\(\int \frac {(-b-a x+x^4) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx\) [2811]

Optimal result

Integrand size = 36, antiderivative size = 276 \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\frac {\sqrt [4]{b x^3+a x^4} \left (-65280 a^4 b+32705 b^4-15360 a^5 x+16420 a b^3 x+10400 a^2 b^2 x^2+8064 a^3 b x^3+6144 a^4 x^4\right )}{30720 a^5}+\frac {\left (19712 a^4 b^2-9843 b^5\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}-\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{23/4}}+\frac {\left (-19712 a^4 b^2+9843 b^5\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}+\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{23/4}} \]

[Out]

1/30720*(a*x^4+b*x^3)^(1/4)*(6144*a^4*x^4+8064*a^3*b*x^3-15360*a^5*x+10400*a^2*b^2*x^2-65280*a^4*b+16420*a*b^3
*x+32705*b^4)/a^5+1/4096*(19712*a^4*b^2-9843*b^5)*arctan(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4)-2*2^(1/4)*(2*
a^4*b^2-b^5)*arctan(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4)+1/4096*(-19712*a^4*b^2+9843*b^5)*arctanh(a
^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4)+2*2^(1/4)*(2*a^4*b^2-b^5)*arctanh(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^3)^(1/4)
)/a^(23/4)

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.98, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.361, Rules used = {2081, 1629, 161, 96, 95, 304, 209, 212, 963, 81, 52, 65, 338} \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=-\frac {x \left (768-\frac {397 b^3}{a^4}\right ) \sqrt [4]{a x^4+b x^3}}{1536}-\frac {8 b x \left (2-\frac {b^3}{a^4}\right ) \sqrt [4]{a x^4+b x^3}}{3 (a x+b)}+\frac {53 b^2 x (a x+b) \sqrt [4]{a x^4+b x^3}}{192 a^4}+\frac {b x^2 (a x+b) \sqrt [4]{a x^4+b x^3}}{16 a^3}+\frac {x^3 (a x+b) \sqrt [4]{a x^4+b x^3}}{5 a^2}+\frac {b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {b \left (19712 a^4-9843 b^3\right ) \sqrt [4]{a x^4+b x^3}}{6144 a^5}-\frac {8 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3}}{3 a^5 (a x+b)} \]

[In]

Int[((-b - a*x + x^4)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x),x]

[Out]

(b*(19712*a^4 - 9843*b^3)*(b*x^3 + a*x^4)^(1/4))/(6144*a^5) - ((768 - (397*b^3)/a^4)*x*(b*x^3 + a*x^4)^(1/4))/
1536 - (8*b^2*(2*a^4 - b^3)*(b*x^3 + a*x^4)^(1/4))/(3*a^5*(b + a*x)) - (8*b*(2 - b^3/a^4)*x*(b*x^3 + a*x^4)^(1
/4))/(3*(b + a*x)) + (53*b^2*x*(b + a*x)*(b*x^3 + a*x^4)^(1/4))/(192*a^4) + (b*x^2*(b + a*x)*(b*x^3 + a*x^4)^(
1/4))/(16*a^3) + (x^3*(b + a*x)*(b*x^3 + a*x^4)^(1/4))/(5*a^2) + (b^2*(19712*a^4 - 9843*b^3)*(b*x^3 + a*x^4)^(
1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(4096*a^(23/4)*x^(3/4)*(b + a*x)^(1/4)) - (2*2^(1/4)*b^2*(2*a^
4 - b^3)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(2^(1/4)*a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(a^(23/4)*x^(3/4)*(b + a*x)^
(1/4)) - (b^2*(19712*a^4 - 9843*b^3)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(4096*a
^(23/4)*x^(3/4)*(b + a*x)^(1/4)) + (2*2^(1/4)*b^2*(2*a^4 - b^3)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(2^(1/4)*a^(1/4)
*x^(1/4))/(b + a*x)^(1/4)])/(a^(23/4)*x^(3/4)*(b + a*x)^(1/4))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 161

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbol]
:> Dist[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/f^(m + n + 2)), Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x
], x] + Dist[1/f^(m + n + 2), Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 2)*(c + d*x)^(m + n
+ 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 304

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

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]

