\(\int \frac {b-3 a x^3+3 x^6}{x^6 (-b+2 a x^3) \sqrt [4]{-b x+a x^4}} \, dx\) [2392]

Optimal result

Integrand size = 43, antiderivative size = 192 \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=-\frac {4 \left (b-a x^3\right ) \left (-b x+a x^4\right )^{3/4}}{21 b^2 x^6}+\frac {\sqrt {2} \left (2 a^2-3 b\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{-b x+a x^4}}{-\sqrt {a} x^2+\sqrt {-b x+a x^4}}\right )}{3 \sqrt [4]{a} b^2}+\frac {\sqrt {2} \left (2 a^2-3 b\right ) \text {arctanh}\left (\frac {\sqrt {a} x^2+\sqrt {-b x+a x^4}}{\sqrt {2} \sqrt [4]{a} x \sqrt [4]{-b x+a x^4}}\right )}{3 \sqrt [4]{a} b^2} \]

[Out]

-4/21*(-a*x^3+b)*(a*x^4-b*x)^(3/4)/b^2/x^6+1/3*2^(1/2)*(2*a^2-3*b)*arctan(2^(1/2)*a^(1/4)*x*(a*x^4-b*x)^(1/4)/
(-a^(1/2)*x^2+(a*x^4-b*x)^(1/2)))/a^(1/4)/b^2+1/3*2^(1/2)*(2*a^2-3*b)*arctanh(1/2*(a^(1/2)*x^2+(a*x^4-b*x)^(1/
2))*2^(1/2)/a^(1/4)/x/(a*x^4-b*x)^(1/4))/a^(1/4)/b^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(617\) vs. \(2(192)=384\).

Time = 1.05 (sec) , antiderivative size = 617, normalized size of antiderivative = 3.21, number of steps used = 20, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.326, Rules used = {2081, 6857, 277, 270, 477, 476, 508, 472, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=-\frac {\sqrt {2} \sqrt [4]{x} \left (2 a^2-3 b\right ) \sqrt [4]{a x^3-b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a} b^2 \sqrt [4]{a x^4-b x}}+\frac {\sqrt {2} \sqrt [4]{x} \left (2 a^2-3 b\right ) \sqrt [4]{a x^3-b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}+1\right )}{3 \sqrt [4]{a} b^2 \sqrt [4]{a x^4-b x}}-\frac {\sqrt [4]{x} \left (2 a^2-3 b\right ) \sqrt [4]{a x^3-b} \log \left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3-b}}-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}+1\right )}{3 \sqrt {2} \sqrt [4]{a} b^2 \sqrt [4]{a x^4-b x}}+\frac {\sqrt [4]{x} \left (2 a^2-3 b\right ) \sqrt [4]{a x^3-b} \log \left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3-b}}+\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}+1\right )}{3 \sqrt {2} \sqrt [4]{a} b^2 \sqrt [4]{a x^4-b x}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )^2}{21 a^2 b^2 x^5 \sqrt [4]{a x^4-b x}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{a x^4-b x}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )}{3 a b^2 x^2 \sqrt [4]{a x^4-b x}}+\frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{a x^4-b x}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{a x^4-b x}} \]

[In]

Int[(b - 3*a*x^3 + 3*x^6)/(x^6*(-b + 2*a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]

[Out]

((2 - b/a^2)*(b - a*x^3))/(7*b*x^5*(-(b*x) + a*x^4)^(1/4)) - ((2*a^2 - 3*b)*(b - a*x^3))/(3*a*b^2*x^2*(-(b*x)
+ a*x^4)^(1/4)) - (2*(b - a*x^3))/(3*a*b*x^2*(-(b*x) + a*x^4)^(1/4)) + (4*a*(2 - b/a^2)*(b - a*x^3))/(21*b^2*x
^2*(-(b*x) + a*x^4)^(1/4)) - ((2*a^2 - 3*b)*(b - a*x^3)^2)/(21*a^2*b^2*x^5*(-(b*x) + a*x^4)^(1/4)) - (Sqrt[2]*
(2*a^2 - 3*b)*x^(1/4)*(-b + a*x^3)^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(3*a^(1/4)*
b^2*(-(b*x) + a*x^4)^(1/4)) + (Sqrt[2]*(2*a^2 - 3*b)*x^(1/4)*(-b + a*x^3)^(1/4)*ArcTan[1 + (Sqrt[2]*a^(1/4)*x^
(3/4))/(-b + a*x^3)^(1/4)])/(3*a^(1/4)*b^2*(-(b*x) + a*x^4)^(1/4)) - ((2*a^2 - 3*b)*x^(1/4)*(-b + a*x^3)^(1/4)
*Log[1 + (Sqrt[a]*x^(3/2))/Sqrt[-b + a*x^3] - (Sqrt[2]*a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(3*Sqrt[2]*a^(1/4
)*b^2*(-(b*x) + a*x^4)^(1/4)) + ((2*a^2 - 3*b)*x^(1/4)*(-b + a*x^3)^(1/4)*Log[1 + (Sqrt[a]*x^(3/2))/Sqrt[-b +
a*x^3] + (Sqrt[2]*a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(3*Sqrt[2]*a^(1/4)*b^2*(-(b*x) + a*x^4)^(1/4))

