3.25.44 \(\int \frac {-b+c x^4+2 a x^8}{\sqrt [4]{-b+a x^4} (b-c x^4+a x^8)} \, dx\) [2444]

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

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(605\) vs. \(2(198)=396\).
time = 1.51, 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 = {6860, 246, 218, 212, 209, 385, 214, 211} \begin {gather*} \frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \text {ArcTan}\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 ) \text {ArcTan}\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 {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}}+\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 {\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 209

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

Rule 211

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 246

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 385

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 6860

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 \text {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+\text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.01, size = 192, normalized size = 0.97 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+\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]

Integrate[(-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)] + 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

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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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 8.60, size = 16736, normalized size = 84.53 \begin {gather*} \text {Too large to display} \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]

3*sqrt(sqrt(1/2)*sqrt((2*a^3*b^2 - 5*a^2*b^2*c - 4*a^2*b*c^2 + 5*a*b*c^3 + a*c^4 - c^5 - (16*a^4*b^2 + 16*a^3*
b^3 - 16*a^3*b^2*c + 8*a^2*b*c^3 - a*c^5 + (a^2 + a*b)*c^4 - 8*(a^3*b + a^2*b^2)*c^2)*sqrt(-(a^4*b^4 + 4*a^4*b
^3*c - 14*a^3*b^2*c^3 + 10*a^2*b*c^5 - 2*a*c^7 + c^8 + (a^2 - 6*a*b)*c^6 - (4*a^3*b - 11*a^2*b^2)*c^4 + 2*(2*a
^4*b^2 - 3*a^3*b^3)*c^2)/(64*a^7*b^3 + 128*a^6*b^4 + 64*a^5*b^5 - a^2*c^8 + 2*(a^3 + a^2*b)*c^7 - (a^4 - 10*a^
3*b + a^2*b^2)*c^6 - 24*(a^4*b + a^3*b^2)*c^5 + 12*(a^5*b - 2*a^4*b^2 + a^3*b^3)*c^4 + 96*(a^5*b^2 + a^4*b^3)*
c^3 - 16*(3*a^6*b^2 + 2*a^5*b^3 + 3*a^4*b^4)*c^2 - 128*(a^6*b^3 + a^5*b^4)*c)))/(16*a^4*b^2 + 16*a^3*b^3 - 16*
