3.32.18 \(\int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx\) [3118]

Optimal. Leaf size=678 \[ -\frac {5}{24 x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/6}}+\frac {7 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{24 b x^2}-\frac {35 \sqrt {2+\sqrt {3}} a^2 \text {ArcTan}\left (\frac {\left (\sqrt {\frac {3}{2}} \sqrt [12]{b}-\frac {\sqrt [12]{b}}{\sqrt {2}}\right ) \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 b^{17/12}}-\frac {35 \sqrt {2-\sqrt {3}} a^2 \text {ArcTan}\left (\frac {\left (\sqrt {\frac {3}{2}} \sqrt [12]{b}+\frac {\sqrt [12]{b}}{\sqrt {2}}\right ) \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 b^{17/12}}+\frac {35 a^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{36 \sqrt {2} b^{17/12}}+\frac {35 a^2 \tanh ^{-1}\left (\frac {\frac {\sqrt [12]{b}}{\sqrt {2}}+\frac {\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2} \sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{36 \sqrt {2} b^{17/12}}-\frac {35 \sqrt {2+\sqrt {3}} a^2 \tanh ^{-1}\left (\frac {\frac {\sqrt {2} \sqrt [12]{b}}{-1+\sqrt {3}}+\frac {\sqrt {2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\left (-1+\sqrt {3}\right ) \sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 b^{17/12}}-\frac {35 \sqrt {2-\sqrt {3}} a^2 \tanh ^{-1}\left (\frac {\frac {\sqrt {2} \sqrt [12]{b}}{1+\sqrt {3}}+\frac {\sqrt {2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\left (1+\sqrt {3}\right ) \sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 b^{17/12}} \]

[Out]

-5/24/x^2/(a*x+(a^2*x^2-b)^(1/2))^(5/6)+7/24*(a*x+(a^2*x^2-b)^(1/2))^(7/6)/b/x^2-35/72*(1/2*6^(1/2)+1/2*2^(1/2
))*a^2*arctan((1/2*6^(1/2)*b^(1/12)-1/2*b^(1/12)*2^(1/2))*(a*x+(a^2*x^2-b)^(1/2))^(1/6)/(-b^(1/6)+(a*x+(a^2*x^
2-b)^(1/2))^(1/3)))/b^(17/12)-35/72*(1/2*6^(1/2)-1/2*2^(1/2))*a^2*arctan((1/2*6^(1/2)*b^(1/12)+1/2*b^(1/12)*2^
(1/2))*(a*x+(a^2*x^2-b)^(1/2))^(1/6)/(-b^(1/6)+(a*x+(a^2*x^2-b)^(1/2))^(1/3)))/b^(17/12)+35/72*a^2*arctan(2^(1
/2)*b^(1/12)*(a*x+(a^2*x^2-b)^(1/2))^(1/6)/(-b^(1/6)+(a*x+(a^2*x^2-b)^(1/2))^(1/3)))*2^(1/2)/b^(17/12)+35/72*a
^2*arctanh((1/2*b^(1/12)*2^(1/2)+1/2*(a*x+(a^2*x^2-b)^(1/2))^(1/3)*2^(1/2)/b^(1/12))/(a*x+(a^2*x^2-b)^(1/2))^(
1/6))*2^(1/2)/b^(17/12)-35/72*(1/2*6^(1/2)+1/2*2^(1/2))*a^2*arctanh((2^(1/2)*b^(1/12)/(3^(1/2)-1)+2^(1/2)*(a*x
+(a^2*x^2-b)^(1/2))^(1/3)/(3^(1/2)-1)/b^(1/12))/(a*x+(a^2*x^2-b)^(1/2))^(1/6))/b^(17/12)-35/72*(1/2*6^(1/2)-1/
2*2^(1/2))*a^2*arctanh((2^(1/2)*b^(1/12)/(1+3^(1/2))+2^(1/2)*(a*x+(a^2*x^2-b)^(1/2))^(1/3)/(1+3^(1/2))/b^(1/12
))/(a*x+(a^2*x^2-b)^(1/2))^(1/6))/b^(17/12)

