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

Optimal. Leaf size=192 \[ \frac {\sqrt {2} \left (2 a^2-3 b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{a x^4-b x}}{\sqrt {a x^4-b x}-\sqrt {a} x^2}\right )}{3 \sqrt [4]{a} b^2}+\frac {\sqrt {2} \left (2 a^2-3 b\right ) \tanh ^{-1}\left (\frac {\sqrt {a x^4-b x}+\sqrt {a} x^2}{\sqrt {2} \sqrt [4]{a} x \sqrt [4]{a x^4-b x}}\right )}{3 \sqrt [4]{a} b^2}-\frac {4 \left (a x^4-b x\right )^{3/4} \left (b-a x^3\right )}{21 b^2 x^6} \]

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Rubi [B]  time = 1.64, antiderivative size = 617, normalized size of antiderivative = 3.21, number of steps used = 20, number of rules used = 14, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.326, Rules used = {2056, 6725, 271, 264, 466, 465, 494, 461, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\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 {\sqrt {2} \sqrt [4]{x} \left (2 a^2-3 b\right ) \sqrt [4]{a x^3-b} \tan ^{-1}\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} \tan ^{-1}\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 {\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}} \end {gather*}

Antiderivative was successfully verified.

[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 204

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

Rule 211

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 264

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 271

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 461

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 465

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 466

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 494

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 617

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 628

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 1162

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 1165

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 2056

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 6725

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*} \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 {\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tan ^{-1}\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*}

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Mathematica [C]  time = 0.92, size = 244, normalized size = 1.27 \begin {gather*} \frac {\frac {\left (2 a^2-3 b\right ) \left (-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 )+8 a x^3 \left (-24 a^2 x^6+10 a b x^3+b^2\right ) \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};\frac {a x^3}{b-a x^3}\right )+5 \left (128 a^3 x^9-144 a^2 b x^6+13 a b^2 x^3+3 b^3\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {a x^3}{b-a x^3}\right )\right )}{a^2 \left (a x^3-b\right )}+15 \left (2-\frac {b}{a^2}\right ) \left (b-a x^3\right ) \left (4 a x^3+3 b\right )+\frac {210 b x^3 \left (a x^3-b\right )}{a}}{315 b^2 x^5 \sqrt [4]{a x^4-b x}} \end {gather*}

Warning: Unable to verify antiderivative.

[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))

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IntegrateAlgebraic [A]  time = 1.06, size = 192, normalized size = 1.00 \begin {gather*} -\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 ) \tan ^{-1}\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 ) \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(b - a*x^3)*(-(b*x) + a*x^4)^(3/4))/(21*b^2*x^6) + (Sqrt[2]*(2*a^2 - 3*b)*ArcTan[(Sqrt[2]*a^(1/4)*x*(-(b*x
) + a*x^4)^(1/4))/(-(Sqrt[a]*x^2) + Sqrt[-(b*x) + a*x^4])])/(3*a^(1/4)*b^2) + (Sqrt[2]*(2*a^2 - 3*b)*ArcTanh[(
Sqrt[a]*x^2 + Sqrt[-(b*x) + a*x^4])/(Sqrt[2]*a^(1/4)*x*(-(b*x) + a*x^4)^(1/4))])/(3*a^(1/4)*b^2)

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fricas [F(-1)]  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((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

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giac [A]  time = 0.51, size = 237, normalized size = 1.23 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {3 x^{6}-3 a \,x^{3}+b}{x^{6} \left (2 a \,x^{3}-b \right ) \left (a \,x^{4}-b x \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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