a^3*b^2*c + 8*a^2*b*c^3 - a*c^5 + (a^2 + a*b)*c^4 - 8*(a^3*b + a^2*b^2)*c^2)))*arctan(1/2*(sqrt(1/2)*((16*a^3*
b^2*c^2 - 8*a^2*b*c^4 + a*c^6 - (a^2 + a*b)*c^5 + 8*(a^3*b + a^2*b^2)*c^3 - 16*(a^4*b^2 + a^3*b^3)*c)*x*sqrt(-
(a^4*b^4 + 4*a^4*b^3*c - 14*a^3*b^2*c^3 + 10*a^2*b*c^5 - 2*a*c^7 + c^8 + (a^2 - 6*a*b)*c^6 - (4*a^3*b - 11*a^2
*b^2)*c^4 + 2*(2*a^4*b^2 - 3*a^3*b^3)*c^2)/(64*a^7*b^3 + 128*a^6*b^4 + 64*a^5*b^5 - a^2*c^8 + 2*(a^3 + a^2*b)*
c^7 - (a^4 - 10*a^3*b + a^2*b^2)*c^6 - 24*(a^4*b + a^3*b^2)*c^5 + 12*(a^5*b - 2*a^4*b^2 + a^3*b^3)*c^4 + 96*(a
^5*b^2 + a^4*b^3)*c^3 - 16*(3*a^6*b^2 + 2*a^5*b^3 + 3*a^4*b^4)*c^2 - 128*(a^6*b^3 + a^5*b^4)*c)) - (4*a^3*b^3
+ 8*a^3*b^2*c - 13*a^2*b^2*c^2 - 6*a^2*b*c^3 + 7*a*b*c^4 + a*c^5 - c^6)*x)*sqrt((sqrt(1/2)*((64*a^12*b^10 + 64
*a^11*b^11 - a^5*b^4*c^13 + (3*a^6*b^4 + a^5*b^5)*c^12 - (3*a^7*b^4 - 14*a^6*b^5)*c^11 + (a^8*b^4 - 45*a^7*b^5
- 16*a^6*b^6)*c^10 + 2*(22*a^8*b^5 - 35*a^7*b^6)*c^9 - (14*a^9*b^5 - 257*a^8*b^6 - 100*a^7*b^7)*c^8 - (243*a^
9*b^6 - 134*a^8*b^7)*c^7 + (72*a^10*b^6 - 685*a^9*b^7 - 305*a^8*b^8)*c^6 + 4*(153*a^10*b^7 - 2*a^9*b^8)*c^5 -
4*(40*a^11*b^7 - 207*a^10*b^8 - 115*a^9*b^9)*c^4 - 16*(41*a^11*b^8 + 14*a^10*b^9)*c^3 + 16*(8*a^12*b^8 - 23*a^
11*b^9 - 19*a^10*b^10)*c^2 + 64*(3*a^12*b^9 + 2*a^11*b^10)*c)*x^2*sqrt(-(a^4*b^4 + 4*a^4*b^3*c - 14*a^3*b^2*c^
3 + 10*a^2*b*c^5 - 2*a*c^7 + c^8 + (a^2 - 6*a*b)*c^6 - (4*a^3*b - 11*a^2*b^2)*c^4 + 2*(2*a^4*b^2 - 3*a^3*b^3)*
c^2)/(64*a^7*b^3 + 128*a^6*b^4 + 64*a^5*b^5 - a^2*c^8 + 2*(a^3 + a^2*b)*c^7 - (a^4 - 10*a^3*b + a^2*b^2)*c^6 -
24*(a^4*b + a^3*b^2)*c^5 + 12*(a^5*b - 2*a^4*b^2 + a^3*b^3)*c^4 + 96*(a^5*b^2 + a^4*b^3)*c^3 - 16*(3*a^6*b^2
+ 2*a^5*b^3 + 3*a^4*b^4)*c^2 - 128*(a^6*b^3 + a^5*b^4)*c)) + (8*a^11*b^10 + 3*a^5*b^4*c^12 - a^4*b^4*c^13 - (3
*a^6*b^4 - 13*a^5*b^5)*c^11 + (a^7*b^4 - 36*a^6*b^5)*c^10 + (33*a^7*b^5 - 65*a^6*b^6)*c^9 - (10*a^8*b^5 - 163*
a^7*b^6)*c^8 - (134*a^8*b^6 - 155*a^7*b^7)*c^7 + 18*(2*a^9*b^6 - 19*a^8*b^7)*c^6 + (244*a^9*b^7 - 175*a^8*b^8)
*c^5 - (56*a^10*b^7 - 321*a^9*b^8)*c^4 - (184*a^10*b^8 - 79*a^9*b^9)*c^3 + 2*(16*a^11*b^8 - 51*a^10*b^9)*c^2 +
4*(8*a^11*b^9 - 3*a^10*b^10)*c)*x^2)*sqrt((2*a^3*b^2 - 5*a^2*b^2*c - 4*a^2*b*c^2 + 5*a*b*c^3 + a*c^4 - c^5 -
(16*a^4*b^2 + 16*a^3*b^3 - 16*a^3*b^2*c + 8*a^2*b*c^3 - a*c^5 + (a^2 + a*b)*c^4 - 8*(a^3*b + a^2*b^2)*c^2)*sqr