________________________________________________________________________________________

Rubi [A]
time = 0.91, antiderivative size = 760, normalized size of antiderivative = 1.12, number of steps used = 25, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2145, 294, 296, 335, 307, 215, 648, 632, 210, 642, 209, 216, 212} \begin {gather*} -\frac {35 a^2 \text {ArcTan}\left (\frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}}-\frac {35 a^2 \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}}{\sqrt {3}}\right )}{24 \sqrt {3} (-b)^{17/12}}+\frac {35 a^2 \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{72 (-b)^{17/12}}+\frac {35 a^2 \text {ArcTan}\left (\frac {\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}+1}{\sqrt {3}}\right )}{24 \sqrt {3} (-b)^{17/12}}-\frac {35 a^2 \text {ArcTan}\left (\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}+\sqrt {3}\right )}{72 (-b)^{17/12}}+\frac {7 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{6 b \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}-\frac {2 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/6}}{\left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )^2}-\frac {35 a^2 \log \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}-\sqrt [12]{-b} \sqrt [6]{\sqrt {a^2 x^2-b}+a x}+\sqrt [6]{-b}\right )}{144 (-b)^{17/12}}+\frac {35 a^2 \log \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+\sqrt [12]{-b} \sqrt [6]{\sqrt {a^2 x^2-b}+a x}+\sqrt [6]{-b}\right )}{144 (-b)^{17/12}}+\frac {35 a^2 \log \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{\sqrt {a^2 x^2-b}+a x}+\sqrt [6]{-b}\right )}{48 \sqrt {3} (-b)^{17/12}}-\frac {35 a^2 \log \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{\sqrt {a^2 x^2-b}+a x}+\sqrt [6]{-b}\right )}{48 \sqrt {3} (-b)^{17/12}}+\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*(a*x + Sqrt[-b + a^2*x^2])^(7/6))/(b + (a*x + Sqrt[-b + a^2*x^2])^2)^2 + (7*a^2*(a*x + Sqrt[-b + a^2*x
^2])^(7/6))/(6*b*(b + (a*x + Sqrt[-b + a^2*x^2])^2)) - (35*a^2*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/6)/(-b)^(1
/12)])/(36*(-b)^(17/12)) - (35*a^2*ArcTan[(1 - (2*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b)^(1/12))/Sqrt[3]])/(24
*Sqrt[3]*(-b)^(17/12)) + (35*a^2*ArcTan[Sqrt[3] - (2*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b)^(1/12)])/(72*(-b)^
(17/12)) + (35*a^2*ArcTan[(1 + (2*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b)^(1/12))/Sqrt[3]])/(24*Sqrt[3]*(-b)^(1
7/12)) - (35*a^2*ArcTan[Sqrt[3] + (2*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b)^(1/12)])/(72*(-b)^(17/12)) + (35*a
^2*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/6)/(-b)^(1/12)])/(36*(-b)^(17/12)) - (35*a^2*Log[(-b)^(1/6) - (-b)^(1
/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3)])/(144*(-b)^(17/12)) + (35*a^2*Log[(-
b)^(1/6) + (-b)^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3)])/(144*(-b)^(17/12)
) + (35*a^2*Log[(-b)^(1/6) - Sqrt[3]*(-b)^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6) + (a*x + Sqrt[-b + a^2*x^2])
^(1/3)])/(48*Sqrt[3]*(-b)^(17/12)) - (35*a^2*Log[(-b)^(1/6) + Sqrt[3]*(-b)^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(
1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3)])/(48*Sqrt[3]*(-b)^(17/12))

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 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 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 215

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] +
Int[(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/
(r^2 + s^2*x^2), x] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 216

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n
]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*C
os[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/(r^2 - s^2*x^2), x] + Dis
t[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

Rule 294

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

Rule 296

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

Rule 307

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2145

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m)))*(i/c)^m, Subst[Int[x^(n - 2*m - p - 2)*((-a)*f^2 + x^2)^p*(a*f^2
+ x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0
] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rubi steps