Rule 963

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rule 1629

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^
(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x} \left (-b-a x+x^4\right )}{-b+a x} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x} \left (-5 a^2 b-5 a^3 x+\frac {15 b^2 x^2}{4}+\frac {5}{4} a b x^3\right )}{-b+a x} \, dx}{5 a^2 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x} \left (-20 a^4 b-\frac {5}{16} a \left (64 a^4-11 b^3\right ) x+\frac {265}{16} a^2 b^2 x^2\right )}{-b+a x} \, dx}{20 a^4 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x} \left (-\frac {5}{64} a^2 b \left (768 a^4-371 b^3\right )-\frac {5}{64} a^3 \left (768 a^4-397 b^3\right ) x\right )}{-b+a x} \, dx}{60 a^6 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \left (-\frac {5}{64} a^5 b^2 \left (5376 a^4-2701 b^3\right )-\frac {5}{32} a^6 b \left (1536 a^4-781 b^3\right ) x-\frac {5}{64} a^7 \left (768 a^4-397 b^3\right ) x^2\right )}{(b+a x)^{7/4}} \, dx}{60 a^9 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 b^3 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4}}{(-b+a x) (b+a x)^{7/4}} \, dx}{a^4 x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {8 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{3 a^5 (b+a x)}-\frac {8 b \left (2-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{3 (b+a x)}+\frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \left (\frac {15}{256} a^5 b^2 \left (1792 a^4-883 b^3\right )-\frac {15}{256} a^6 b \left (768 a^4-397 b^3\right ) x\right )}{(b+a x)^{3/4}} \, dx}{45 a^9 b x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 b^3 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x) (b+a x)^{3/4}} \, dx}{a^5 x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {\left (768-\frac {397 b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{1536}-\frac {8 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{3 a^5 (b+a x)}-\frac {8 b \left (2-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{3 (b+a x)}+\frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}+\frac {\left (b \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4}}{(b+a x)^{3/4}} \, dx}{6144 a^4 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (8 b^3 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-b+2 a b x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {b \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {\left (768-\frac {397 b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{1536}-\frac {8 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{3 a^5 (b+a x)}-\frac {8 b \left (2-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{3 (b+a x)}+\frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}-\frac {\left (b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{8192 a^5 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \sqrt {2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{11/2} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \sqrt {2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{11/2} x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {b \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {\left (768-\frac {397 b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{1536}-\frac {8 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{3 a^5 (b+a x)}-\frac {8 b \left (2-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{3 (b+a x)}+\frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}-\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {b \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {\left (768-\frac {397 b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{1536}-\frac {8 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{3 a^5 (b+a x)}-\frac {8 b \left (2-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{3 (b+a x)}+\frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}-\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {b \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {\left (768-\frac {397 b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{1536}-\frac {8 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{3 a^5 (b+a x)}-\frac {8 b \left (2-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{3 (b+a x)}+\frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}-\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {b \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {\left (768-\frac {397 b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{1536}-\frac {8 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{3 a^5 (b+a x)}-\frac {8 b \left (2-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}}{3 (b+a x)}+\frac {53 b^2 x (b+a x) \sqrt [4]{b x^3+a x^4}}{192 a^4}+\frac {b x^2 (b+a x) \sqrt [4]{b x^3+a x^4}}{16 a^3}+\frac {x^3 (b+a x) \sqrt [4]{b x^3+a x^4}}{5 a^2}+\frac {b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (19712 a^4-9843 b^3\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{2} b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{23/4} x^{3/4} \sqrt [4]{b+a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\frac {x^{9/4} (b+a x)^{3/4} \left (2 a^{3/4} x^{3/4} \sqrt [4]{b+a x} \left (32705 b^4-15360 a^5 x+16420 a b^3 x+10400 a^2 b^2 x^2+8064 a^3 b x^3+768 a^4 \left (-85 b+8 x^4\right )\right )+15 b^2 \left (19712 a^4-9843 b^3\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+122880 \sqrt [4]{2} b^2 \left (-2 a^4+b^3\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+15 b^2 \left (-19712 a^4+9843 b^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-122880 \sqrt [4]{2} b^2 \left (-2 a^4+b^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{61440 a^{23/4} \left (x^3 (b+a x)\right )^{3/4}} \]

[In]

Integrate[((-b - a*x + x^4)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x),x]

[Out]

