\(\int \frac {x^4 (A+B x^2)}{(c+d x^2) \sqrt {a-c x^4}} \, dx\) [8]

Optimal result

Integrand size = 32, antiderivative size = 268 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {B x \sqrt {a-c x^4}}{3 c d}-\frac {a^{3/4} (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} d^2 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (3 B c^3-3 A c^2 d+a B d^2+3 \sqrt {a} \sqrt {c} d (B c-A d)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{5/4} d^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} c^{3/4} (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^3 \sqrt {a-c x^4}} \] Output:

-1/3*B*x*(-c*x^4+a)^(1/2)/c/d-a^(3/4)*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*Ellipti 
cE(c^(1/4)*x/a^(1/4),I)/c^(3/4)/d^2/(-c*x^4+a)^(1/2)+1/3*a^(1/4)*(3*B*c^3- 
3*A*c^2*d+B*a*d^2+3*a^(1/2)*c^(1/2)*d*(-A*d+B*c))*(1-c*x^4/a)^(1/2)*Ellipt 
icF(c^(1/4)*x/a^(1/4),I)/c^(5/4)/d^3/(-c*x^4+a)^(1/2)-a^(1/4)*c^(3/4)*(-A* 
d+B*c)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*d/c^(3/2),I 
)/d^3/(-c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.98 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.37 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^2 x+B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^2 x^5+3 i \sqrt {a} \sqrt {c} d (B c-A d) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \left (-3 A \sqrt {c} d \left (c^{3/2}+\sqrt {a} d\right )+B \left (3 c^3+3 \sqrt {a} c^{3/2} d+a d^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i B c^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i A c^2 d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{3 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^3 \sqrt {a-c x^4}} \] Input:

Integrate[(x^4*(A + B*x^2))/((c + d*x^2)*Sqrt[a - c*x^4]),x]
 

Output:

(-(a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2*x) + B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2* 
x^5 + (3*I)*Sqrt[a]*Sqrt[c]*d*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[I* 
ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*(-3*A*Sqrt[c]*d*(c^(3/2) + Sq 
rt[a]*d) + B*(3*c^3 + 3*Sqrt[a]*c^(3/2)*d + a*d^2))*Sqrt[1 - (c*x^4)/a]*El 
lipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (3*I)*B*c^3*Sqrt[1 - 
(c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqr 
t[a])]*x], -1] - (3*I)*A*c^2*d*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d 
)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(3*Sqrt[-(Sqrt[c]/ 
Sqrt[a])]*c*d^3*Sqrt[a - c*x^4])
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2237, 25, 2235, 25, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}

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.

\(\displaystyle \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-c x^4} \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 2237

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2235

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d (B c-A d)+a B d^2-3 A c^2 d+3 B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-3 \sqrt {c} d (B c-A d) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{d^2}-\frac {3 c^3 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}-\frac {B x \sqrt {a-c x^4}}{3 c d}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d (B c-A d)+a B d^2-3 A c^2 d+3 B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 \sqrt {c} d \sqrt {1-\frac {c x^4}{a}} (B c-A d) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}}{d^2}-\frac {3 c^3 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}-\frac {B x \sqrt {a-c x^4}}{3 c d}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d (B c-A d)+a B d^2-3 A c^2 d+3 B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 \sqrt {a} \sqrt {c} d \sqrt {1-\frac {c x^4}{a}} (B c-A d) \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {a-c x^4}}}{d^2}-\frac {3 c^3 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}-\frac {B x \sqrt {a-c x^4}}{3 c d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d (B c-A d)+a B d^2-3 A c^2 d+3 B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 a^{3/4} \sqrt [4]{c} d \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-c x^4}}}{d^2}-\frac {3 c^3 (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {a-c x^4}}dx}{d^2}}{3 c d}-\frac {B x \sqrt {a-c x^4}}{3 c d}\)

\(\Big \downarrow \) 1543

\(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d (B c-A d)+a B d^2-3 A c^2 d+3 B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 a^{3/4} \sqrt [4]{c} d \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-c x^4}}}{d^2}-\frac {3 c^3 \sqrt {1-\frac {c x^4}{a}} (B c-A d) \int \frac {1}{\left (d x^2+c\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{d^2 \sqrt {a-c x^4}}}{3 c d}-\frac {B x \sqrt {a-c x^4}}{3 c d}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d (B c-A d)+a B d^2-3 A c^2 d+3 B c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 a^{3/4} \sqrt [4]{c} d \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-c x^4}}}{d^2}-\frac {3 \sqrt [4]{a} c^{7/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^2 \sqrt {a-c x^4}}}{3 c d}-\frac {B x \sqrt {a-c x^4}}{3 c d}\)

Input:

Int[(x^4*(A + B*x^2))/((c + d*x^2)*Sqrt[a - c*x^4]),x]
 

