\(\int \frac {x^2 (A+B x^2)}{(c+d x^2) \sqrt {a-c x^4}} \, dx\) [9]

Optimal result
Mathematica [C] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [F(-1)]
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 32, antiderivative size = 212 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {a^{3/4} B \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (A d-B \left (c+\frac {\sqrt {a} d}{\sqrt {c}}\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d^2 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d^2 \sqrt {a-c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.52 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {i \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B d E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-\left (B c^{3/2}+\sqrt {a} B d-A \sqrt {c} d\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {c} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {a} \left (-\frac {\sqrt {c}}{\sqrt {a}}\right )^{3/2} d^2 \sqrt {a-c x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2235, 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.

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

\(\Big \downarrow \) 2235

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

-((-((a^(3/4)*B*d*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4) 
], -1])/(c^(3/4)*Sqrt[a - c*x^4])) - (a^(1/4)*(A*d - B*(c + (Sqrt[a]*d)/Sq 
rt[c]))*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) + (a^(1/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*E 
llipticPi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/ 
4)*d^2*Sqrt[a - c*x^4])
 

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 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]
 
Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.55

method result size
default \(\frac {\frac {A d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {B d \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {B c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{d^{2}}-\frac {\left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) \(329\)
elliptic \(\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) A}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) B c}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {B \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {B \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) A}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) B}{d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) \(475\)

Input:

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

Output:

1/d^2*(A*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*d*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))-B*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))-1/d^2*(A*d-B*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)*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))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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