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

Optimal. Leaf size=198 \[ \frac {3}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+\text {$\#$1}^4 c+a^2+a b-a c\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)-c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+a \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-a \log (x)+c \log (x)}{-2 \text {$\#$1}^5+2 \text {$\#$1} a-\text {$\#$1} c}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}} \]

________________________________________________________________________________________

Rubi [B]  time = 2.22, antiderivative size = 605, normalized size of antiderivative = 3.06, number of steps used = 16, number of rules used = 8, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6728, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} \frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}{\sqrt [4]{\sqrt {c^2-4 a b}-c} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {c^2-4 a b}-c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}-\frac {3 \left (c-\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}{\sqrt [4]{\sqrt {c^2-4 a b}+c} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}+\frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}{\sqrt [4]{\sqrt {c^2-4 a b}-c} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {c^2-4 a b}-c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}-\frac {3 \left (c-\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}{\sqrt [4]{\sqrt {c^2-4 a b}+c} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-b + c*x^4 + 2*a*x^8)/((-b + a*x^4)^(1/4)*(b - c*x^4 + a*x^8)),x]

[Out]

ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + (3*(c + (2*a*b - c^2)/Sqrt[-4*a*b + c^2])*ArcTan[(a^(1/4)*(2*
b - c + Sqrt[-4*a*b + c^2])^(1/4)*x)/((-c + Sqrt[-4*a*b + c^2])^(1/4)*(-b + a*x^4)^(1/4))])/(2*a^(1/4)*(-c + S
qrt[-4*a*b + c^2])^(3/4)*(2*b - c + Sqrt[-4*a*b + c^2])^(1/4)) - (3*(c - (2*a*b - c^2)/Sqrt[-4*a*b + c^2])*Arc
Tan[(a^(1/4)*(-2*b + c + Sqrt[-4*a*b + c^2])^(1/4)*x)/((c + Sqrt[-4*a*b + c^2])^(1/4)*(-b + a*x^4)^(1/4))])/(2
*a^(1/4)*(c + Sqrt[-4*a*b + c^2])^(3/4)*(-2*b + c + Sqrt[-4*a*b + c^2])^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x
^4)^(1/4)]/a^(1/4) + (3*(c + (2*a*b - c^2)/Sqrt[-4*a*b + c^2])*ArcTanh[(a^(1/4)*(2*b - c + Sqrt[-4*a*b + c^2])
^(1/4)*x)/((-c + Sqrt[-4*a*b + c^2])^(1/4)*(-b + a*x^4)^(1/4))])/(2*a^(1/4)*(-c + Sqrt[-4*a*b + c^2])^(3/4)*(2
*b - c + Sqrt[-4*a*b + c^2])^(1/4)) - (3*(c - (2*a*b - c^2)/Sqrt[-4*a*b + c^2])*ArcTanh[(a^(1/4)*(-2*b + c + S
qrt[-4*a*b + c^2])^(1/4)*x)/((c + Sqrt[-4*a*b + c^2])^(1/4)*(-b + a*x^4)^(1/4))])/(2*a^(1/4)*(c + Sqrt[-4*a*b
+ c^2])^(3/4)*(-2*b + c + Sqrt[-4*a*b + c^2])^(1/4))

