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

Optimal. Leaf size=122 \[ \frac {\left (a x^4-b\right )^{3/4} \left (-8 a^2 x+4 a x^5+5 b x\right )}{16 a^2}+\frac {\left (5 b^2-24 a^2 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}}+\frac {\left (5 b^2-24 a^2 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}} \]

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Rubi [A]  time = 0.08, antiderivative size = 130, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1411, 388, 240, 212, 206, 203} \begin {gather*} -\frac {1}{16} x \left (8-\frac {5 b}{a^2}\right ) \left (a x^4-b\right )^{3/4}-\frac {b \left (24 a^2-5 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}}-\frac {b \left (24 a^2-5 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}}+\frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*((8 - (5*b)/a^2)*x*(-b + a*x^4)^(3/4)) + (x^5*(-b + a*x^4)^(3/4))/(4*a) - ((24*a^2 - 5*b)*b*ArcTan[(a^(1
/4)*x)/(-b + a*x^4)^(1/4)])/(32*a^(9/4)) - ((24*a^2 - 5*b)*b*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(32*a^(9
/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 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 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 388

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

Rule 1411

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Simp[(c*x^(n + 1)*
(d + e*x^n)^(q + 1))/(e*(n*(q + 2) + 1)), x] + Dist[1/(e*(n*(q + 2) + 1)), Int[(d + e*x^n)^q*(a*e*(n*(q + 2) +
1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && N
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps

\begin {align*} \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx &=\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}+\frac {\int \frac {-8 a b-\left (16 a^2-10 b\right ) x^4}{\sqrt [4]{-b+a x^4}} \, dx}{8 a}\\ &=-\frac {1}{16} \left (8-\frac {5 b}{a^2}\right ) x \left (-b+a x^4\right )^{3/4}+\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}-\frac {\left (32 a^2 b-b \left (-16 a^2+10 b\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx}{32 a^2}\\ &=-\frac {1}{16} \left (8-\frac {5 b}{a^2}\right ) x \left (-b+a x^4\right )^{3/4}+\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}-\frac {\left (32 a^2 b-b \left (-16 a^2+10 b\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^2}\\ &=-\frac {1}{16} \left (8-\frac {5 b}{a^2}\right ) x \left (-b+a x^4\right )^{3/4}+\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}-\frac {\left (32 a^2 b-b \left (-16 a^2+10 b\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{64 a^2}-\frac {\left (32 a^2 b-b \left (-16 a^2+10 b\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{64 a^2}\\ &=-\frac {1}{16} \left (8-\frac {5 b}{a^2}\right ) x \left (-b+a x^4\right )^{3/4}+\frac {x^5 \left (-b+a x^4\right )^{3/4}}{4 a}-\frac {\left (24 a^2-5 b\right ) b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}}-\frac {\left (24 a^2-5 b\right ) b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 112, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt [4]{a} x \left (a x^4-b\right )^{3/4} \left (-8 a^2+4 a x^4+5 b\right )-b \left (24 a^2-5 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )-b \left (24 a^2-5 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^(1/4)*x*(-b + a*x^4)^(3/4)*(-8*a^2 + 5*b + 4*a*x^4) - (24*a^2 - 5*b)*b*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1
/4)] - (24*a^2 - 5*b)*b*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(32*a^(9/4))

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IntegrateAlgebraic [A]  time = 0.82, size = 122, normalized size = 1.00 \begin {gather*} \frac {\left (-b+a x^4\right )^{3/4} \left (-8 a^2 x+5 b x+4 a x^5\right )}{16 a^2}+\frac {\left (-24 a^2 b+5 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}}+\frac {\left (-24 a^2 b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-b + a*x^4)^(3/4)*(-8*a^2*x + 5*b*x + 4*a*x^5))/(16*a^2) + ((-24*a^2*b + 5*b^2)*ArcTan[(a^(1/4)*x)/(-b + a*x
^4)^(1/4)])/(32*a^(9/4)) + ((-24*a^2*b + 5*b^2)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(32*a^(9/4))