(x^(9/4)*(b + a*x)^(3/4)*(2*a^(3/4)*x^(3/4)*(b + a*x)^(1/4)*(32705*b^4 - 15360*a^5*x + 16420*a*b^3*x + 10400*a
^2*b^2*x^2 + 8064*a^3*b*x^3 + 768*a^4*(-85*b + 8*x^4)) + 15*b^2*(19712*a^4 - 9843*b^3)*ArcTan[(a^(1/4)*x^(1/4)
)/(b + a*x)^(1/4)] + 122880*2^(1/4)*b^2*(-2*a^4 + b^3)*ArcTan[(2^(1/4)*a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] + 15*
b^2*(-19712*a^4 + 9843*b^3)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] - 122880*2^(1/4)*b^2*(-2*a^4 + b^3)*Arc
Tanh[(2^(1/4)*a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]))/(61440*a^(23/4)*(x^3*(b + a*x))^(3/4))

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {77 \left (-\frac {32 \left (a^{4}-\frac {b^{3}}{2}\right ) b^{2} 2^{\frac {1}{4}} \ln \left (\frac {-2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )}{77}+\frac {\left (a^{4} b^{2}-\frac {9843}{19712} b^{5}\right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )}{2}-\frac {64 \left (a^{4}-\frac {b^{3}}{2}\right ) b^{2} 2^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )}{77}+\left (a^{4} b^{2}-\frac {9843}{19712} b^{5}\right ) \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )-\frac {65 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} \left (\frac {24 \left (\frac {8 x^{4}}{5}-17 b \right ) a^{\frac {19}{4}}}{65}+a^{\frac {11}{4}} b^{2} x^{2}+\frac {252 a^{\frac {15}{4}} b \,x^{3}}{325}+\frac {6541 a^{\frac {3}{4}} b^{4}}{2080}+\frac {821 a^{\frac {7}{4}} b^{3} x}{520}-\frac {96 a^{\frac {23}{4}} x}{65}\right )}{924}\right )}{16 a^{\frac {23}{4}}}\)	\(274\)

[In]

int((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x,method=_RETURNVERBOSE)

[Out]

-77/16/a^(23/4)*(-32/77*(a^4-1/2*b^3)*b^2*2^(1/4)*ln((-2^(1/4)*a^(1/4)*x-(x^3*(a*x+b))^(1/4))/(2^(1/4)*a^(1/4)
*x-(x^3*(a*x+b))^(1/4)))+1/2*(a^4*b^2-9843/19712*b^5)*ln((-a^(1/4)*x-(x^3*(a*x+b))^(1/4))/(a^(1/4)*x-(x^3*(a*x
+b))^(1/4)))-64/77*(a^4-1/2*b^3)*b^2*2^(1/4)*arctan(1/2*(x^3*(a*x+b))^(1/4)/x*2^(3/4)/a^(1/4))+(a^4*b^2-9843/1
9712*b^5)*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1/4))-65/924*(x^3*(a*x+b))^(1/4)*(24/65*(8/5*x^4-17*b)*a^(19/4)+a^
(11/4)*b^2*x^2+252/325*a^(15/4)*b*x^3+6541/2080*a^(3/4)*b^4+821/520*a^(7/4)*b^3*x-96/65*a^(23/4)*x))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 1152, normalized size of antiderivative = 4.17 \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\text {Too large to display} \]

[In]

integrate((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x, algorithm="fricas")

[Out]