Rule 210

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

Rule 217

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

Rule 270

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

Rule 277

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

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 476

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 508

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[k*(a^(p + (m + 1)/n)/n), Subst[Int[x^(k*((m + 1)/n) - 1)*((c - (b*c - a*d)*x^k)^q/(1 - b*x^k)^(p +
q + (m + 1)/n + 1)), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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]

Rule 6857

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {b-3 a x^3+3 x^6}{x^{25/4} \sqrt [4]{-b+a x^3} \left (-b+2 a x^3\right )} \, dx}{\sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \left (-\frac {3 \left (2-\frac {b}{a^2}\right )}{4 x^{25/4} \sqrt [4]{-b+a x^3}}+\frac {3}{2 a x^{13/4} \sqrt [4]{-b+a x^3}}+\frac {-2 a^2 b+3 b^2}{4 a^2 x^{25/4} \sqrt [4]{-b+a x^3} \left (-b+2 a x^3\right )}\right ) \, dx}{\sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (3 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{-b+a x^3}} \, dx}{2 a \sqrt [4]{-b x+a x^4}}-\frac {\left (3 \left (2-\frac {b}{a^2}\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{x^{25/4} \sqrt [4]{-b+a x^3}} \, dx}{4 \sqrt [4]{-b x+a x^4}}+\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{x^{25/4} \sqrt [4]{-b+a x^3} \left (-b+2 a x^3\right )} \, dx}{4 a^2 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (3 a \left (2-\frac {b}{a^2}\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{-b+a x^3}} \, dx}{7 b \sqrt [4]{-b x+a x^4}}+\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{x^{22} \sqrt [4]{-b+a x^{12}} \left (-b+2 a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{a^2 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{-b x+a x^4}}+\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{-b+a x^4} \left (-b+2 a x^4\right )} \, dx,x,x^{3/4}\right )}{3 a^2 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{-b x+a x^4}}+\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {\left (1-a x^4\right )^2}{x^8 \left (-b-a b x^4\right )} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 a^2 b^2 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{-b x+a x^4}}+\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \left (-\frac {1}{b x^8}+\frac {3 a}{b x^4}-\frac {4 a^2}{b \left (1+a x^4\right )}\right ) \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 a^2 b^2 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )}{3 a b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )^2}{21 a^2 b^2 x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (4 \left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1+a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 b^3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )}{3 a b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )^2}{21 a^2 b^2 x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 \left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1-\sqrt {a} x^2}{1+a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 b^3 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 \left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1+\sqrt {a} x^2}{1+a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 b^3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )}{3 a b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )^2}{21 a^2 b^2 x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {a}}-\frac {\sqrt {2} x}{\sqrt [4]{a}}+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt {a} b^3 \sqrt [4]{-b x+a x^4}}-\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {a}}+\frac {\sqrt {2} x}{\sqrt [4]{a}}+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt {a} b^3 \sqrt [4]{-b x+a x^4}}+\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{a}}+2 x}{-\frac {1}{\sqrt {a}}-\frac {\sqrt {2} x}{\sqrt [4]{a}}-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt {2} \sqrt [4]{a} b^3 \sqrt [4]{-b x+a x^4}}+\frac {\left (\left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{a}}-2 x}{-\frac {1}{\sqrt {a}}+\frac {\sqrt {2} x}{\sqrt [4]{a}}-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt {2} \sqrt [4]{a} b^3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )}{3 a b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )^2}{21 a^2 b^2 x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3} \log \left (1+\frac {\sqrt {a} x^{3/2}}{\sqrt {-b+a x^3}}-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt {2} \sqrt [4]{a} b^2 \sqrt [4]{-b x+a x^4}}+\frac {\left (2 a^2-3 b\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3} \log \left (1+\frac {\sqrt {a} x^{3/2}}{\sqrt {-b+a x^3}}+\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt {2} \sqrt [4]{a} b^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (\sqrt {2} \left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{a} b^3 \sqrt [4]{-b x+a x^4}}+\frac {\left (\sqrt {2} \left (-2 a^2 b+3 b^2\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{a} b^3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {\left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{7 b x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )}{3 a b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {2 \left (b-a x^3\right )}{3 a b x^2 \sqrt [4]{-b x+a x^4}}+\frac {4 a \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right )}{21 b^2 x^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \left (b-a x^3\right )^2}{21 a^2 b^2 x^5 \sqrt [4]{-b x+a x^4}}-\frac {\sqrt {2} \left (2 a^2-3 b\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{a} b^2 \sqrt [4]{-b x+a x^4}}+\frac {\sqrt {2} \left (2 a^2-3 b\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{a} b^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (2 a^2-3 b\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3} \log \left (1+\frac {\sqrt {a} x^{3/2}}{\sqrt {-b+a x^3}}-\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt {2} \sqrt [4]{a} b^2 \sqrt [4]{-b x+a x^4}}+\frac {\left (2 a^2-3 b\right ) \sqrt [4]{x} \sqrt [4]{-b+a x^3} \log \left (1+\frac {\sqrt {a} x^{3/2}}{\sqrt {-b+a x^3}}+\frac {\sqrt {2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt {2} \sqrt [4]{a} b^2 \sqrt [4]{-b x+a x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.68 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.27 \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {\frac {210 b x^3 \left (-b+a x^3\right )}{a}+15 \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right ) \left (3 b+4 a x^3\right )+\frac {\left (2 a^2-3 b\right ) \left (5 \left (3 b^3+13 a b^2 x^3-144 a^2 b x^6+128 a^3 x^9\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {a x^3}{b-a x^3}\right )+8 a x^3 \left (b^2+10 a b x^3-24 a^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{4},2,\frac {9}{4},\frac {a x^3}{b-a x^3}\right )-16 a x^3 \left (b-2 a x^3\right )^2 \, _3F_2\left (\frac {5}{4},2,2;1,\frac {9}{4};\frac {a x^3}{b-a x^3}\right )\right )}{a^2 \left (-b+a x^3\right )}}{315 b^2 x^5 \sqrt [4]{-b x+a x^4}} \]

