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

Optimal. Leaf size=540 \[ \frac {1}{4} \sqrt {2-\sqrt {2}} a b^{9/8} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} a b^{9/8} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )+\frac {1}{4} \sqrt {2-\sqrt {2}} a b^{9/8} \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )+\frac {1}{4} \sqrt {2+\sqrt {2}} a b^{9/8} \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )+\frac {112 a^6 x^6-1652 a^4 b x^4+1034 a^2 b^2 x^2+\sqrt {a^2 x^2-b} \left (112 a^5 x^5-1596 a^3 b x^3+250 a b^2 x\right )+63 b^3}{63 x \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}} \]

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Rubi [A]  time = 0.70, antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2120, 466, 468, 570, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\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}}-\frac {1}{2} a (-b)^{9/8} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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 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 211

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

Rule 212

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] &&
!GtQ[a/b, 0]

Rule 214

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 466

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

Rule 468

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)), I
nt[(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 570

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 617

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

Rule 628

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

Rule 1162

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

Rule 1165

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

Rule 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 {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx &=\frac {1}{4} a \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {1}{2} a (-b)^{9/8} \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {\left (a (-b)^{9/8}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {1}{2} a (-b)^{9/8} \tanh ^{-1}\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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [C]  time = 3.90, size = 226, normalized size = 0.42 \begin {gather*} \frac {\left (a^2 x^2-b\right ) \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (56 a^5 x^5-812 a^3 b x^3-63 b^2 \sqrt {a^2 x^2-b}+126 a b x \left (2 a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right ) \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^2}{b}\right )-784 a^2 b x^2 \sqrt {a^2 x^2-b}+56 a^4 x^4 \sqrt {a^2 x^2-b}+313 a b^2 x\right )}{63 x \left (b-a x \left (\sqrt {a^2 x^2-b}+a x\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-b + a^2*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(313*a*b^2*x - 812*a^3*b*x^3 + 56*a^5*x^5 - 63*b^2*Sqrt[-b +
a^2*x^2] - 784*a^2*b*x^2*Sqrt[-b + a^2*x^2] + 56*a^4*x^4*Sqrt[-b + a^2*x^2] + 126*a*b*x*(-b + 2*a*x*(a*x + Sqr
t[-b + a^2*x^2]))*Hypergeometric2F1[1/8, 1, 9/8, -((a*x + Sqrt[-b + a^2*x^2])^2/b)]))/(63*x*(b - a*x*(a*x + Sq
rt[-b + a^2*x^2]))^2)

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IntegrateAlgebraic [A]  time = 2.03, size = 514, normalized size = 0.95 \begin {gather*} \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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{b}+\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt {a x+\sqrt {-b+a^2 x^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} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{b}+\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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]*(250*a*b^2*x - 1596*a^3*b*x^3 +
112*a^5*x^5))/(63*x*(a*x + Sqrt[-b + a^2*x^2])^(11/4)) - (Sqrt[2 - Sqrt[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]])])/4 - (Sqrt[2 + Sqrt[
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]])])/4 + (Sqrt[2 + Sqrt[2]]*a*b^(9/8)*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]
*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/4 + (Sqrt[2 - Sqrt[2]]*a*b^(9/8)*
ArcTanh[(Sqrt[1 + 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sq
rt[-b + a^2*x^2])^(1/4)])/4

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fricas [A]  time = 0.53, size = 770, normalized size = 1.43 \begin {gather*} \frac {252 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \arctan \left (-\frac {a^{8} b^{9} + \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {7}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {7}{8}} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} - \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \left (-a^{8} b^{9}\right )^{\frac {1}{4}}}}{a^{8} b^{9}}\right ) + 252 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \arctan \left (\frac {a^{8} b^{9} - \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {7}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {7}{8}} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} + \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \left (-a^{8} b^{9}\right )^{\frac {1}{4}}}}{a^{8} b^{9}}\right ) + 63 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} + 4 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + 4 \, \left (-a^{8} b^{9}\right )^{\frac {1}{4}}\right ) - 63 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} - 4 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + 4 \, \left (-a^{8} b^{9}\right )^{\frac {1}{4}}\right ) + 504 \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \arctan \left (-\frac {\left (-a^{8} b^{9}\right )^{\frac {7}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - \left (-a^{8} b^{9}\right )^{\frac {7}{8}} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} + \left (-a^{8} b^{9}\right )^{\frac {1}{4}}}}{a^{8} b^{9}}\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 \, \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(252*sqrt(2)*(-a^8*b^9)^(1/8)*x*arctan(-(a^8*b^9 + sqrt(2)*(-a^8*b^9)^(7/8)*(a*x + sqrt(a^2*x^2 - b))^(1
/4)*a*b - sqrt(2)*(-a^8*b^9)^(7/8)*sqrt(sqrt(a*x + sqrt(a^2*x^2 - b))*a^2*b^2 - sqrt(2)*(-a^8*b^9)^(1/8)*(a*x
+ sqrt(a^2*x^2 - b))^(1/4)*a*b + (-a^8*b^9)^(1/4)))/(a^8*b^9)) + 252*sqrt(2)*(-a^8*b^9)^(1/8)*x*arctan((a^8*b^
9 - sqrt(2)*(-a^8*b^9)^(7/8)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b + sqrt(2)*(-a^8*b^9)^(7/8)*sqrt(sqrt(a*x + sq
rt(a^2*x^2 - b))*a^2*b^2 + sqrt(2)*(-a^8*b^9)^(1/8)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b + (-a^8*b^9)^(1/4)))/(
a^8*b^9)) + 63*sqrt(2)*(-a^8*b^9)^(1/8)*x*log(4*sqrt(a*x + sqrt(a^2*x^2 - b))*a^2*b^2 + 4*sqrt(2)*(-a^8*b^9)^(
1/8)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b + 4*(-a^8*b^9)^(1/4)) - 63*sqrt(2)*(-a^8*b^9)^(1/8)*x*log(4*sqrt(a*x
+ sqrt(a^2*x^2 - b))*a^2*b^2 - 4*sqrt(2)*(-a^8*b^9)^(1/8)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b + 4*(-a^8*b^9)^(
1/4)) + 504*(-a^8*b^9)^(1/8)*x*arctan(-((-a^8*b^9)^(7/8)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*a*b - (-a^8*b^9)^(7/8
)*sqrt(sqrt(a*x + sqrt(a^2*x^2 - b))*a^2*b^2 + (-a^8*b^9)^(1/4)))/(a^8*b^9)) + 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*(-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)) - 8*(4*a^3*x^3 + 439*a*b*x - (32*a^2*x^2 + 63*b)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2
*x^2 - b))^(1/4))/x

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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^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

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} x^{2}-b \right )^{\frac {3}{2}} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 )}^{1/4}\,{\left (a^2\,x^2-b\right )}^{3/2}}{x^2} \,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))^(1/4)*(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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