\(\int \frac {x^4}{(\frac {e (a+b x^2)}{c+d x^2})^{3/2}} \, dx\) [92]

Optimal result

Integrand size = 26, antiderivative size = 477 \[ \int \frac {x^4}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) x}{5 b^3 d e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}+\frac {(7 b c-8 a d) x \left (a+b x^2\right )}{5 b^3 e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}+\frac {6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}-\frac {x^3 \left (c+d x^2\right )}{b e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}-\frac {\sqrt {a} \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{5 b^{7/2} d e \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}-\frac {a^{3/2} (7 b c-8 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{5 b^{7/2} e \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}} \] Output:

1/5*(16*a^2*d^2-16*a*b*c*d+b^2*c^2)*x/b^3/d/e/(b*e/d-(-a*d+b*c)*e/d/(d*x^2 
+c))^(1/2)+1/5*(-8*a*d+7*b*c)*x*(b*x^2+a)/b^3/e/(b*e/d-(-a*d+b*c)*e/d/(d*x 
^2+c))^(1/2)+6/5*d*x^3*(b*x^2+a)/b^2/e/(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1 
/2)-x^3*(d*x^2+c)/b/e/(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)-1/5*a^(1/2)*( 
16*a^2*d^2-16*a*b*c*d+b^2*c^2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/ 
2),(1-a*d/b/c)^(1/2))/b^(7/2)/d/e/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(b*e/d-( 
-a*d+b*c)*e/d/(d*x^2+c))^(1/2)-1/5*a^(3/2)*(-8*a*d+7*b*c)*InverseJacobiAM( 
arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(7/2)/e/(a*(d*x^2+c)/c/(b*x 
^2+a))^(1/2)/(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.83 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.57 \[ \int \frac {x^4}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} d x \left (c+d x^2\right ) \left (-8 a^2 d+a b \left (7 c-2 d x^2\right )+b^2 x^2 \left (2 c+d x^2\right )\right )-i c \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (b^2 c^2-9 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{5 b^3 \sqrt {\frac {b}{a}} d e^2 \left (a+b x^2\right )} \] Input:

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

Output:

(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(c + d*x^2)*(-8*a^2*d + 
a*b*(7*c - 2*d*x^2) + b^2*x^2*(2*c + d*x^2)) - I*c*(b^2*c^2 - 16*a*b*c*d + 
 16*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[S 
qrt[b/a]*x], (a*d)/(b*c)] + I*c*(b^2*c^2 - 9*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + 
 (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b 
*c)]))/(5*b^3*Sqrt[b/a]*d*e^2*(a + b*x^2))
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2058, 369, 27, 443, 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.

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 369

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 443

\(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {3 \left (\frac {\int \frac {x^2 \left (d (7 b c-8 a d) x^2+c (5 b c-6 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b}+\frac {2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}\right )}{b}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {3 \left (\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 b}-\frac {\int \frac {d \left (a c (7 b c-8 a d)-\left (b^2 c^2-16 a b d c+16 a^2 d^2\right ) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{5 b}+\frac {2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}\right )}{b}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {3 \left (\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 b}-\frac {\int \frac {a c (7 b c-8 a d)-\left (b^2 c^2-16 a b d c+16 a^2 d^2\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}}{5 b}+\frac {2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}\right )}{b}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {3 \left (\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 b}-\frac {a c (7 b c-8 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}}{5 b}+\frac {2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}\right )}{b}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {3 \left (\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 b}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} (7 b c-8 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}}{5 b}+\frac {2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}\right )}{b}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {3 \left (\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 b}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} (7 b c-8 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 b}}{5 b}+\frac {2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}\right )}{b}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {3 \left (\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 b}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} (7 b c-8 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b}}{5 b}+\frac {2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}\right )}{b}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

Input:

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

Output:

(Sqrt[a + b*x^2]*(-((x^3*(c + d*x^2)^(3/2))/(b*Sqrt[a + b*x^2])) + (3*((2* 
d*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b) + (((7*b*c - 8*a*d)*x*Sqrt[a 
+ b*x^2]*Sqrt[c + d*x^2])/(3*b) - (-((b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*( 
(x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Ellipti 
cE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + 
b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))) + (c^(3/2)*(7*b*c - 8*a*d)*Sqr 
t[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqr 
t[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*b))/(5*b)) 
)/b))/(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])
 

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 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 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]
 

Maple [A] (verified)

Time = 13.16 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.64

method	result	size
risch	\(-\frac {x \left (-b d \,x^{2}+3 a d -2 b c \right ) \left (b \,x^{2}+a \right )}{5 b^{3} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (-\frac {2 \left (11 a^{2} d^{2}-11 a b c d +b^{2} c^{2}\right ) a c e \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}-\frac {a \left (5 a^{2} d^{2}-13 a b c d +7 b^{2} c^{2}\right ) \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{b \sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {5 a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\left (b d \,x^{2} e +b c e \right ) x}{a \left (a d -b c \right ) e \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2} e +b c e \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{a \left (a d -b c \right )}\right ) \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 d b c e \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}\right )}{b}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{5 b^{3} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(780\)
default	\(-\frac {\left (b \,x^{2}+a \right ) \left (-\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}-3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+5 \sqrt {-\frac {b}{a}}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, a^{2} d^{3} x^{3}-5 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}+8 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-9 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x +5 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -5 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{5 d \,b^{3} {\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )}^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\)	\(936\)

