\(\int \frac {(-b+a x^6)^{3/4}}{x} \, dx\) [2037]

Optimal result

Integrand size = 17, antiderivative size = 145 \[ \int \frac {\left (-b+a x^6\right )^{3/4}}{x} \, dx=\frac {2}{9} \left (-b+a x^6\right )^{3/4}+\frac {b^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}}{-\sqrt {b}+\sqrt {-b+a x^6}}\right )}{3 \sqrt {2}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^6}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^6}}\right )}{3 \sqrt {2}} \]

[Out]

2/9*(a*x^6-b)^(3/4)+1/6*b^(3/4)*arctan(2^(1/2)*b^(1/4)*(a*x^6-b)^(1/4)/(-b^(1/2)+(a*x^6-b)^(1/2)))*2^(1/2)+1/6
*b^(3/4)*arctanh((1/2*b^(1/4)*2^(1/2)+1/2*(a*x^6-b)^(1/2)*2^(1/2)/b^(1/4))/(a*x^6-b)^(1/4))*2^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {272, 52, 65, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (-b+a x^6\right )^{3/4}}{x} \, dx=\frac {b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^6-b}}{\sqrt [4]{b}}\right )}{3 \sqrt {2}}-\frac {b^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^6-b}}{\sqrt [4]{b}}+1\right )}{3 \sqrt {2}}-\frac {b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^6-b}+\sqrt {a x^6-b}+\sqrt {b}\right )}{6 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^6-b}+\sqrt {a x^6-b}+\sqrt {b}\right )}{6 \sqrt {2}}+\frac {2}{9} \left (a x^6-b\right )^{3/4} \]

[In]

Int[(-b + a*x^6)^(3/4)/x,x]

[Out]

(2*(-b + a*x^6)^(3/4))/9 + (b^(3/4)*ArcTan[1 - (Sqrt[2]*(-b + a*x^6)^(1/4))/b^(1/4)])/(3*Sqrt[2]) - (b^(3/4)*A
rcTan[1 + (Sqrt[2]*(-b + a*x^6)^(1/4))/b^(1/4)])/(3*Sqrt[2]) - (b^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*(-b + a*
x^6)^(1/4) + Sqrt[-b + a*x^6]])/(6*Sqrt[2]) + (b^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*(-b + a*x^6)^(1/4) + Sqrt
[-b + a*x^6]])/(6*Sqrt[2])

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 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 272

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

Rule 303

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {(-b+a x)^{3/4}}{x} \, dx,x,x^6\right ) \\ & = \frac {2}{9} \left (-b+a x^6\right )^{3/4}-\frac {1}{6} b \text {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^6\right ) \\ & = \frac {2}{9} \left (-b+a x^6\right )^{3/4}-\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 a} \\ & = \frac {2}{9} \left (-b+a x^6\right )^{3/4}+\frac {b \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 a}-\frac {b \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 a} \\ & = \frac {2}{9} \left (-b+a x^6\right )^{3/4}-\frac {b^{3/4} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{6 \sqrt {2}}-\frac {b^{3/4} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{6 \sqrt {2}}-\frac {1}{6} b \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^6}\right )-\frac {1}{6} b \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^6}\right ) \\ & = \frac {2}{9} \left (-b+a x^6\right )^{3/4}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}+\sqrt {-b+a x^6}\right )}{6 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}+\sqrt {-b+a x^6}\right )}{6 \sqrt {2}}-\frac {b^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )}{3 \sqrt {2}}+\frac {b^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )}{3 \sqrt {2}} \\ & = \frac {2}{9} \left (-b+a x^6\right )^{3/4}+\frac {b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )}{3 \sqrt {2}}-\frac {b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )}{3 \sqrt {2}}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}+\sqrt {-b+a x^6}\right )}{6 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}+\sqrt {-b+a x^6}\right )}{6 \sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-b+a x^6\right )^{3/4}}{x} \, dx=\frac {1}{18} \left (4 \left (-b+a x^6\right )^{3/4}-3 \sqrt {2} b^{3/4} \arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^6}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}}\right )+3 \sqrt {2} b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}}{\sqrt {b}+\sqrt {-b+a x^6}}\right )\right ) \]

[In]

Integrate[(-b + a*x^6)^(3/4)/x,x]

[Out]

(4*(-b + a*x^6)^(3/4) - 3*Sqrt[2]*b^(3/4)*ArcTan[(-Sqrt[b] + Sqrt[-b + a*x^6])/(Sqrt[2]*b^(1/4)*(-b + a*x^6)^(
1/4))] + 3*Sqrt[2]*b^(3/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^6)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^6])])/18

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {2 \left (a \,x^{6}-b \right )^{\frac {3}{4}}}{9}-\frac {\ln \left (\frac {\sqrt {a \,x^{6}-b}-b^{\frac {1}{4}} \left (a \,x^{6}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}{\sqrt {a \,x^{6}-b}+b^{\frac {1}{4}} \left (a \,x^{6}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}\right ) b^{\frac {3}{4}} \sqrt {2}}{12}-\frac {\arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) b^{\frac {3}{4}} \sqrt {2}}{6}-\frac {\arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}-b \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) b^{\frac {3}{4}} \sqrt {2}}{6}\)	\(160\)

