\(\int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx\) [1805]

Optimal result

Integrand size = 29, antiderivative size = 122 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {\left (-b+a x^4\right )^{3/4} \left (-8 a^2 x+5 b x+4 a x^5\right )}{16 a^2}+\frac {\left (-24 a^2 b+5 b^2\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}}+\frac {\left (-24 a^2 b+5 b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \]

[Out]

1/16*(a*x^4-b)^(3/4)*(4*a*x^5-8*a^2*x+5*b*x)/a^2+1/32*(-24*a^2*b+5*b^2)*arctan(a^(1/4)*x/(a*x^4-b)^(1/4))/a^(9
/4)+1/32*(-24*a^2*b+5*b^2)*arctanh(a^(1/4)*x/(a*x^4-b)^(1/4))/a^(9/4)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1425, 396, 246, 218, 212, 209} \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=-\frac {1}{16} x \left (8-\frac {5 b}{a^2}\right ) \left (a x^4-b\right )^{3/4}-\frac {b \left (24 a^2-5 b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}}-\frac {b \left (24 a^2-5 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{32 a^{9/4}}+\frac {x^5 \left (a x^4-b\right )^{3/4}}{4 a} \]

[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 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 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 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 396

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 1425

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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 \arctan \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 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {2 \sqrt [4]{a} x \left (-b+a x^4\right )^{3/4} \left (-8 a^2+5 b+4 a x^4\right )-\left (24 a^2-5 b\right ) b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )-\left (24 a^2-5 b\right ) b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{32 a^{9/4}} \]

[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))

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.63

method	result	size
pseudoelliptic	\(\frac {-32 a^{\frac {9}{4}} x \left (a \,x^{4}-b \right )^{\frac {3}{4}}+16 a^{\frac {5}{4}} x^{5} \left (a \,x^{4}-b \right )^{\frac {3}{4}}+20 b x \,a^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {3}{4}}+48 \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2} b -24 \ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right ) a^{2} b -10 \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{2}+5 \ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right ) b^{2}}{64 a^{\frac {9}{4}}}\)	\(199\)

[In]

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

[Out]

1/64*(-32*a^(9/4)*x*(a*x^4-b)^(3/4)+16*a^(5/4)*x^5*(a*x^4-b)^(3/4)+20*b*x*a^(1/4)*(a*x^4-b)^(3/4)+48*arctan(1/
a^(1/4)/x*(a*x^4-b)^(1/4))*a^2*b-24*ln((a^(1/4)*x+(a*x^4-b)^(1/4))/(-a^(1/4)*x+(a*x^4-b)^(1/4)))*a^2*b-10*arct
an(1/a^(1/4)/x*(a*x^4-b)^(1/4))*b^2+5*ln((a^(1/4)*x+(a*x^4-b)^(1/4))/(-a^(1/4)*x+(a*x^4-b)^(1/4)))*b^2)/a^(9/4
)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 627, normalized size of antiderivative = 5.14 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=-\frac {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 ) + i \, 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 {i \, 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 ) - i \, 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 {-i \, 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}} \]

[In]

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

[Out]

-1/64*(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 - 86
40*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 - 1200
0*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
) + I*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((I*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 - 86
40*a^4*b^4 + 1800*a^2*b^5 - 125*b^6)*(a*x^4 - b)^(1/4))/x) - I*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((-I*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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {a x^{5} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {9}{4}\right )} - \frac {b^{\frac {3}{4}} x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {x^{9} e^{- \frac {i \pi }{4}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {13}{4}\right )} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (102) = 204\).

Time = 0.29 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.96 \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\frac {1}{8} \, a {\left (\frac {b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{a} - \frac {4 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} b}{{\left (a^{2} - \frac {{\left (a x^{4} - b\right )} a}{x^{4}}\right )} x^{3}}\right )} + \frac {1}{4} \, b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )} - \frac {5 \, b^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{64 \, a^{2}} + \frac {\frac {9 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} a b^{2}}{x^{3}} - \frac {5 \, {\left (a x^{4} - b\right )}^{\frac {7}{4}} b^{2}}{x^{7}}}{16 \, {\left (a^{4} - \frac {2 \, {\left (a x^{4} - b\right )} a^{3}}{x^{4}} + \frac {{\left (a x^{4} - b\right )}^{2} a^{2}}{x^{8}}\right )}} \]

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {-b-2 a x^4+2 x^8}{\sqrt [4]{-b+a x^4}} \, dx=\int -\frac {-2\,x^8+2\,a\,x^4+b}{{\left (a\,x^4-b\right )}^{1/4}} \,d x \]

[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)