\begin {align*} \int \frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{x^3 \sqrt {-b+a^2 x^2}} \, dx &=\left (8 a^2\right ) \text {Subst}\left (\int \frac {x^{13/6}}{\left (b+x^2\right )^3} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=-\frac {2 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{\left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {1}{3} \left (7 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [6]{x}}{\left (b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=-\frac {2 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{\left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{6 b \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [6]{x}}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{36 b}\\ &=-\frac {2 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{\left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{6 b \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {x^6}{b+x^{12}} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{6 b}\\ &=-\frac {2 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{\left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{6 b \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b}-x^6} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{12 b}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b}+x^6} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{12 b}\\ &=-\frac {2 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{\left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{6 b \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [12]{-b}-\frac {x}{2}}{\sqrt [6]{-b}-\sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{36 (-b)^{17/12}}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [12]{-b}+\frac {x}{2}}{\sqrt [6]{-b}+\sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{36 (-b)^{17/12}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [12]{-b}-\frac {\sqrt {3} x}{2}}{\sqrt [6]{-b}-\sqrt {3} \sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{36 (-b)^{17/12}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [12]{-b}+\frac {\sqrt {3} x}{2}}{\sqrt [6]{-b}+\sqrt {3} \sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{36 (-b)^{17/12}}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-b}-x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{36 (-b)^{4/3}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-b}+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{36 (-b)^{4/3}}\\ &=-\frac {2 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{\left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{6 b \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {35 a^2 \tan ^{-1}\left (\frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}}+\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {-\sqrt [12]{-b}+2 x}{\sqrt [6]{-b}-\sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{144 (-b)^{17/12}}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [12]{-b}+2 x}{\sqrt [6]{-b}+\sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{144 (-b)^{17/12}}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [12]{-b}+2 x}{\sqrt [6]{-b}-\sqrt {3} \sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{48 \sqrt {3} (-b)^{17/12}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [12]{-b}+2 x}{\sqrt [6]{-b}+\sqrt {3} \sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{48 \sqrt {3} (-b)^{17/12}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-b}-\sqrt {3} \sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{144 (-b)^{4/3}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-b}+\sqrt {3} \sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{144 (-b)^{4/3}}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-b}-\sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{48 (-b)^{4/3}}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-b}+\sqrt [12]{-b} x+x^2} \, dx,x,\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}\right )}{48 (-b)^{4/3}}\\ &=-\frac {2 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{\left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{6 b \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {35 a^2 \tan ^{-1}\left (\frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}}+\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}}-\frac {35 a^2 \log \left (\sqrt [6]{-b}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{144 (-b)^{17/12}}+\frac {35 a^2 \log \left (\sqrt [6]{-b}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{144 (-b)^{17/12}}+\frac {35 a^2 \log \left (\sqrt [6]{-b}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{48 \sqrt {3} (-b)^{17/12}}-\frac {35 a^2 \log \left (\sqrt [6]{-b}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{48 \sqrt {3} (-b)^{17/12}}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )}{24 (-b)^{17/12}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )}{24 (-b)^{17/12}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt {3} \sqrt [12]{-b}}\right )}{72 \sqrt {3} (-b)^{17/12}}+\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt {3} \sqrt [12]{-b}}\right )}{72 \sqrt {3} (-b)^{17/12}}\\ &=-\frac {2 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{\left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{6 b \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {35 a^2 \tan ^{-1}\left (\frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}}+\frac {35 a^2 \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )\right )}{72 (-b)^{17/12}}-\frac {35 a^2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}}{\sqrt {3}}\right )}{24 \sqrt {3} (-b)^{17/12}}+\frac {35 a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}}{\sqrt {3}}\right )}{24 \sqrt {3} (-b)^{17/12}}-\frac {35 a^2 \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )\right )}{72 (-b)^{17/12}}+\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}}-\frac {35 a^2 \log \left (\sqrt [6]{-b}-\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{144 (-b)^{17/12}}+\frac {35 a^2 \log \left (\sqrt [6]{-b}+\sqrt [12]{-b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{144 (-b)^{17/12}}+\frac {35 a^2 \log \left (\sqrt [6]{-b}-\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{48 \sqrt {3} (-b)^{17/12}}-\frac {35 a^2 \log \left (\sqrt [6]{-b}+\sqrt {3} \sqrt [12]{-b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{48 \sqrt {3} (-b)^{17/12}}\\ \end {align*}