1/122880*(122880*2^(1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4)*log(-
(2^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4) + (2*a^4*b^2 - b^5)
*(a*x^4 + b*x^3)^(1/4))/x) - 122880*2^(1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20
)/a^23)^(1/4)*log((2^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4) -
(2*a^4*b^2 - b^5)*(a*x^4 + b*x^3)^(1/4))/x) - 122880*I*2^(1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14
- 8*a^4*b^17 + b^20)/a^23)^(1/4)*log((I*2^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17
+ b^20)/a^23)^(1/4) - (2*a^4*b^2 - b^5)*(a*x^4 + b*x^3)^(1/4))/x) + 122880*I*2^(1/4)*a^5*((16*a^16*b^8 - 32*a
^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4)*log((-I*2^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12*b^11 + 2
4*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4) - (2*a^4*b^2 - b^5)*(a*x^4 + b*x^3)^(1/4))/x) - 15*a^5*((150981161
449947136*a^16*b^8 - 301564036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 +
9386635211853201*b^20)/a^23)^(1/4)*log(-(a^6*x*((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 +
225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4) + (19712*a^4*b^
2 - 9843*b^5)*(a*x^4 + b*x^3)^(1/4))/x) + 15*a^5*((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11
+ 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4)*log((a^6*x*((1
50981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^
4*b^17 + 9386635211853201*b^20)/a^23)^(1/4) - (19712*a^4*b^2 - 9843*b^5)*(a*x^4 + b*x^3)^(1/4))/x) + 15*I*a^5*
((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736
*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4)*log((I*a^6*x*((150981161449947136*a^16*b^8 - 301564036556783616
*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4) - (
19712*a^4*b^2 - 9843*b^5)*(a*x^4 + b*x^3)^(1/4))/x) - 15*I*a^5*((150981161449947136*a^16*b^8 - 301564036556783
616*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4)*
log((-I*a^6*x*((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 751
92259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4) - (19712*a^4*b^2 - 9843*b^5)*(a*x^4 + b*x^3)^(1/4
))/x) + 4*(6144*a^4*x^4 + 8064*a^3*b*x^3 + 10400*a^2*b^2*x^2 - 65280*a^4*b + 32705*b^4 - 20*(768*a^5 - 821*a*b
^3)*x)*(a*x^4 + b*x^3)^(1/4))/a^5

Sympy [F]

\[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (- a x - b + x^{4}\right )}{a x - b}\, dx \]

[In]

integrate((x**4-a*x-b)*(a*x**4+b*x**3)**(1/4)/(a*x-b),x)

[Out]

Integral((x**3*(a*x + b))**(1/4)*(-a*x - b + x**4)/(a*x - b), x)

Maxima [F]

\[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - a x - b\right )}}{a x - b} \,d x } \]

[In]

integrate((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + b*x^3)^(1/4)*(x^4 - a*x - b)/(a*x - b), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (236) = 472\).

Time = 0.38 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.55 \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\text {Too large to display} \]

[In]

integrate((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x, algorithm="giac")

[Out]

1/8192*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/
4))/((-a)^(3/4)*a^5) + 1/8192*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(
a + b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^5) + 1/16384*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*log(sqrt(2)*(-a)^(1/
4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^5) - 1/16384*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*l
og(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^5) + 1/2*sqrt(2)*(2*2^(1/4)*(
-a)^(1/4)*a^4*b^2 - 2^(1/4)*(-a)^(1/4)*b^5)*log(2^(3/4)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a
+ b/x))/a^6 - 1/2*sqrt(2)*(2*2^(1/4)*(-a)^(1/4)*a^4*b^2 - 2^(1/4)*(-a)^(1/4)*b^5)*log(-2^(3/4)*(-a)^(1/4)*(a
+ b/x)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x))/a^6 - 1/30720*(65280*(a + b/x)^(17/4)*a^4*b^2 - 245760*(a + b
/x)^(13/4)*a^5*b^2 + 345600*(a + b/x)^(9/4)*a^6*b^2 - 215040*(a + b/x)^(5/4)*a^7*b^2 + 49920*(a + b/x)^(1/4)*a
^8*b^2 - 32705*(a + b/x)^(17/4)*b^5 + 114400*(a + b/x)^(13/4)*a*b^5 - 157370*(a + b/x)^(9/4)*a^2*b^5 + 94296*(
a + b/x)^(5/4)*a^3*b^5 - 24765*(a + b/x)^(1/4)*a^4*b^5)*x^5/(a^5*b^5) + (2*2^(3/4)*(-a)^(1/4)*a^4*b^2 - 2^(3/4
)*(-a)^(1/4)*b^5)*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^6 + (2*2^(3/4)*(-a
)^(1/4)*a^4*b^2 - 2^(3/4)*(-a)^(1/4)*b^5)*arctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1
/4))/a^6

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}\,\left (-x^4+a\,x+b\right )}{b-a\,x} \,d x \]

[In]

int(((a*x^4 + b*x^3)^(1/4)*(b + a*x - x^4))/(b - a*x),x)

[Out]

int(((a*x^4 + b*x^3)^(1/4)*(b + a*x - x^4))/(b - a*x), x)