[In]

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

[Out]

2/9*(a*x^6-b)^(3/4)-1/12*ln(((a*x^6-b)^(1/2)-b^(1/4)*(a*x^6-b)^(1/4)*2^(1/2)+b^(1/2))/((a*x^6-b)^(1/2)+b^(1/4)
*(a*x^6-b)^(1/4)*2^(1/2)+b^(1/2)))*b^(3/4)*2^(1/2)-1/6*arctan((2^(1/2)*(a*x^6-b)^(1/4)+b^(1/4))/b^(1/4))*b^(3/
4)*2^(1/2)-1/6*arctan((2^(1/2)*(a*x^6-b)^(1/4)-b^(1/4))/b^(1/4))*b^(3/4)*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.05 \[ \int \frac {\left (-b+a x^6\right )^{3/4}}{x} \, dx=-\frac {1}{6} \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{2} + \left (-b^{3}\right )^{\frac {3}{4}}\right ) + \frac {1}{6} i \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{2} + i \, \left (-b^{3}\right )^{\frac {3}{4}}\right ) - \frac {1}{6} i \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{2} - i \, \left (-b^{3}\right )^{\frac {3}{4}}\right ) + \frac {1}{6} \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{2} - \left (-b^{3}\right )^{\frac {3}{4}}\right ) + \frac {2}{9} \, {\left (a x^{6} - b\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

-1/6*(-b^3)^(1/4)*log((a*x^6 - b)^(1/4)*b^2 + (-b^3)^(3/4)) + 1/6*I*(-b^3)^(1/4)*log((a*x^6 - b)^(1/4)*b^2 + I
*(-b^3)^(3/4)) - 1/6*I*(-b^3)^(1/4)*log((a*x^6 - b)^(1/4)*b^2 - I*(-b^3)^(3/4)) + 1/6*(-b^3)^(1/4)*log((a*x^6
- b)^(1/4)*b^2 - (-b^3)^(3/4)) + 2/9*(a*x^6 - b)^(3/4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.98 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.33 \[ \int \frac {\left (-b+a x^6\right )^{3/4}}{x} \, dx=- \frac {a^{\frac {3}{4}} x^{\frac {9}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{6}}} \right )}}{6 \Gamma \left (\frac {1}{4}\right )} \]

[In]

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

[Out]

-a**(3/4)*x**(9/2)*gamma(-3/4)*hyper((-3/4, -3/4), (1/4,), b*exp_polar(2*I*pi)/(a*x**6))/(6*gamma(1/4))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-b+a x^6\right )^{3/4}}{x} \, dx=-\frac {1}{12} \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{6} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{6} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{6} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{6} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} b + \frac {2}{9} \, {\left (a x^{6} - b\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

-1/12*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^6 - b)^(1/4))/b^(1/4))/b^(1/4) + 2*sqrt(2)*arcta
n(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^6 - b)^(1/4))/b^(1/4))/b^(1/4) - sqrt(2)*log(sqrt(2)*(a*x^6 - b)^(1/4
)*b^(1/4) + sqrt(a*x^6 - b) + sqrt(b))/b^(1/4) + sqrt(2)*log(-sqrt(2)*(a*x^6 - b)^(1/4)*b^(1/4) + sqrt(a*x^6 -
b) + sqrt(b))/b^(1/4))*b + 2/9*(a*x^6 - b)^(3/4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-b+a x^6\right )^{3/4}}{x} \, dx=-\frac {1}{6} \, \sqrt {2} b^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{6} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {1}{6} \, \sqrt {2} b^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{6} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) + \frac {1}{12} \, \sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {2} {\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{6} - b} + \sqrt {b}\right ) - \frac {1}{12} \, \sqrt {2} b^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{6} - b} + \sqrt {b}\right ) + \frac {2}{9} \, {\left (a x^{6} - b\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

-1/6*sqrt(2)*b^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^6 - b)^(1/4))/b^(1/4)) - 1/6*sqrt(2)*b^(3/4)
*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^6 - b)^(1/4))/b^(1/4)) + 1/12*sqrt(2)*b^(3/4)*log(sqrt(2)*(a*x^
6 - b)^(1/4)*b^(1/4) + sqrt(a*x^6 - b) + sqrt(b)) - 1/12*sqrt(2)*b^(3/4)*log(-sqrt(2)*(a*x^6 - b)^(1/4)*b^(1/4
) + sqrt(a*x^6 - b) + sqrt(b)) + 2/9*(a*x^6 - b)^(3/4)

Mupad [B] (verification not implemented)

Time = 6.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.44 \[ \int \frac {\left (-b+a x^6\right )^{3/4}}{x} \, dx=\frac {2\,{\left (a\,x^6-b\right )}^{3/4}}{9}+\frac {{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (a\,x^6-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{3}-\frac {{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (a\,x^6-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{3} \]

[In]

int((a*x^6 - b)^(3/4)/x,x)

[Out]

(2*(a*x^6 - b)^(3/4))/9 + ((-b)^(3/4)*atan((a*x^6 - b)^(1/4)/(-b)^(1/4)))/3 - ((-b)^(3/4)*atanh((a*x^6 - b)^(1
/4)/(-b)^(1/4)))/3