Output:

-1/3*(B*x*Sqrt[a - c*x^4])/(c*d) + (((-3*a^(3/4)*c^(1/4)*d*(B*c - A*d)*Sqr 
t[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - c*x^ 
4] + (a^(1/4)*(3*B*c^3 - 3*A*c^2*d + a*B*d^2 + 3*Sqrt[a]*Sqrt[c]*d*(B*c - 
A*d))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^( 
1/4)*Sqrt[a - c*x^4]))/d^2 - (3*a^(1/4)*c^(7/4)*(B*c - A*d)*Sqrt[1 - (c*x^ 
4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1]) 
/(d^2*Sqrt[a - c*x^4]))/(3*c*d)
 

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 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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

rule 1513

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[-c/a, 2]}, Simp[(d*q - e)/q   Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q 
  Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg 
Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

rule 2235

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> 
With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Si 
mp[-(e^2)^(-1)   Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x], x] + Simp[( 
C*d^2 - B*d*e + A*e^2)/e^2   Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] / 
; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0]
 

rule 2237

Int[(Px_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> W 
ith[{q = Expon[Px, x]}, Simp[Coeff[Px, x, q]*x^(q - 5)*(Sqrt[a + c*x^4]/(c* 
e*(q - 3))), x] + Simp[1/(c*e*(q - 3))   Int[(c*e*(q - 3)*Px - Coeff[Px, x, 
 q]*x^(q - 6)*(d + e*x^2)*(a*(q - 5) + c*(q - 3)*x^4))/((d + e*x^2)*Sqrt[a 
+ c*x^4]), x], x] /; GtQ[q, 4]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x]
 

Maple [A] (verified)

Time = 4.68 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.60

method	result	size
default	\(\frac {c \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{3} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\frac {d \left (A d -B c \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {A c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {B \,c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-B \,d^{2} \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{d^{3}}\)	\(430\)
risch	\(-\frac {B x \sqrt {-c \,x^{4}+a}}{3 c d}+\frac {\frac {-\frac {3 \sqrt {c}\, d \left (A d -B c \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {B a \,d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 B \,c^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 A \,c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{d^{2}}+\frac {3 c^{2} \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 c d}\)	\(442\)
elliptic	\(-\frac {B x \sqrt {-c \,x^{4}+a}}{3 c d}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) c A}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, d^{2}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) c^{2} B}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, d^{3}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B a}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, d c}-\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) A}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) A}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {\sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {c}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) A}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {c^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) B}{d^{3} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\)	\(720\)

Input:

int(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

c*(A*d-B*c)/d^3/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c 
^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(c^(1/2)/a^(1/2))^ 
(1/2),-a^(1/2)*d/c^(3/2),(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/2)) 
-1/d^3*(d*(A*d-B*c)*a^(1/2)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2) 
)^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(EllipticF( 
x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))+A*c*d 
/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1 
/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-B*c^2/( 
c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2 
))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-B*d^2*(-1 
/3/c*x*(-c*x^4+a)^(1/2)+1/3*a/c/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^( 
1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^ 
(1/2)/a^(1/2))^(1/2),I)))
 

Fricas [F]

\[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:

integrate(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="fricas")
 

Output:

integral(-(B*x^6 + A*x^4)*sqrt(-c*x^4 + a)/(c*d*x^6 + c^2*x^4 - a*d*x^2 - 
a*c), x)
 

Sympy [F]

\[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int \frac {x^{4} \left (A + B x^{2}\right )}{\sqrt {a - c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:

integrate(x**4*(B*x**2+A)/(d*x**2+c)/(-c*x**4+a)**(1/2),x)
 

Output:

Integral(x**4*(A + B*x**2)/(sqrt(a - c*x**4)*(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:

integrate(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)*x^4/(sqrt(-c*x^4 + a)*(d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {x^4 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{\sqrt {-c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:

integrate(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)*x^4/(sqrt(-c*x^4 + a)*(d*x^2 + c)), x)
 

Mupad [F(-1)]

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

int((x^4*(A + B*x^2))/((a - c*x^4)^(1/2)*(c + d*x^2)),x)
 

Output:

int((x^4*(A + B*x^2))/((a - c*x^4)^(1/2)*(c + d*x^2)), x)
 

Reduce [F]

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

int(x^4*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

( - sqrt(a - c*x**4)*b*x + int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x** 
4 - c*d*x**6),x)*a*b*c + 3*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x**2 - c 
**2*x**4 - c*d*x**6),x)*a*c*d - 3*int((sqrt(a - c*x**4)*x**4)/(a*c + a*d*x 
**2 - c**2*x**4 - c*d*x**6),x)*b*c**2 + int((sqrt(a - c*x**4)*x**2)/(a*c + 
 a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a*b*d)/(3*c*d)