\(\int \frac {x^4 (e+f x^2)}{(a-b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [276]

Optimal result

Integrand size = 34, antiderivative size = 354 \[ \int \frac {x^4 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {(b e+a f) x^3 \sqrt {c+d x^2}}{b (b c+a d) \sqrt {a-b x^2}}+\frac {(3 b d e+b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 b^2 d (b c+a d)}-\frac {\sqrt {a} \left (8 a^2 d^2 f+b^2 c (3 d e-2 c f)+3 a b d (2 d e+c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 b^{5/2} d^2 (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} c (3 b d e-2 b c f+4 a d f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 b^{5/2} d^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.55 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {-\frac {b}{a}} d x \left (c+d x^2\right ) \left (4 a^2 d f-b^2 c f x^2+a b \left (3 d e+c f-d f x^2\right )\right )+i c \left (8 a^2 d^2 f+b^2 c (3 d e-2 c f)+3 a b d (2 d e+c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c (b c+a d) (3 b d e-2 b c f+4 a d f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{3 b^2 \sqrt {-\frac {b}{a}} d^2 (b c+a d) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[-(b/a)]*d*x*(c + d*x^2)*(4*a^2*d*f - b^2*c*f*x^2 + a*b*(3*d*e + c*f 
- d*f*x^2)) + I*c*(8*a^2*d^2*f + b^2*c*(3*d*e - 2*c*f) + 3*a*b*d*(2*d*e + 
c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b 
/a)]*x], -((a*d)/(b*c))] - I*c*(b*c + a*d)*(3*b*d*e - 2*b*c*f + 4*a*d*f)*S 
qrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x] 
, -((a*d)/(b*c))])/(3*b^2*Sqrt[-(b/a)]*d^2*(b*c + a*d)*Sqrt[a - b*x^2]*Sqr 
t[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {440, 25, 444, 399, 323, 323, 321, 331, 330, 327}

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.

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

\(\Big \downarrow \) 440

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 444

\(\displaystyle \frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {\left (c (3 d e-2 c f) b^2+3 a d (2 d e+c f) b+8 a^2 d^2 f\right ) x^2+a c (3 b d e+b c f+4 a d f)}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c (a d+b c) (4 a d f-2 b c f+3 b d e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 331

\(\displaystyle \frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\)

Input:

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

Output:

((b*e + a*f)*x^3*Sqrt[c + d*x^2])/(b*(b*c + a*d)*Sqrt[a - b*x^2]) - (-1/3* 
((3*b*d*e + b*c*f + 4*a*d*f)*x*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + (( 
Sqrt[a]*(8*a^2*d^2*f + b^2*c*(3*d*e - 2*c*f) + 3*a*b*d*(2*d*e + c*f))*Sqrt 
[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -(( 
a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a]*c 
*(b*c + a*d)*(3*b*d*e - 2*b*c*f + 4*a*d*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d 
*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]* 
d*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]))/(3*b*d))/(b*(b*c + a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 331

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 440

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

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]
 

Maple [A] (verified)

Time = 10.68 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.25

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) a x \left (a f +b e \right )}{b^{3} \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {f x \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{3 d \,b^{2}}+\frac {\left (-\frac {c a \left (a f +b e \right )}{b^{2} \left (a d +b c \right )}-\frac {f a c}{3 b^{2} d}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (-\frac {a f +b e}{b^{2}}-\frac {d a \left (a f +b e \right )}{b^{2} \left (a d +b c \right )}-\frac {f \left (2 a d -2 b c \right )}{3 d \,b^{2}}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(441\)
risch	\(\frac {f x \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b^{2} d}-\frac {\left (-\frac {\left (5 a d f -2 b c f +3 b d e \right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}+\frac {a \left (3 a d f +b c f +3 b d e \right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{b \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {3 d \,a^{2} \left (a f +b e \right ) \left (\frac {\left (-b d \,x^{2}-b c \right ) x}{a \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {\left (-\frac {1}{a}+\frac {b c}{a \left (a d +b c \right )}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{a \left (a d +b c \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{b}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b^{2} d \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(643\)
default	\(\frac {\left (-\sqrt {\frac {b}{a}}\, a b \,d^{3} f \,x^{5}-\sqrt {\frac {b}{a}}\, b^{2} c \,d^{2} f \,x^{5}+4 \sqrt {\frac {b}{a}}\, a^{2} d^{3} f \,x^{3}+3 \sqrt {\frac {b}{a}}\, a b \,d^{3} e \,x^{3}-\sqrt {\frac {b}{a}}\, b^{2} c^{2} d f \,x^{3}+4 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a^{2} c \,d^{2} f +2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,c^{2} d f +3 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b c \,d^{2} e -2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c^{3} f +3 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c^{2} d e -8 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a^{2} c \,d^{2} f -3 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,c^{2} d f -6 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b c \,d^{2} e +2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c^{3} f -3 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c^{2} d e +4 \sqrt {\frac {b}{a}}\, a^{2} c \,d^{2} f x +\sqrt {\frac {b}{a}}\, a b \,c^{2} d f x +3 \sqrt {\frac {b}{a}}\, a b c \,d^{2} e x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{3 b^{2} \left (a d +b c \right ) \sqrt {\frac {b}{a}}\, d^{2} \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(801\)