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Mathematica [A]
time = 2.60, size = 606, normalized size = 0.89 \begin {gather*} \frac {1}{72} \left (-\frac {15}{x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/6}}+\frac {21 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/6}}{b x^2}+\frac {35 \sqrt {2+\sqrt {3}} a^2 \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {3}} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [6]{b}-\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}+\frac {35 \sqrt {2-\sqrt {3}} a^2 \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {3}} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [6]{b}-\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}+\frac {35 \sqrt {2} a^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}+\frac {35 \sqrt {2} a^2 \tanh ^{-1}\left (\frac {\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2} \sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}-\frac {35 \sqrt {2-\sqrt {3}} a^2 \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {3}} \left (\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}-\frac {35 \sqrt {2+\sqrt {3}} a^2 \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [6]{b}+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [12]{b} \sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{17/12}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15/(x^2*(a*x + Sqrt[-b + a^2*x^2])^(5/6)) + (21*(a*x + Sqrt[-b + a^2*x^2])^(7/6))/(b*x^2) + (35*Sqrt[2 + Sqr
t[3]]*a^2*ArcTan[(Sqrt[2 - Sqrt[3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(b^(1/6) - (a*x + Sqrt[-b + a^2
*x^2])^(1/3))])/b^(17/12) + (35*Sqrt[2 - Sqrt[3]]*a^2*ArcTan[(Sqrt[2 + Sqrt[3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*
x^2])^(1/6))/(b^(1/6) - (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/b^(17/12) + (35*Sqrt[2]*a^2*ArcTan[(Sqrt[2]*b^(1/1
2)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/b^(17/12) + (35*Sqrt[2]*a
^2*ArcTanh[(b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))/(Sqrt[2]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))])/
b^(17/12) - (35*Sqrt[2 - Sqrt[3]]*a^2*ArcTanh[(Sqrt[2 - Sqrt[3]]*(b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3)))
/(b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))])/b^(17/12) - (35*Sqrt[2 + Sqrt[3]]*a^2*ArcTanh[(Sqrt[2 + Sqrt[3]
]*(b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3)))/(b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))])/b^(17/12))/72

