\(\int \frac {A+B x}{(d+e x) \sqrt {-a+c x^4}} \, dx\) [229]

Optimal result

Integrand size = 26, antiderivative size = 219 \[ \int \frac {A+B x}{(d+e x) \sqrt {-a+c x^4}} \, dx=-\frac {(B d-A e) \text {arctanh}\left (\frac {a e^2-c d^2 x^2}{\sqrt {c d^4-a e^4} \sqrt {-a+c x^4}}\right )}{2 \sqrt {c d^4-a e^4}}+\frac {\sqrt [4]{a} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} e \sqrt {-a+c x^4}}-\frac {\sqrt [4]{a} (B d-A e) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d e \sqrt {-a+c x^4}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.71 (sec) , antiderivative size = 719, normalized size of antiderivative = 3.28 \[ \int \frac {A+B x}{(d+e x) \sqrt {-a+c x^4}} \, dx=\frac {-\frac {i B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} e}+\frac {i A \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \sqrt {-\frac {(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{i \sqrt [4]{a}+\sqrt [4]{c} x}} \sqrt {\frac {(1+i) \left (\sqrt [4]{a}+i \sqrt [4]{c} x\right ) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2}} \left (\left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right ),2\right )-(1-i) \sqrt [4]{a} e \operatorname {EllipticPi}\left (\frac {(1-i) \left (\sqrt [4]{c} d-i \sqrt [4]{a} e\right )}{\sqrt [4]{c} d-\sqrt [4]{a} e},\arcsin \left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right ),2\right )\right )}{\sqrt [4]{a} \left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) \left (i \sqrt [4]{c} d+\sqrt [4]{a} e\right )}+\frac {B d \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \sqrt {-\frac {(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{i \sqrt [4]{a}+\sqrt [4]{c} x}} \sqrt {\frac {(1+i) \left (\sqrt [4]{a}+i \sqrt [4]{c} x\right ) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2}} \left (i \left (\sqrt [4]{c} d-\sqrt [4]{a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right ),2\right )+(1+i) \sqrt [4]{a} e \operatorname {EllipticPi}\left (\frac {(1-i) \left (\sqrt [4]{c} d-i \sqrt [4]{a} e\right )}{\sqrt [4]{c} d-\sqrt [4]{a} e},\arcsin \left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right ),2\right )\right )}{\sqrt [4]{a} e \left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) \left (i \sqrt [4]{c} d+\sqrt [4]{a} e\right )}}{\sqrt {-a+c x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2280, 27, 1577, 488, 219, 2229, 765, 762, 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.

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

\(\Big \downarrow \) 2280

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1577

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 2229

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

-1/2*((B*d - A*e)*ArcTanh[(a*e^2 - c*d^2*x^2)/(Sqrt[c*d^4 - a*e^4]*Sqrt[-a 
 + c*x^4])])/Sqrt[c*d^4 - a*e^4] + (a^(1/4)*B*Sqrt[1 - (c*x^4)/a]*Elliptic 
F[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*e*Sqrt[-a + c*x^4]) - (a^(1/4 
)*(B*d - A*e)*Sqrt[1 - (c*x^4)/a]*EllipticPi[(Sqrt[a]*e^2)/(Sqrt[c]*d^2), 
ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*e*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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

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 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 1577

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, c, d, e, p, q}, x]
 

rule 2229

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

rule 2280

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Wit 
h[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff 
[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sqrt[a 
+ c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt 
[a + c*x^4]), x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px 
, x], 3] && NeQ[c*d^4 + a*e^4, 0]
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {B \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}+\frac {\left (A e -B d \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}-2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}\, \sqrt {c \,x^{4}-a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}}+\frac {e \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}-a}}\right )}{e^{2}}\)	\(247\)
elliptic	\(\frac {B \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}+\frac {\left (A e -B d \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}-2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}\, \sqrt {c \,x^{4}-a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}}+\frac {e \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}-a}}\right )}{e^{2}}\)	\(247\)

Input:

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

Output:

B/e/(-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)+(A* 
e-B*d)/e^2*(-1/2/(c*d^4/e^4-a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2-2*a)/(c* 
d^4/e^4-a)^(1/2)/(c*x^4-a)^(1/2))+1/(-c^(1/2)/a^(1/2))^(1/2)/d*e*(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)*Ellipti 
cPi(x*(-c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*e^2/c^(1/2)/d^2,(c^(1/2)/a^(1/2))^ 
(1/2)/(-c^(1/2)/a^(1/2))^(1/2)))
 

Fricas [F]

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

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

Output:

integral(sqrt(c*x^4 - a)*(B*x + A)/(c*e*x^5 + c*d*x^4 - a*e*x - a*d), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x}{(d+e x) \sqrt {-a+c x^4}} \, dx=-\left (\int \frac {\sqrt {c \,x^{4}-a}}{-c e \,x^{5}-c d \,x^{4}+a e x +a d}d x \right ) a -\left (\int \frac {\sqrt {c \,x^{4}-a}\, x}{-c e \,x^{5}-c d \,x^{4}+a e x +a d}d x \right ) b \] Input:

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

Output:

 - (int(sqrt( - a + c*x**4)/(a*d + a*e*x - c*d*x**4 - c*e*x**5),x)*a + int 
((sqrt( - a + c*x**4)*x)/(a*d + a*e*x - c*d*x**4 - c*e*x**5),x)*b)