[In]

Integrate[(b - 3*a*x^3 + 3*x^6)/(x^6*(-b + 2*a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]

[Out]

((210*b*x^3*(-b + a*x^3))/a + 15*(2 - b/a^2)*(b - a*x^3)*(3*b + 4*a*x^3) + ((2*a^2 - 3*b)*(5*(3*b^3 + 13*a*b^2
*x^3 - 144*a^2*b*x^6 + 128*a^3*x^9)*Hypergeometric2F1[1/4, 1, 5/4, (a*x^3)/(b - a*x^3)] + 8*a*x^3*(b^2 + 10*a*
b*x^3 - 24*a^2*x^6)*Hypergeometric2F1[5/4, 2, 9/4, (a*x^3)/(b - a*x^3)] - 16*a*x^3*(b - 2*a*x^3)^2*Hypergeomet
ricPFQ[{5/4, 2, 2}, {1, 9/4}, (a*x^3)/(b - a*x^3)]))/(a^2*(-b + a*x^3)))/(315*b^2*x^5*(-(b*x) + a*x^4)^(1/4))

Maple [A] (verified)

Time = 1.68 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(-\frac {2 \left (\frac {\sqrt {2}\, x^{7} \left (a^{2}-\frac {3 b}{2}\right ) \ln \left (\frac {-a^{\frac {1}{4}} {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}\, x^{2}+\sqrt {x \left (a \,x^{3}-b \right )}}{a^{\frac {1}{4}} {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}\, x^{2}+\sqrt {x \left (a \,x^{3}-b \right )}}\right )}{2}+\sqrt {2}\, x^{7} \left (a^{2}-\frac {3 b}{2}\right ) \arctan \left (\frac {\sqrt {2}\, {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}-a^{\frac {1}{4}} x}{a^{\frac {1}{4}} x}\right )+\sqrt {2}\, x^{7} \left (a^{2}-\frac {3 b}{2}\right ) \arctan \left (\frac {\sqrt {2}\, {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}+a^{\frac {1}{4}} x}{a^{\frac {1}{4}} x}\right )-\frac {2 {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {7}{4}} a^{\frac {1}{4}}}{7}\right )}{3 a^{\frac {1}{4}} x^{7} b^{2}}\)	\(226\)

[In]

int((3*x^6-3*a*x^3+b)/x^6/(2*a*x^3-b)/(a*x^4-b*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-2/3/a^(1/4)*(1/2*2^(1/2)*x^7*(a^2-3/2*b)*ln((-a^(1/4)*(x*(a*x^3-b))^(1/4)*2^(1/2)*x+a^(1/2)*x^2+(x*(a*x^3-b))
^(1/2))/(a^(1/4)*(x*(a*x^3-b))^(1/4)*2^(1/2)*x+a^(1/2)*x^2+(x*(a*x^3-b))^(1/2)))+2^(1/2)*x^7*(a^2-3/2*b)*arcta
n((2^(1/2)*(x*(a*x^3-b))^(1/4)-a^(1/4)*x)/a^(1/4)/x)+2^(1/2)*x^7*(a^2-3/2*b)*arctan((2^(1/2)*(x*(a*x^3-b))^(1/
4)+a^(1/4)*x)/a^(1/4)/x)-2/7*(x*(a*x^3-b))^(7/4)*a^(1/4))/x^7/b^2