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{6}}}{x^{3} \sqrt {a^{2} x^{2}-b}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 1.20, size = 1065, normalized size = 1.57 \begin {gather*} -\frac {70 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{\frac {3}{2}} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} + 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{\frac {3}{2}}\right ) - 70 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{\frac {3}{2}} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} - 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{\frac {3}{2}}\right ) - 35 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} {\left (-i \, \sqrt {3} - 1\right )} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} + \frac {64339296875}{2} \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} {\left (-i \, \sqrt {3} - 1\right )}\right ) + 35 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} {\left (-i \, \sqrt {3} - 1\right )} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} - \frac {64339296875}{2} \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} {\left (-i \, \sqrt {3} - 1\right )}\right ) + 70 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} \sqrt {\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} + 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} \sqrt {\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}}\right ) - 70 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} \sqrt {\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} - 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} \sqrt {\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}}\right ) - 70 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} + 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10}\right ) + 70 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} - 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10}\right ) + 70 \, {\left (\left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{\frac {3}{2}} - \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} \sqrt {\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}}\right )} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} + 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{\frac {3}{2}} - 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} \sqrt {\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}}\right ) - 70 \, {\left (\left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{\frac {3}{2}} - \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} \sqrt {\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}}\right )} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} - 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{\frac {3}{2}} + 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} \sqrt {\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}}\right ) - 35 \, {\left (\left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} {\left (-i \, \sqrt {3} - 1\right )} + 2 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2}\right )} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} + \frac {64339296875}{2} \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} {\left (-i \, \sqrt {3} - 1\right )} + 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10}\right ) + 35 \, {\left (\left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2} {\left (-i \, \sqrt {3} - 1\right )} + 2 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {1}{12}} b x^{2}\right )} \log \left (64339296875 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}} a^{14} - \frac {64339296875}{2} \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10} {\left (-i \, \sqrt {3} - 1\right )} - 64339296875 \, \left (-\frac {a^{24}}{b^{17}}\right )^{\frac {7}{12}} b^{10}\right ) - 12 \, {\left (a x + 6 \, \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}}}{144 \, b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(70*(-a^24/b^17)^(1/12)*b*x^2*(1/2*I*sqrt(3) + 1/2)^(3/2)*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/
6)*a^14 + 64339296875*(-a^24/b^17)^(7/12)*b^10*(1/2*I*sqrt(3) + 1/2)^(3/2)) - 70*(-a^24/b^17)^(1/12)*b*x^2*(1/
2*I*sqrt(3) + 1/2)^(3/2)*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 - 64339296875*(-a^24/b^17)^(7/12
)*b^10*(1/2*I*sqrt(3) + 1/2)^(3/2)) - 35*(-a^24/b^17)^(1/12)*b*x^2*(-I*sqrt(3) - 1)*log(64339296875*(a*x + sqr
t(a^2*x^2 - b))^(1/6)*a^14 + 64339296875/2*(-a^24/b^17)^(7/12)*b^10*(-I*sqrt(3) - 1)) + 35*(-a^24/b^17)^(1/12)
*b*x^2*(-I*sqrt(3) - 1)*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 - 64339296875/2*(-a^24/b^17)^(7/1
2)*b^10*(-I*sqrt(3) - 1)) + 70*(-a^24/b^17)^(1/12)*b*x^2*sqrt(1/2*I*sqrt(3) + 1/2)*log(64339296875*(a*x + sqrt
(a^2*x^2 - b))^(1/6)*a^14 + 64339296875*(-a^24/b^17)^(7/12)*b^10*sqrt(1/2*I*sqrt(3) + 1/2)) - 70*(-a^24/b^17)^
(1/12)*b*x^2*sqrt(1/2*I*sqrt(3) + 1/2)*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 - 64339296875*(-a^
24/b^17)^(7/12)*b^10*sqrt(1/2*I*sqrt(3) + 1/2)) - 70*(-a^24/b^17)^(1/12)*b*x^2*log(64339296875*(a*x + sqrt(a^2
*x^2 - b))^(1/6)*a^14 + 64339296875*(-a^24/b^17)^(7/12)*b^10) + 70*(-a^24/b^17)^(1/12)*b*x^2*log(64339296875*(
a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 - 64339296875*(-a^24/b^17)^(7/12)*b^10) + 70*((-a^24/b^17)^(1/12)*b*x^2*(1
/2*I*sqrt(3) + 1/2)^(3/2) - (-a^24/b^17)^(1/12)*b*x^2*sqrt(1/2*I*sqrt(3) + 1/2))*log(64339296875*(a*x + sqrt(a
^2*x^2 - b))^(1/6)*a^14 + 64339296875*(-a^24/b^17)^(7/12)*b^10*(1/2*I*sqrt(3) + 1/2)^(3/2) - 64339296875*(-a^2
4/b^17)^(7/12)*b^10*sqrt(1/2*I*sqrt(3) + 1/2)) - 70*((-a^24/b^17)^(1/12)*b*x^2*(1/2*I*sqrt(3) + 1/2)^(3/2) - (
-a^24/b^17)^(1/12)*b*x^2*sqrt(1/2*I*sqrt(3) + 1/2))*log(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 - 643
39296875*(-a^24/b^17)^(7/12)*b^10*(1/2*I*sqrt(3) + 1/2)^(3/2) + 64339296875*(-a^24/b^17)^(7/12)*b^10*sqrt(1/2*
I*sqrt(3) + 1/2)) - 35*((-a^24/b^17)^(1/12)*b*x^2*(-I*sqrt(3) - 1) + 2*(-a^24/b^17)^(1/12)*b*x^2)*log(64339296
875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 + 64339296875/2*(-a^24/b^17)^(7/12)*b^10*(-I*sqrt(3) - 1) + 643392968
75*(-a^24/b^17)^(7/12)*b^10) + 35*((-a^24/b^17)^(1/12)*b*x^2*(-I*sqrt(3) - 1) + 2*(-a^24/b^17)^(1/12)*b*x^2)*l
og(64339296875*(a*x + sqrt(a^2*x^2 - b))^(1/6)*a^14 - 64339296875/2*(-a^24/b^17)^(7/12)*b^10*(-I*sqrt(3) - 1)
- 64339296875*(-a^24/b^17)^(7/12)*b^10) - 12*(a*x + 6*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(1/6))/(b*x
^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [6]{a x + \sqrt {a^{2} x^{2} - b}}}{x^{3} \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
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((a*x+(a^2*x^2-b)^(1/2))^(1/6)/x^3/(a^2*x^2-b)^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/6}}{x^3\,\sqrt {a^2\,x^2-b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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