Input:

int(x^4*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-(-b*d*x^2- 
b*c)/b^3*a/(a*d+b*c)*x*(a*f+b*e)/((x^2-a/b)*(-b*d*x^2-b*c))^(1/2)+1/3*f/d/ 
b^2*x*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)+(-1/b^2*c*a/(a*d+b*c)*(a*f+b*e) 
-1/3*f/b^2/d*a*c)/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^ 
4+a*d*x^2-b*c*x^2+a*c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1 
/2))-(-1/b^2*(a*f+b*e)-1/b^2*d*a*(a*f+b*e)/(a*d+b*c)-1/3*f/d/b^2*(2*a*d-2* 
b*c))*c/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2- 
b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))-El 
lipticE(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.56 \[ \int \frac {x^4 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {{\left ({\left (3 \, {\left (a b^{3} c d + 2 \, a^{2} b^{2} d^{2}\right )} e - {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d - 8 \, a^{3} b d^{2}\right )} f\right )} x^{3} - {\left (3 \, {\left (a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} e - {\left (2 \, a^{2} b^{2} c^{2} - 3 \, a^{3} b c d - 8 \, a^{4} d^{2}\right )} f\right )} x\right )} \sqrt {-b d} \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - {\left ({\left (3 \, {\left (2 \, a^{2} b^{2} d^{2} + {\left (a b^{3} + b^{4}\right )} c d\right )} e + {\left (8 \, a^{3} b d^{2} - {\left (2 \, a b^{3} - b^{4}\right )} c^{2} + {\left (3 \, a^{2} b^{2} + 4 \, a b^{3}\right )} c d\right )} f\right )} x^{3} - {\left (3 \, {\left (2 \, a^{3} b d^{2} + {\left (a^{2} b^{2} + a b^{3}\right )} c d\right )} e + {\left (8 \, a^{4} d^{2} - {\left (2 \, a^{2} b^{2} - a b^{3}\right )} c^{2} + {\left (3 \, a^{3} b + 4 \, a^{2} b^{2}\right )} c d\right )} f\right )} x\right )} \sqrt {-b d} \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + {\left ({\left (b^{4} c d + a b^{3} d^{2}\right )} f x^{4} + {\left (3 \, {\left (b^{4} c d + a b^{3} d^{2}\right )} e - 2 \, {\left (b^{4} c^{2} - a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} f\right )} x^{2} - 3 \, {\left (a b^{3} c d + 2 \, a^{2} b^{2} d^{2}\right )} e + {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d - 8 \, a^{3} b d^{2}\right )} f\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left ({\left (b^{6} c d^{2} + a b^{5} d^{3}\right )} x^{3} - {\left (a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x\right )}} \] Input:

integrate(x^4*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fri 
cas")
 

Output:

1/3*(((3*(a*b^3*c*d + 2*a^2*b^2*d^2)*e - (2*a*b^3*c^2 - 3*a^2*b^2*c*d - 8* 
a^3*b*d^2)*f)*x^3 - (3*(a^2*b^2*c*d + 2*a^3*b*d^2)*e - (2*a^2*b^2*c^2 - 3* 
a^3*b*c*d - 8*a^4*d^2)*f)*x)*sqrt(-b*d)*sqrt(a/b)*elliptic_e(arcsin(sqrt(a 
/b)/x), -b*c/(a*d)) - ((3*(2*a^2*b^2*d^2 + (a*b^3 + b^4)*c*d)*e + (8*a^3*b 
*d^2 - (2*a*b^3 - b^4)*c^2 + (3*a^2*b^2 + 4*a*b^3)*c*d)*f)*x^3 - (3*(2*a^3 
*b*d^2 + (a^2*b^2 + a*b^3)*c*d)*e + (8*a^4*d^2 - (2*a^2*b^2 - a*b^3)*c^2 + 
 (3*a^3*b + 4*a^2*b^2)*c*d)*f)*x)*sqrt(-b*d)*sqrt(a/b)*elliptic_f(arcsin(s 
qrt(a/b)/x), -b*c/(a*d)) + ((b^4*c*d + a*b^3*d^2)*f*x^4 + (3*(b^4*c*d + a* 
b^3*d^2)*e - 2*(b^4*c^2 - a*b^3*c*d - 2*a^2*b^2*d^2)*f)*x^2 - 3*(a*b^3*c*d 
 + 2*a^2*b^2*d^2)*e + (2*a*b^3*c^2 - 3*a^2*b^2*c*d - 8*a^3*b*d^2)*f)*sqrt( 
-b*x^2 + a)*sqrt(d*x^2 + c))/((b^6*c*d^2 + a*b^5*d^3)*x^3 - (a*b^5*c*d^2 + 
 a^2*b^4*d^3)*x)
 

Sympy [F]

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

integrate(x**4*(f*x**2+e)/(-b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral(x**4*(e + f*x**2)/((a - b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
 

Maxima [F]

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

integrate(x^4*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="max 
ima")
 

Output:

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

Giac [F]

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

integrate(x^4*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

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

int((x^4*(e + f*x^2))/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((x^4*(e + f*x^2))/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^4 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {3 \sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, c f x -2 \sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, d f \,x^{3}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} d^{2} f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b c d f +6 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b \,d^{2} e -8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b \,d^{2} f \,x^{2}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} c d f \,x^{2}-6 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} d^{2} e \,x^{2}-3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} c^{2} f +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b \,c^{2} f \,x^{2}}{6 b \,d^{2} \left (-b \,x^{2}+a \right )} \] Input:

int(x^4*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
 

Output:

(3*sqrt(c + d*x**2)*sqrt(a - b*x**2)*c*f*x - 2*sqrt(c + d*x**2)*sqrt(a - b 
*x**2)*d*f*x**3 + 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*d**2*f - 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* 
c*d*f + 6*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*d**2* 
e - 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*b*d**2*f*x* 
*2 + 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**2*c*d*f*x** 
2 - 6*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**2*d**2*e*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**2*c**2*f + 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*b*c**2*f*x**2)/(6*b*d* 
*2*(a - b*x**2))