Rule 203

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

Rule 205

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

Rule 206

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

Rule 208

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

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-b+c x^4+2 a x^8}{\sqrt [4]{-b+a x^4} \left (b-c x^4+a x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{-b+a x^4}}-\frac {3 \left (b-c x^4\right )}{\sqrt [4]{-b+a x^4} \left (b-c x^4+a x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx-3 \int \frac {b-c x^4}{\sqrt [4]{-b+a x^4} \left (b-c x^4+a x^8\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-3 \int \left (\frac {-c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}}{\sqrt [4]{-b+a x^4} \left (-c-\sqrt {-4 a b+c^2}+2 a x^4\right )}+\frac {-c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}}{\sqrt [4]{-b+a x^4} \left (-c+\sqrt {-4 a b+c^2}+2 a x^4\right )}\right ) \, dx\\ &=\left (3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-c-\sqrt {-4 a b+c^2}+2 a x^4\right )} \, dx+\left (3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-c+\sqrt {-4 a b+c^2}+2 a x^4\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\left (3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c-\sqrt {-4 a b+c^2}-\left (2 a b+a \left (-c-\sqrt {-4 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c+\sqrt {-4 a b+c^2}-\left (2 a b+a \left (-c+\sqrt {-4 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {-4 a b+c^2}}-\sqrt {a} \sqrt {2 b-c+\sqrt {-4 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {-c+\sqrt {-4 a b+c^2}}}+\frac {\left (3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {-4 a b+c^2}}+\sqrt {a} \sqrt {2 b-c+\sqrt {-4 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {-c+\sqrt {-4 a b+c^2}}}-\frac {\left (3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {-4 a b+c^2}}-\sqrt {a} \sqrt {-2 b+c+\sqrt {-4 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {c+\sqrt {-4 a b+c^2}}}-\frac {\left (3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {-4 a b+c^2}}+\sqrt {a} \sqrt {-2 b+c+\sqrt {-4 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {c+\sqrt {-4 a b+c^2}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{2 b-c+\sqrt {-4 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {-4 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a} \left (-c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{2 b-c+\sqrt {-4 a b+c^2}}}-\frac {3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-2 b+c+\sqrt {-4 a b+c^2}} x}{\sqrt [4]{c+\sqrt {-4 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a} \left (c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{-2 b+c+\sqrt {-4 a b+c^2}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{2 b-c+\sqrt {-4 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {-4 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a} \left (-c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{2 b-c+\sqrt {-4 a b+c^2}}}-\frac {3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-2 b+c+\sqrt {-4 a b+c^2}} x}{\sqrt [4]{c+\sqrt {-4 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a} \left (c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{-2 b+c+\sqrt {-4 a b+c^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.87, size = 594, normalized size = 3.00 \begin {gather*} \frac {\frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}{\sqrt [4]{\sqrt {c^2-4 a b}-c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {c^2-4 a b}-c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}+\frac {3 \left (2 a b-c \left (\sqrt {c^2-4 a b}+c\right )\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}{\sqrt [4]{\sqrt {c^2-4 a b}+c} \sqrt [4]{a x^4-b}}\right )}{\sqrt {c^2-4 a b} \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}+\frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}{\sqrt [4]{\sqrt {c^2-4 a b}-c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {c^2-4 a b}-c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}+\frac {3 \left (2 a b-c \left (\sqrt {c^2-4 a b}+c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}{\sqrt [4]{\sqrt {c^2-4 a b}+c} \sqrt [4]{a x^4-b}}\right )}{\sqrt {c^2-4 a b} \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}+2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b + c*x^4 + 2*a*x^8)/((-b + a*x^4)^(1/4)*(b - c*x^4 + a*x^8)),x]

[Out]