Input:

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

Output:

-1/5*x*(-b*d*x^2+3*a*d-2*b*c)*(b*x^2+a)/b^3/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2 
)+1/5/b^3*(-2*(11*a^2*d^2-11*a*b*c*d+b^2*c^2)*a*c*e/(-b/a)^(1/2)*(1+1/a*x^ 
2*b)^(1/2)*(1+1/c*x^2*d)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2) 
/(a*d*e+b*c*e+e*(a*d-b*c))*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b 
/e)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2)))-a*(5* 
a^2*d^2-13*a*b*c*d+7*b^2*c^2)/b/(-b/a)^(1/2)*(1+1/a*x^2*b)^(1/2)*(1+1/c*x^ 
2*d)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^ 
(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))+5*a^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b* 
(-(b*d*e*x^2+b*c*e)/a/(a*d-b*c)*x/e/((x^2+a/b)*(b*d*e*x^2+b*c*e))^(1/2)+(1 
/a+b*c/a/(a*d-b*c))/(-b/a)^(1/2)*(1+1/a*x^2*b)^(1/2)*(1+1/c*x^2*d)^(1/2)/( 
b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a 
*d*e+b*c*e)/c/b/e)^(1/2))-2*d*b/(a*d-b*c)*c*e/(-b/a)^(1/2)*(1+1/a*x^2*b)^( 
1/2)*(1+1/c*x^2*d)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)/(a*d* 
e+b*c*e+e*(a*d-b*c))*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1 
/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2)))))/e/(e*(b*x 
^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*(b*x^2+a)*e)^(1/2)/(d*x^2+c)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 425, normalized size of antiderivative = 0.89 \[ \int \frac {x^4}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=-\frac {{\left ({\left (b^{3} c^{3} - 16 \, a b^{2} c^{2} d + 16 \, a^{2} b c d^{2}\right )} x^{3} + {\left (a b^{2} c^{3} - 16 \, a^{2} b c^{2} d + 16 \, a^{3} c d^{2}\right )} x\right )} \sqrt {\frac {b e}{d}} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (b^{3} c^{3} - 16 \, a b^{2} c^{2} d + 8 \, a^{2} b d^{3} + {\left (16 \, a^{2} b - 7 \, a b^{2}\right )} c d^{2}\right )} x^{3} + {\left (a b^{2} c^{3} - 16 \, a^{2} b c^{2} d + 8 \, a^{3} d^{3} + {\left (16 \, a^{3} - 7 \, a^{2} b\right )} c d^{2}\right )} x\right )} \sqrt {\frac {b e}{d}} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b^{3} d^{3} x^{8} + {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{6} + a b^{2} c^{3} - 16 \, a^{2} b c^{2} d + 16 \, a^{3} c d^{2} + {\left (3 \, b^{3} c^{2} d - 11 \, a b^{2} c d^{2} + 8 \, a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - 8 \, a b^{2} c^{2} d - 8 \, a^{2} b c d^{2} + 16 \, a^{3} d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{5 \, {\left (b^{5} d e^{2} x^{3} + a b^{4} d e^{2} x\right )}} \] Input:

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

Output:

-1/5*(((b^3*c^3 - 16*a*b^2*c^2*d + 16*a^2*b*c*d^2)*x^3 + (a*b^2*c^3 - 16*a 
^2*b*c^2*d + 16*a^3*c*d^2)*x)*sqrt(b*e/d)*sqrt(-c/d)*elliptic_e(arcsin(sqr 
t(-c/d)/x), a*d/(b*c)) - ((b^3*c^3 - 16*a*b^2*c^2*d + 8*a^2*b*d^3 + (16*a^ 
2*b - 7*a*b^2)*c*d^2)*x^3 + (a*b^2*c^3 - 16*a^2*b*c^2*d + 8*a^3*d^3 + (16* 
a^3 - 7*a^2*b)*c*d^2)*x)*sqrt(b*e/d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/ 
d)/x), a*d/(b*c)) - (b^3*d^3*x^8 + (3*b^3*c*d^2 - 2*a*b^2*d^3)*x^6 + a*b^2 
*c^3 - 16*a^2*b*c^2*d + 16*a^3*c*d^2 + (3*b^3*c^2*d - 11*a*b^2*c*d^2 + 8*a 
^2*b*d^3)*x^4 + (b^3*c^3 - 8*a*b^2*c^2*d - 8*a^2*b*c*d^2 + 16*a^3*d^3)*x^2 
)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^5*d*e^2*x^3 + a*b^4*d*e^2*x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d*x - 2*sqrt(c + d*x**2) 
*sqrt(a + b*x**2)*a*d**2*x**3 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c**2 
*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*d*x**3 + sqrt(c + d*x**2)*sqr 
t(a + b*x**2)*b*d**2*x**5 + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4) 
/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2* 
d*x**6),x)*a**3*d**3 - 12*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a* 
*2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x* 
*6),x)*a**2*b*c*d**2 + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a** 
2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x** 
6),x)*a**2*b*d**3*x**2 + 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a 
**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x 
**6),x)*a*b**2*c**2*d - 12*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a 
**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x 
**6),x)*a*b**2*c*d**2*x**2 + 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4 
)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2 
*d*x**6),x)*b**3*c**2*d*x**2 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/( 
a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d* 
x**6),x)*a**3*c**2*d + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + 
 a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x) 
*a**2*b*c**3 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2...