Fricas [F(-1)]

Timed out. \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\text {Timed out} \]

[In]

integrate((3*x^6-3*a*x^3+b)/x^6/(2*a*x^3-b)/(a*x^4-b*x)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {- 3 a x^{3} + b + 3 x^{6}}{x^{6} \sqrt [4]{x \left (a x^{3} - b\right )} \left (2 a x^{3} - b\right )}\, dx \]

[In]

integrate((3*x**6-3*a*x**3+b)/x**6/(2*a*x**3-b)/(a*x**4-b*x)**(1/4),x)

[Out]

Integral((-3*a*x**3 + b + 3*x**6)/(x**6*(x*(a*x**3 - b))**(1/4)*(2*a*x**3 - b)), x)

Maxima [F]

\[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int { \frac {3 \, x^{6} - 3 \, a x^{3} + b}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} {\left (2 \, a x^{3} - b\right )} x^{6}} \,d x } \]

[In]

integrate((3*x^6-3*a*x^3+b)/x^6/(2*a*x^3-b)/(a*x^4-b*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((3*x^6 - 3*a*x^3 + b)/((a*x^4 - b*x)^(1/4)*(2*a*x^3 - b)*x^6), x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.23 \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {7}{4}}}{21 \, b^{2}} + \frac {\sqrt {2} {\left (2 \, a^{\frac {11}{4}} - 3 \, a^{\frac {3}{4}} b\right )} \log \left (\sqrt {2} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {a - \frac {b}{x^{3}}} + \sqrt {a}\right )}{6 \, a b^{2}} - \frac {\sqrt {2} {\left (2 \, a^{\frac {11}{4}} - 3 \, a^{\frac {3}{4}} b\right )} \log \left (-\sqrt {2} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {a - \frac {b}{x^{3}}} + \sqrt {a}\right )}{6 \, a b^{2}} - \frac {{\left (2 \, \sqrt {2} a^{\frac {11}{4}} - 3 \, \sqrt {2} a^{\frac {3}{4}} b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{3 \, a b^{2}} - \frac {{\left (2 \, \sqrt {2} a^{\frac {11}{4}} - 3 \, \sqrt {2} a^{\frac {3}{4}} b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{3 \, a b^{2}} \]

[In]

integrate((3*x^6-3*a*x^3+b)/x^6/(2*a*x^3-b)/(a*x^4-b*x)^(1/4),x, algorithm="giac")

[Out]

4/21*(a - b/x^3)^(7/4)/b^2 + 1/6*sqrt(2)*(2*a^(11/4) - 3*a^(3/4)*b)*log(sqrt(2)*(a - b/x^3)^(1/4)*a^(1/4) + sq
rt(a - b/x^3) + sqrt(a))/(a*b^2) - 1/6*sqrt(2)*(2*a^(11/4) - 3*a^(3/4)*b)*log(-sqrt(2)*(a - b/x^3)^(1/4)*a^(1/
4) + sqrt(a - b/x^3) + sqrt(a))/(a*b^2) - 1/3*(2*sqrt(2)*a^(11/4) - 3*sqrt(2)*a^(3/4)*b)*arctan(1/2*sqrt(2)*(s
qrt(2)*a^(1/4) + 2*(a - b/x^3)^(1/4))/a^(1/4))/(a*b^2) - 1/3*(2*sqrt(2)*a^(11/4) - 3*sqrt(2)*a^(3/4)*b)*arctan
(-1/2*sqrt(2)*(sqrt(2)*a^(1/4) - 2*(a - b/x^3)^(1/4))/a^(1/4))/(a*b^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {b-3 a x^3+3 x^6}{x^6 \left (-b+2 a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=-\int \frac {3\,x^6-3\,a\,x^3+b}{x^6\,{\left (a\,x^4-b\,x\right )}^{1/4}\,\left (b-2\,a\,x^3\right )} \,d x \]

[In]

int(-(b - 3*a*x^3 + 3*x^6)/(x^6*(a*x^4 - b*x)^(1/4)*(b - 2*a*x^3)),x)

[Out]

-int((b - 3*a*x^3 + 3*x^6)/(x^6*(a*x^4 - b*x)^(1/4)*(b - 2*a*x^3)), x)