(2*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] + (3*(c + (2*a*b - c^2)/Sqrt[-4*a*b + c^2])*ArcTan[(a^(1/4)*(2*b - c
+ Sqrt[-4*a*b + c^2])^(1/4)*x)/((-c + Sqrt[-4*a*b + c^2])^(1/4)*(-b + a*x^4)^(1/4))])/((-c + Sqrt[-4*a*b + c^
2])^(3/4)*(2*b - c + Sqrt[-4*a*b + c^2])^(1/4)) + (3*(2*a*b - c*(c + Sqrt[-4*a*b + c^2]))*ArcTan[(a^(1/4)*(-2*
b + c + Sqrt[-4*a*b + c^2])^(1/4)*x)/((c + Sqrt[-4*a*b + c^2])^(1/4)*(-b + a*x^4)^(1/4))])/(Sqrt[-4*a*b + c^2]
*(c + Sqrt[-4*a*b + c^2])^(3/4)*(-2*b + c + Sqrt[-4*a*b + c^2])^(1/4)) + 2*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1
/4)] + (3*(c + (2*a*b - c^2)/Sqrt[-4*a*b + c^2])*ArcTanh[(a^(1/4)*(2*b - c + Sqrt[-4*a*b + c^2])^(1/4)*x)/((-c
+ Sqrt[-4*a*b + c^2])^(1/4)*(-b + a*x^4)^(1/4))])/((-c + Sqrt[-4*a*b + c^2])^(3/4)*(2*b - c + Sqrt[-4*a*b + c
^2])^(1/4)) + (3*(2*a*b - c*(c + Sqrt[-4*a*b + c^2]))*ArcTanh[(a^(1/4)*(-2*b + c + Sqrt[-4*a*b + c^2])^(1/4)*x
)/((c + Sqrt[-4*a*b + c^2])^(1/4)*(-b + a*x^4)^(1/4))])/(Sqrt[-4*a*b + c^2]*(c + Sqrt[-4*a*b + c^2])^(3/4)*(-2
*b + c + Sqrt[-4*a*b + c^2])^(1/4)))/(2*a^(1/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.00, size = 197, normalized size = 0.99 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3}{4} \text {RootSum}\left [a^2+a b-a c-2 a \text {$\#$1}^4+c \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-c \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 a \text {$\#$1}+c \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b + c*x^4 + 2*a*x^8)/((-b + a*x^4)^(1/4)*(b - c*x^4 + a*x^8)),x]

[Out]

ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + (3*RootSum[
a^2 + a*b - a*c - 2*a*#1^4 + c*#1^4 + #1^8 & , (a*Log[x] - c*Log[x] - a*Log[(-b + a*x^4)^(1/4) - x*#1] + c*Log
[(-b + a*x^4)^(1/4) - x*#1] - Log[x]*#1^4 + Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-2*a*#1 + c*#1 + 2*#1^5) & ]
)/4

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^8+c*x^4-b)/(a*x^4-b)^(1/4)/(a*x^8-c*x^4+b),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{8} + c x^{4} - b}{{\left (a x^{8} - c x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^8+c*x^4-b)/(a*x^4-b)^(1/4)/(a*x^8-c*x^4+b),x, algorithm="giac")

[Out]

integrate((2*a*x^8 + c*x^4 - b)/((a*x^8 - c*x^4 + b)*(a*x^4 - b)^(1/4)), x)

________________________________________________________________________________________

maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{8}+c \,x^{4}-b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}-c \,x^{4}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*a*x^8+c*x^4-b)/(a*x^4-b)^(1/4)/(a*x^8-c*x^4+b),x)

[Out]

int((2*a*x^8+c*x^4-b)/(a*x^4-b)^(1/4)/(a*x^8-c*x^4+b),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{8} + c x^{4} - b}{{\left (a x^{8} - c x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^8+c*x^4-b)/(a*x^4-b)^(1/4)/(a*x^8-c*x^4+b),x, algorithm="maxima")

[Out]

integrate((2*a*x^8 + c*x^4 - b)/((a*x^8 - c*x^4 + b)*(a*x^4 - b)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,a\,x^8+c\,x^4-b}{{\left (a\,x^4-b\right )}^{1/4}\,\left (a\,x^8-c\,x^4+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*a*x^8 - b + c*x^4)/((a*x^4 - b)^(1/4)*(b + a*x^8 - c*x^4)),x)

[Out]

int((2*a*x^8 - b + c*x^4)/((a*x^4 - b)^(1/4)*(b + a*x^8 - c*x^4)), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x**8+c*x**4-b)/(a*x**4-b)**(1/4)/(a*x**8-c*x**4+b),x)

[Out]

Timed out

________________________________________________________________________________________