^{2} + 16 \, a b^{2} c\right )} d x^{3} + {\left (a^{3} c^{4} + 17 \, a^{2} b c^{3} + 32 \, a b^{2} c^{2} + 16 \, b^{3} c\right )} x\right )} \sqrt {a} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left ({\left (a^{3} c^{2} + 9 \, a^{2} b c + 8 \, a b^{2}\right )} d^{2} + {\left (a^{3} c^{3} + 16 \, a^{2} b c^{2} + 16 \, a b^{2} c\right )} d\right )} x^{3} + {\left (a^{3} c^{4} + 17 \, a^{2} b c^{3} + 32 \, a b^{2} c^{2} + 16 \, b^{3} c + {\left (a^{3} c^{3} + 10 \, a^{2} b c^{2} + 17 \, a b^{2} c + 8 \, b^{3}\right )} d\right )} x\right )} \sqrt {a} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{3} d^{4} x^{8} + {\left (a^{3} c - 2 \, a^{2} b\right )} d^{3} x^{6} + a^{3} c^{4} + {\left (5 \, a^{2} b c + 8 \, a b^{2}\right )} d^{2} x^{4} + 17 \, a^{2} b c^{3} + 32 \, a b^{2} c^{2} + 16 \, b^{3} c + {\left (a^{3} c^{3} + 24 \, a^{2} b c^{2} + 40 \, a b^{2} c + 16 \, b^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{5 \, {\left (a^{5} d^{4} x^{3} + {\left (a^{5} c + a^{4} b\right )} d^{3} x\right )}} \]

input

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

output

-1/5*(((a^3*c^3 + 16*a^2*b*c^2 + 16*a*b^2*c)*d*x^3 + (a^3*c^4 + 17*a^2*b*c 
^3 + 32*a*b^2*c^2 + 16*b^3*c)*x)*sqrt(a)*sqrt(-c/d)*elliptic_e(arcsin(sqrt 
(-c/d)/x), (a*c + b)/(a*c)) - (((a^3*c^2 + 9*a^2*b*c + 8*a*b^2)*d^2 + (a^3 
*c^3 + 16*a^2*b*c^2 + 16*a*b^2*c)*d)*x^3 + (a^3*c^4 + 17*a^2*b*c^3 + 32*a* 
b^2*c^2 + 16*b^3*c + (a^3*c^3 + 10*a^2*b*c^2 + 17*a*b^2*c + 8*b^3)*d)*x)*s 
qrt(a)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), (a*c + b)/(a*c)) - (a^3 
*d^4*x^8 + (a^3*c - 2*a^2*b)*d^3*x^6 + a^3*c^4 + (5*a^2*b*c + 8*a*b^2)*d^2 
*x^4 + 17*a^2*b*c^3 + 32*a*b^2*c^2 + 16*b^3*c + (a^3*c^3 + 24*a^2*b*c^2 + 
40*a*b^2*c + 16*b^3)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^5*d^ 
4*x^3 + (a^5*c + a^4*b)*d^3*x)
 

3.4.61.6 Sympy [F]

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

input

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

output

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

3.4.61.7 Maxima [F]

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

input

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

output

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

3.4.61.8 Giac [F]

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

input

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

output

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

3.4.61.9 Mupad [F(-1)]

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

input

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

output

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