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

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

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Rubi [A]  time = 1.26, 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 = {2120, 288, 290, 329, 301, 209, 634, 618, 204, 628, 203, 210, 206} \begin {gather*} \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 \tan ^{-1}\left (\frac {\sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}\right )}{36 (-b)^{17/12}}-\frac {35 a^2 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\left (\frac {2 \sqrt [6]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [12]{-b}}+\sqrt {3}\right )}{72 (-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 203

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

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 206

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

Rule 209

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*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + 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 210

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*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 - s^2*x^2), x])/(a*n) +
Dist[(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 288

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 290

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] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 301

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 329

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 618

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 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 634

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 2120

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1*(i/c)^m)/(2^(2*m + p + 1)*e^(p + 1)*f^(2*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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [C]  time = 2.36, size = 317, normalized size = 0.47 \begin {gather*} \frac {12 \sqrt {a^2 x^2-b} \left (b-2 a x \left (\sqrt {a^2 x^2-b}+a x\right )\right )^4 \left (4 a^2 x^2 \left (2 a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right ) \, _2F_1\left (\frac {7}{12},3;\frac {19}{12};-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^2}{b}\right )-b^2\right )}{17 b^2 x^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/6} \left (128 a^9 x^9-320 a^7 b x^7+272 a^5 b^2 x^5-88 a^3 b^3 x^3+b^4 \sqrt {a^2 x^2-b}-32 a^2 b^3 x^2 \sqrt {a^2 x^2-b}+128 a^8 x^8 \sqrt {a^2 x^2-b}-256 a^6 b x^6 \sqrt {a^2 x^2-b}+160 a^4 b^2 x^4 \sqrt {a^2 x^2-b}+8 a b^4 x\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]

(12*Sqrt[-b + a^2*x^2]*(b - 2*a*x*(a*x + Sqrt[-b + a^2*x^2]))^4*(-b^2 + 4*a^2*x^2*(-b + 2*a*x*(a*x + Sqrt[-b +
a^2*x^2]))*Hypergeometric2F1[7/12, 3, 19/12, -((a*x + Sqrt[-b + a^2*x^2])^2/b)]))/(17*b^2*x^2*(a*x + Sqrt[-b
+ a^2*x^2])^(5/6)*(8*a*b^4*x - 88*a^3*b^3*x^3 + 272*a^5*b^2*x^5 - 320*a^7*b*x^7 + 128*a^9*x^9 + b^4*Sqrt[-b +
a^2*x^2] - 32*a^2*b^3*x^2*Sqrt[-b + a^2*x^2] + 160*a^4*b^2*x^4*Sqrt[-b + a^2*x^2] - 256*a^6*b*x^6*Sqrt[-b + a^
2*x^2] + 128*a^8*x^8*Sqrt[-b + a^2*x^2]))

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IntegrateAlgebraic [A]  time = 7.51, size = 1211, normalized size = 1.79 \begin {gather*} -\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 a^2 \tan ^{-1}\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 \tan ^{-1}\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 )}{144 \sqrt {2} b^{17/12}}-\frac {35 a^2 \tan ^{-1}\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 )}{48 \sqrt {6} b^{17/12}}-\frac {35 \sqrt {2+\sqrt {3}} a^2 \tan ^{-1}\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 )}{144 b^{17/12}}+\frac {35 a^2 \tan ^{-1}\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 )}{144 \sqrt {2} b^{17/12}}-\frac {35 a^2 \tan ^{-1}\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 )}{48 \sqrt {6} b^{17/12}}-\frac {35 \sqrt {2-\sqrt {3}} a^2 \tan ^{-1}\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 )}{144 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 a^2 \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {3}} \sqrt [12]{b}+\frac {\sqrt {2-\sqrt {3}} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 \sqrt {2} b^{17/12}}-\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {3}} \sqrt [12]{b}+\frac {\sqrt {2-\sqrt {3}} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{24 \sqrt {6} b^{17/12}}-\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {3}} \sqrt [12]{b}+\frac {\sqrt {2+\sqrt {3}} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{72 \sqrt {2} b^{17/12}}-\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {3}} \sqrt [12]{b}+\frac {\sqrt {2+\sqrt {3}} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [12]{b}}}{\sqrt [6]{a x+\sqrt {-b+a^2 x^2}}}\right )}{24 \sqrt {6} b^{17/12}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-5/(24*x^2*(a*x + Sqrt[-b + a^2*x^2])^(5/6)) + (7*(a*x + Sqrt[-b + a^2*x^2])^(7/6))/(24*b*x^2) + (35*a^2*ArcTa
n[(Sqrt[2]*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))])/(36*Sqrt
[2]*b^(17/12)) - (35*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))])/(144*Sqrt[2]*b^(17/12)) - (35*a^2*ArcTan[(Sqrt[2 - Sqrt[3]]*b^(1/12)*(a*x + Sq
rt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/(48*Sqrt[6]*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))])/(144*b^(17/12)) + (35*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))])/(144*Sqrt[2]*b^(17/12)) - (35*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))])/(48*Sqrt[6]*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))])/(144*b^(17/12)) + (35*a^2*ArcTanh[(b^(1/12)/Sqrt[2] + (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)])/(36*Sqrt[2]*b^(17/12)) + (35*a^2*Ar
cTanh[(Sqrt[2 - Sqrt[3]]*b^(1/12) + (Sqrt[2 - Sqrt[3]]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/b^(1/12))/(a*x + Sqrt
[-b + a^2*x^2])^(1/6)])/(72*Sqrt[2]*b^(17/12)) - (35*a^2*ArcTanh[(Sqrt[2 - Sqrt[3]]*b^(1/12) + (Sqrt[2 - Sqrt[
3]]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/b^(1/12))/(a*x + Sqrt[-b + a^2*x^2])^(1/6)])/(24*Sqrt[6]*b^(17/12)) - (3
5*a^2*ArcTanh[(Sqrt[2 + Sqrt[3]]*b^(1/12) + (Sqrt[2 + Sqrt[3]]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/b^(1/12))/(a*
x + Sqrt[-b + a^2*x^2])^(1/6)])/(72*Sqrt[2]*b^(17/12)) - (35*a^2*ArcTanh[(Sqrt[2 + Sqrt[3]]*b^(1/12) + (Sqrt[2
+ Sqrt[3]]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/b^(1/12))/(a*x + Sqrt[-b + a^2*x^2])^(1/6)])/(24*Sqrt[6]*b^(17/1
2))

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fricas [C]  time = 1.29, size = 1065, normalized size = 1.57

result too large to display

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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giac [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((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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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*} \int \frac {{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{6}}}{\sqrt {a^{2} x^{2} - b} x^{3}}\,{d x} \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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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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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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