t(-(a^4*b^4 + 4*a^4*b^3*c - 14*a^3*b^2*c^3 + 10*a^2*b*c^5 - 2*a*c^7 + c^8 + (a^2 - 6*a*b)*c^6 - (4*a^3*b - 11*
a^2*b^2)*c^4 + 2*(2*a^4*b^2 - 3*a^3*b^3)*c^2)/(64*a^7*b^3 + 128*a^6*b^4 + 64*a^5*b^5 - a^2*c^8 + 2*(a^3 + a^2*
b)*c^7 - (a^4 - 10*a^3*b + a^2*b^2)*c^6 - 24*(a^4*b + a^3*b^2)*c^5 + 12*(a^5*b - 2*a^4*b^2 + a^3*b^3)*c^4 + 96
*(a^5*b^2 + a^4*b^3)*c^3 - 16*(3*a^6*b^2 + 2*a^5*b^3 + 3*a^4*b^4)*c^2 - 128*(a^6*b^3 + a^5*b^4)*c)))/(16*a^4*b
^2 + 16*a^3*b^3 - 16*a^3*b^2*c + 8*a^2*b*c^3 - a*c^5 + (a^2 + a*b)*c^4 - 8*(a^3*b + a^2*b^2)*c^2)) + 2*(a^10*b
^10 + 4*a^10*b^9*c - 14*a^9*b^8*c^3 + 10*a^8*b^7*c^5 - 2*a^7*b^6*c^7 + a^6*b^6*c^8 + (a^8*b^6 - 6*a^7*b^7)*c^6
- (4*a^9*b^7 - 11*a^8*b^8)*c^4 + 2*(2*a^10*b^8 - 3*a^9*b^9)*c^2)*sqrt(a*x^4 - b))/x^2) + (4*a^8*b^8 + 16*a^8*
b^7*c - 60*a^7*b^6*c^3 + 54*a^6*b^5*c^5 - 18*a^5*b^4*c^7 + 2*a^4*b^3*c^9 - a^3*b^3*c^10 - (a^5*b^3 - 10*a^4*b^
4)*c^8 + (8*a^6*b^4 - 35*a^5*b^5)*c^6 - 10*(2*a^7*b^5 - 5*a^6*b^6)*c^4 + (16*a^8*b^6 - 25*a^7*b^7)*c^2 - (a^4*
b^3*c^10 - (2*a^5*b^3 + a^4*b^4)*c^9 + (a^6*b^3 - 10*a^5*b^4)*c^8 + (21*a^6*b^4 + 11*a^5*b^5)*c^7 - (10*a^7*b^
4 - 31*a^6*b^5)*c^6 - (73*a^7*b^5 + 41*a^6*b^6)*c^5 + 8*(4*a^8*b^5 - 3*a^7*b^6)*c^4 + 8*(11*a^8*b^6 + 7*a^7*b^
7)*c^3 - 16*(2*a^9*b^6 + a^8*b^7)*c^2 - 16*(a^9*b^7 + a^8*b^8)*c)*sqrt(-(a^4*b^4 + 4*a^4*b^3*c - 14*a^3*b^2*c^
3 + 10*a^2*b*c^5 - 2*a*c^7 + c^8 + (a^2 - 6*a*b)*c^6 - (4*a^3*b - 11*a^2*b^2)*c^4 + 2*(2*a^4*b^2 - 3*a^3*b^3)*
c^2)/(64*a^7*b^3 + 128*a^6*b^4 + 64*a^5*b^5 - a^2*c^8 + 2*(a^3 + a^2*b)*c^7 - (a^4 - 10*a^3*b + a^2*b^2)*c^6 -
24*(a^4*b + a^3*b^2)*c^5 + 12*(a^5*b - 2*a^4*b^2 + a^3*b^3)*c^4 + 96*(a^5*b^2 + a^4*b^3)*c^3 - 16*(3*a^6*b^2
+ 2*a^5*b^3 + 3*a^4*b^4)*c^2 - 128*(a^6*b^3 + a^5*b^4)*c)))*(a*x^4 - b)^(1/4))*sqrt(sqrt(1/2)*sqrt((2*a^3*b^2
- 5*a^2*b^2*c - 4*a^2*b*c^2 + 5*a*b*c^3 + a*c^4 - c^5 - (16*a^4*b^2 + 16*a^3*b^3 - 16*a^3*b^2*c + 8*a^2*b*c^3
- a*c^5 + (a^2 + a*b)*c^4 - 8*(a^3*b + a^2*b^2)*c^2)*sqrt(-(a^4*b^4 + 4*a^4*b^3*c - 14*a^3*b^2*c^3 + 10*a^2*b*
c^5 - 2*a*c^7 + c^8 + (a^2 - 6*a*b)*c^6 - (4*a^...

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Sympy [F(-1)] Timed out
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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)

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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)

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