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fricas [B]  time = 0.89, size = 726, normalized size = 5.95 \begin {gather*} \frac {4 \, a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{2} x \sqrt {\frac {{\left (331776 \, a^{13} b^{4} - 276480 \, a^{11} b^{5} + 86400 \, a^{9} b^{6} - 12000 \, a^{7} b^{7} + 625 \, a^{5} b^{8}\right )} x^{2} \sqrt {\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}} + {\left (191102976 \, a^{12} b^{6} - 238878720 \, a^{10} b^{7} + 124416000 \, a^{8} b^{8} - 34560000 \, a^{6} b^{9} + 5400000 \, a^{4} b^{10} - 450000 \, a^{2} b^{11} + 15625 \, b^{12}\right )} \sqrt {a x^{4} - b}}{x^{2}}} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} + {\left (13824 \, a^{8} b^{3} - 8640 \, a^{6} b^{4} + 1800 \, a^{4} b^{5} - 125 \, a^{2} b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}}}{{\left (331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}\right )} x}\right ) - a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} + {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + a^{2} \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {a^{7} x \left (\frac {331776 \, a^{8} b^{4} - 276480 \, a^{6} b^{5} + 86400 \, a^{4} b^{6} - 12000 \, a^{2} b^{7} + 625 \, b^{8}}{a^{9}}\right )^{\frac {3}{4}} - {\left (13824 \, a^{6} b^{3} - 8640 \, a^{4} b^{4} + 1800 \, a^{2} b^{5} - 125 \, b^{6}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + 4 \, {\left (4 \, a x^{5} - {\left (8 \, a^{2} - 5 \, b\right )} x\right )} {\left (a x^{4} - b\right )}^{\frac {3}{4}}}{64 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(4*a^2*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(1/4)*arctan((a^
2*x*sqrt(((331776*a^13*b^4 - 276480*a^11*b^5 + 86400*a^9*b^6 - 12000*a^7*b^7 + 625*a^5*b^8)*x^2*sqrt((331776*a
^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9) + (191102976*a^12*b^6 - 238878720*a^10
*b^7 + 124416000*a^8*b^8 - 34560000*a^6*b^9 + 5400000*a^4*b^10 - 450000*a^2*b^11 + 15625*b^12)*sqrt(a*x^4 - b)
)/x^2)*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(1/4) + (13824*a^8*b^
3 - 8640*a^6*b^4 + 1800*a^4*b^5 - 125*a^2*b^6)*(a*x^4 - b)^(1/4)*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4
*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(1/4))/((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7
+ 625*b^8)*x)) - a^2*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(1/4)*l
og(-(a^7*x*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(3/4) + (13824*a^
6*b^3 - 8640*a^4*b^4 + 1800*a^2*b^5 - 125*b^6)*(a*x^4 - b)^(1/4))/x) + a^2*((331776*a^8*b^4 - 276480*a^6*b^5 +
86400*a^4*b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(1/4)*log((a^7*x*((331776*a^8*b^4 - 276480*a^6*b^5 + 86400*a^4*
b^6 - 12000*a^2*b^7 + 625*b^8)/a^9)^(3/4) - (13824*a^6*b^3 - 8640*a^4*b^4 + 1800*a^2*b^5 - 125*b^6)*(a*x^4 - b
)^(1/4))/x) + 4*(4*a*x^5 - (8*a^2 - 5*b)*x)*(a*x^4 - b)^(3/4))/a^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{8} - 2 \, a x^{4} - b}{{\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*x^8-2*a*x^4-b)/(a*x^4-b)^(1/4),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-2 a \,x^{4}-b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.73, size = 361, normalized size = 2.96 \begin {gather*} \frac {1}{8} \, a {\left (\frac {b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{a} - \frac {4 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} b}{{\left (a^{2} - \frac {{\left (a x^{4} - b\right )} a}{x^{4}}\right )} x^{3}}\right )} + \frac {1}{4} \, b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )} - \frac {5 \, b^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{64 \, a^{2}} + \frac {\frac {9 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} a b^{2}}{x^{3}} - \frac {5 \, {\left (a x^{4} - b\right )}^{\frac {7}{4}} b^{2}}{x^{7}}}{16 \, {\left (a^{4} - \frac {2 \, {\left (a x^{4} - b\right )} a^{3}}{x^{4}} + \frac {{\left (a x^{4} - b\right )}^{2} a^{2}}{x^{8}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*(b*(2*arctan((a*x^4 - b)^(1/4)/(a^(1/4)*x))/a^(1/4) + log(-(a^(1/4) - (a*x^4 - b)^(1/4)/x)/(a^(1/4) + (a
*x^4 - b)^(1/4)/x))/a^(1/4))/a - 4*(a*x^4 - b)^(3/4)*b/((a^2 - (a*x^4 - b)*a/x^4)*x^3)) + 1/4*b*(2*arctan((a*x
^4 - b)^(1/4)/(a^(1/4)*x))/a^(1/4) + log(-(a^(1/4) - (a*x^4 - b)^(1/4)/x)/(a^(1/4) + (a*x^4 - b)^(1/4)/x))/a^(
1/4)) - 5/64*b^2*(2*arctan((a*x^4 - b)^(1/4)/(a^(1/4)*x))/a^(1/4) + log(-(a^(1/4) - (a*x^4 - b)^(1/4)/x)/(a^(1
/4) + (a*x^4 - b)^(1/4)/x))/a^(1/4))/a^2 + 1/16*(9*(a*x^4 - b)^(3/4)*a*b^2/x^3 - 5*(a*x^4 - b)^(7/4)*b^2/x^7)/
(a^4 - 2*(a*x^4 - b)*a^3/x^4 + (a*x^4 - b)^2*a^2/x^8)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 3.36, size = 122, normalized size = 1.00 \begin {gather*} \frac {a x^{5} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {9}{4}\right )} - \frac {b^{\frac {3}{4}} x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {x^{9} e^{- \frac {i \pi }{4}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**5*exp(3*I*pi/4)*gamma(5/4)*hyper((1/4, 5/4), (9/4,), a*x**4/b)/(2*b**(1/4)*gamma(9/4)) - b**(3/4)*x*exp(-
I*pi/4)*gamma(1/4)*hyper((1/4, 1/4), (5/4,), a*x**4/b)/(4*gamma(5/4)) + x**9*exp(-I*pi/4)*gamma(9/4)*hyper((1/
4, 9/4), (13/4,), a*x**4/b)/(2*b**(1/4)*gamma(13/4))

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