\(\int \frac {x}{(a+b x^8) \sqrt {c+d x^8}} \, dx\) [103]

Optimal result

Integrand size = 22, antiderivative size = 635 \[ \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{8 (-a)^{3/4} \sqrt {-b c+a d}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt {-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{8 (-a)^{3/4} \sqrt {-b c+a d}}+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} (b c+a d) \sqrt {c+d x^8}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{16 a \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^8}}+\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{16 a \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^8}} \] Output:

-1/8*b^(1/4)*arctan((a*d-b*c)^(1/2)*x^2/(-a)^(1/4)/b^(1/4)/(d*x^8+c)^(1/2) 
)/(-a)^(3/4)/(a*d-b*c)^(1/2)-1/8*b^(1/4)*arctanh((a*d-b*c)^(1/2)*x^2/(-a)^ 
(1/4)/b^(1/4)/(d*x^8+c)^(1/2))/(-a)^(3/4)/(a*d-b*c)^(1/2)+1/4*d^(3/4)*(c^( 
1/2)+d^(1/2)*x^4)*((d*x^8+c)/(c^(1/2)+d^(1/2)*x^4)^2)^(1/2)*InverseJacobiA 
M(2*arctan(d^(1/4)*x^2/c^(1/4)),1/2*2^(1/2))/c^(1/4)/(a*d+b*c)/(d*x^8+c)^( 
1/2)+1/16*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))*(c^(1/2)+d^(1/2)*x^4)*((d*x 
^8+c)/(c^(1/2)+d^(1/2)*x^4)^2)^(1/2)*EllipticPi(sin(2*arctan(d^(1/4)*x^2/c 
^(1/4))),-1/4*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^ 
(1/2)/d^(1/2),1/2*2^(1/2))/a/c^(1/4)/(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))/ 
d^(1/4)/(d*x^8+c)^(1/2)+1/16*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*(c^(1/2) 
+d^(1/2)*x^4)*((d*x^8+c)/(c^(1/2)+d^(1/2)*x^4)^2)^(1/2)*EllipticPi(sin(2*a 
rctan(d^(1/4)*x^2/c^(1/4))),1/4*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))^2/(-a 
)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))/a/c^(1/4)/(b^(1/2)*c^(1/2)+(- 
a)^(1/2)*d^(1/2))/d^(1/4)/(d*x^8+c)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.10 \[ \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\frac {x^2 \sqrt {\frac {c+d x^8}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{2 a \sqrt {c+d x^8}} \] Input:

Integrate[x/((a + b*x^8)*Sqrt[c + d*x^8]),x]
 

Output:

(x^2*Sqrt[(c + d*x^8)/c]*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8 
)/a)])/(2*a*Sqrt[c + d*x^8])
 

Rubi [A] (warning: unable to verify)

Time = 2.04 (sec) , antiderivative size = 876, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {965, 925, 1541, 27, 761, 2221, 2223}

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.

\(\displaystyle \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx\)

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 1541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^4}{\sqrt {-a}}\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8} (a d+b c)}}{2 a}+\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (\frac {\sqrt {b} x^4}{\sqrt {-a}}+1\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {a \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8} (a d+b c)}}{2 a}\right )\)

\(\Big \downarrow \) 2221

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^4+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^4}{\sqrt {-a}}\right ) \sqrt {d x^8+c}}dx^2}{a d+b c}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8} (a d+b c)}}{2 a}+\frac {\frac {a \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^8} (a d+b c)}+\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \left (\frac {(-a)^{3/4} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}-\sqrt {d}\right ) \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^8}}\right )}{2 \sqrt [4]{b} \sqrt {b c-a d}}+\frac {\left (\sqrt {c}+\sqrt {d} x^4\right ) \sqrt {\frac {c+d x^8}{\left (\sqrt {c}+\sqrt {d} x^4\right )^2}} \left (\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}+\sqrt {c}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^8}}\right )}{a d+b c}}{2 a}\right )\)

