\(\int \frac {(-b+a^2 x^2)^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx\) [3092]

Optimal result

Integrand size = 42, antiderivative size = 540 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\frac {63 b^3+1034 a^2 b^2 x^2-1652 a^4 b x^4+112 a^6 x^6+\sqrt {-b+a^2 x^2} \left (250 a b^2 x-1596 a^3 b x^3+112 a^5 x^5\right )}{63 x \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\frac {1}{4} \sqrt {2-\sqrt {2}} a b^{9/8} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} a b^{9/8} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {2}} a b^{9/8} \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {2}} a b^{9/8} \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \]

[Out]

1/63*(63*b^3+1034*a^2*b^2*x^2-1652*a^4*b*x^4+112*a^6*x^6+(a^2*x^2-b)^(1/2)*(112*a^5*x^5-1596*a^3*b*x^3+250*a*b
^2*x))/x/(a*x+(a^2*x^2-b)^(1/2))^(11/4)+1/4*(2-2^(1/2))^(1/2)*a*b^(9/8)*arctan((2^(1/2)/(2-2^(1/2))^(1/2)*b^(1
/8)-2*b^(1/8)/(2-2^(1/2))^(1/2))*(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(-b^(1/4)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)))-1/4*(
2+2^(1/2))^(1/2)*a*b^(9/8)*arctan((2+2^(1/2))^(1/2)*b^(1/8)*(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(-b^(1/4)+(a*x+(a^2*
x^2-b)^(1/2))^(1/2)))+1/4*(2-2^(1/2))^(1/2)*a*b^(9/8)*arctanh((b^(1/8)/(2-2^(1/2))^(1/2)+(a*x+(a^2*x^2-b)^(1/2
))^(1/2)/(2-2^(1/2))^(1/2)/b^(1/8))/(a*x+(a^2*x^2-b)^(1/2))^(1/4))+1/4*(2+2^(1/2))^(1/2)*a*b^(9/8)*arctanh((b^
(1/8)/(2+2^(1/2))^(1/2)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(2+2^(1/2))^(1/2)/b^(1/8))/(a*x+(a^2*x^2-b)^(1/2))^(1/4)
)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2145, 477, 479, 584, 220, 218, 212, 209, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=-\frac {1}{2} a (-b)^{9/8} \arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{2 \sqrt {2}}-\frac {a (-b)^{9/8} \arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )}{2 \sqrt {2}}-\frac {1}{2} a (-b)^{9/8} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )-\frac {11 a b^2}{28 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/4}}+\frac {a \left (b-\left (\sqrt {a^2 x^2-b}+a x\right )^2\right )^3}{4 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/4} \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}+\frac {13}{36} a \left (\sqrt {a^2 x^2-b}+a x\right )^{9/4}-7 a b \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\frac {a (-b)^{9/8} \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{4 \sqrt {2}}-\frac {a (-b)^{9/8} \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{4 \sqrt {2}} \]

[In]

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

[Out]

(-11*a*b^2)/(28*(a*x + Sqrt[-b + a^2*x^2])^(7/4)) - 7*a*b*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + (13*a*(a*x + Sqrt
[-b + a^2*x^2])^(9/4))/36 + (a*(b - (a*x + Sqrt[-b + a^2*x^2])^2)^3)/(4*(a*x + Sqrt[-b + a^2*x^2])^(7/4)*(b +
(a*x + Sqrt[-b + a^2*x^2])^2)) - (a*(-b)^(9/8)*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)])/2 + (a*(-b
)^(9/8)*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^(1/8)])/(2*Sqrt[2]) - (a*(-b)^(9/8)*ArcTan[
1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^(1/8)])/(2*Sqrt[2]) - (a*(-b)^(9/8)*ArcTanh[(a*x + Sqrt[-b
+ a^2*x^2])^(1/4)/(-b)^(1/8)])/2 + (a*(-b)^(9/8)*Log[(-b)^(1/4) - Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2
])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(4*Sqrt[2]) - (a*(-b)^(9/8)*Log[(-b)^(1/4) + Sqrt[2]*(-b)^(1/8)*(a
*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(4*Sqrt[2])

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 217

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

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 220

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]
}, Dist[r/(2*a), Int[1/(r - s*x^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b},
x] && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]