\(\Big \downarrow \) 2223

\(\displaystyle \frac {1}{2} \left (\frac {\frac {a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \left (\frac {(-a)^{3/4} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}-\sqrt {d}\right ) \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{2 \sqrt [4]{b} \sqrt {b c-a d}}+\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^8+c}}\right )}{b c+a d}}{2 a}+\frac {\frac {\left (\sqrt {d} a+\sqrt {-a} \sqrt {b} \sqrt {c}\right ) \sqrt [4]{d} \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {d x^8+c}}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \left (\frac {\sqrt [4]{-a} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^8+c}}\right )}{2 \sqrt [4]{b} \sqrt {b c-a d}}+\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \left (\sqrt {d} x^4+\sqrt {c}\right ) \sqrt {\frac {d x^8+c}{\left (\sqrt {d} x^4+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^8+c}}\right )}{b c+a d}}{2 a}\right )\)

Input:

Int[x/((a + b*x^8)*Sqrt[c + d*x^8]),x]
 

Output:

(((a*((Sqrt[b]*Sqrt[c])/Sqrt[-a] + Sqrt[d])*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4 
)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)* 
x^2)/c^(1/4)], 1/2])/(2*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*(S 
qrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(((-a)^(3/4)*((Sqrt[b]*Sqrt[c])/Sqrt[-a 
] - Sqrt[d])*ArcTan[(Sqrt[b*c - a*d]*x^2)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x 
^8])])/(2*b^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b 
])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ell 
ipticPi[-1/4*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(Sqrt[-a]*Sqrt[b]*Sqrt 
[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(4*c^(1/4)*d^(1/4)*Sq 
rt[c + d*x^8])))/(b*c + a*d))/(2*a) + (((Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt 
[d])*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x 
^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(2*c^(1/4)*(b*c + 
a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(((- 
a)^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*ArcTanh[(Sqrt[b*c - a*d]*x^2 
)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^8])])/(2*b^(1/4)*Sqrt[b*c - a*d]) + ((S 
qrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x 
^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt 
[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4 
)], 1/2])/(4*c^(1/4)*d^(1/4)*Sqrt[c + d*x^8])))/(b*c + a*d))/(2*a))/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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 925

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

rule 965

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 
 x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 
 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free 
Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
 

rule 1541

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

rule 2221

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e 
) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] 
+ Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* 
d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x 
], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po 
sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
 

rule 2223

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

Maple [F]

\[\int \frac {x}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}d x\]

Input:

int(x/(b*x^8+a)/(d*x^8+c)^(1/2),x)
 

Output:

int(x/(b*x^8+a)/(d*x^8+c)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\text {Timed out} \] Input:

integrate(x/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {x}{\left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \] Input:

integrate(x/(b*x**8+a)/(d*x**8+c)**(1/2),x)
 

Output:

Integral(x/((a + b*x**8)*sqrt(c + d*x**8)), x)
 

Maxima [F]

\[ \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {x}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}} \,d x } \] Input:

integrate(x/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x/((b*x^8 + a)*sqrt(d*x^8 + c)), x)
 

Giac [F]

\[ \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int { \frac {x}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}} \,d x } \] Input:

integrate(x/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(x/((b*x^8 + a)*sqrt(d*x^8 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {x}{\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \] Input:

int(x/((a + b*x^8)*(c + d*x^8)^(1/2)),x)
 

Output:

int(x/((a + b*x^8)*(c + d*x^8)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx=\int \frac {\sqrt {d \,x^{8}+c}\, x}{b d \,x^{16}+a d \,x^{8}+b c \,x^{8}+a c}d x \] Input:

int(x/(b*x^8+a)/(d*x^8+c)^(1/2),x)
 

Output:

int((sqrt(c + d*x**8)*x)/(a*c + a*d*x**8 + b*c*x**8 + b*d*x**16),x)