Rule 477

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

Rule 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 584

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 631

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

Rule 642

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

Rule 1176

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

Rule 1179

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

Rule 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*} \text {integral}& = \frac {1}{4} a \text {Subst}\left (\int \frac {\left (-b+x^2\right )^4}{x^{11/4} \left (b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right ) \\ & = a \text {Subst}\left (\int \frac {\left (-b+x^8\right )^4}{x^8 \left (b+x^8\right )^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = \frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {a \text {Subst}\left (\int \frac {\left (-b+x^8\right )^2 \left (-22 b^2-26 b x^8\right )}{x^8 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{8 b} \\ & = \frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {a \text {Subst}\left (\int \left (56 b^2-\frac {22 b^3}{x^8}-26 b x^8-\frac {16 b^3}{b+x^8}\right ) \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{8 b} \\ & = -\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\left (2 a b^2\right ) \text {Subst}\left (\int \frac {1}{b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\left (a (-b)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\left (a (-b)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {1}{2} \left (a (-b)^{5/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\frac {1}{2} \left (a (-b)^{5/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\frac {1}{2} \left (a (-b)^{5/4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-b}-x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\frac {1}{2} \left (a (-b)^{5/4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-b}+x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {1}{2} a (-b)^{9/8} \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {1}{2} a (-b)^{9/8} \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {\left (a (-b)^{9/8}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}+2 x}{-\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2}}+\frac {\left (a (-b)^{9/8}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}-2 x}{-\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2}}-\frac {1}{4} \left (a (-b)^{5/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\frac {1}{4} \left (a (-b)^{5/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {1}{2} a (-b)^{9/8} \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {1}{2} a (-b)^{9/8} \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2}}-\frac {a (-b)^{9/8} \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2}}-\frac {\left (a (-b)^{9/8}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{2 \sqrt {2}}+\frac {\left (a (-b)^{9/8}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{2 \sqrt {2}} \\ & = -\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {1}{2} a (-b)^{9/8} \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{2 \sqrt {2}}-\frac {a (-b)^{9/8} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{2 \sqrt {2}}-\frac {1}{2} a (-b)^{9/8} \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2}}-\frac {a (-b)^{9/8} \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{4 \sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 481, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\frac {1}{252} \left (\frac {4 \left (63 b^3+1034 a^2 b^2 x^2-1652 a^4 b x^4+112 a^6 x^6+2 \sqrt {-b+a^2 x^2} \left (125 a b^2 x-798 a^3 b x^3+56 a^5 x^5\right )\right )}{x \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+63 \sqrt {2-\sqrt {2}} a b^{9/8} \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}-\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+63 \sqrt {2+\sqrt {2}} a b^{9/8} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}-\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+63 \sqrt {2+\sqrt {2}} a b^{9/8} \text {arctanh}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+63 \sqrt {2-\sqrt {2}} a b^{9/8} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )\right ) \]

[In]

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

[Out]

((4*(63*b^3 + 1034*a^2*b^2*x^2 - 1652*a^4*b*x^4 + 112*a^6*x^6 + 2*Sqrt[-b + a^2*x^2]*(125*a*b^2*x - 798*a^3*b*
x^3 + 56*a^5*x^5)))/(x*(a*x + Sqrt[-b + a^2*x^2])^(11/4)) + 63*Sqrt[2 - Sqrt[2]]*a*b^(9/8)*ArcTan[(Sqrt[2 - Sq
rt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(b^(1/4) - Sqrt[a*x + Sqrt[-b + a^2*x^2]])] + 63*Sqrt[2 + Sqr
t[2]]*a*b^(9/8)*ArcTan[(Sqrt[2 + Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(b^(1/4) - Sqrt[a*x + Sqrt
[-b + a^2*x^2]])] + 63*Sqrt[2 + Sqrt[2]]*a*b^(9/8)*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*(b^(1/4) + Sqrt[a*x + Sqrt[-b
+ a^2*x^2]]))/(b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))] + 63*Sqrt[2 - Sqrt[2]]*a*b^(9/8)*ArcTanh[(Sqrt[1 + 1
/Sqrt[2]]*(b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]))/(b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))])/252

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 486, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\frac {\left (63 i + 63\right ) \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \left (i + 1\right ) \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) - \left (63 i - 63\right ) \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - \left (i - 1\right ) \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) + \left (63 i - 63\right ) \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \left (i - 1\right ) \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) - \left (63 i + 63\right ) \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - \left (i + 1\right ) \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) + 126 \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) + 126 i \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + i \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) - 126 i \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - i \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) - 126 \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) - 8 \, {\left (4 \, a^{3} x^{3} + 439 \, a b x - {\left (32 \, a^{2} x^{2} + 63 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{504 \, x} \]

[In]

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

[Out]

1/504*((63*I + 63)*sqrt(2)*(-a^8*b^9)^(1/8)*x*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b + (I + 1)*sqrt(2)*(-a^
8*b^9)^(1/8)) - (63*I - 63)*sqrt(2)*(-a^8*b^9)^(1/8)*x*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b - (I - 1)*sqr
t(2)*(-a^8*b^9)^(1/8)) + (63*I - 63)*sqrt(2)*(-a^8*b^9)^(1/8)*x*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b + (I
- 1)*sqrt(2)*(-a^8*b^9)^(1/8)) - (63*I + 63)*sqrt(2)*(-a^8*b^9)^(1/8)*x*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)
*a*b - (I + 1)*sqrt(2)*(-a^8*b^9)^(1/8)) + 126*(-a^8*b^9)^(1/8)*x*log((a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b + (-
a^8*b^9)^(1/8)) + 126*I*(-a^8*b^9)^(1/8)*x*log((a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b + I*(-a^8*b^9)^(1/8)) - 126
*I*(-a^8*b^9)^(1/8)*x*log((a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b - I*(-a^8*b^9)^(1/8)) - 126*(-a^8*b^9)^(1/8)*x*l
og((a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b - (-a^8*b^9)^(1/8)) - 8*(4*a^3*x^3 + 439*a*b*x - (32*a^2*x^2 + 63*b)*sq
rt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(1/4))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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