3.25.65 \(\int \frac {-d+c x^2}{(d+c x^2) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2465]

Optimal. Leaf size=200 \[ -\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\& ,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}^3-2 a^2 d \text {$\#$1}^3-c \text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1359\) vs. \(2(200)=400\).
time = 3.38, antiderivative size = 1359, normalized size of antiderivative = 6.80, number of steps used = 35, number of rules used = 12, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6857, 2142, 14, 2144, 1642, 842, 840, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {-c} \left (\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}-\sqrt {2} \sqrt [4]{-c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {-b^2} (-c)^{5/4}}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {-c} \left (\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}+\sqrt {2} \sqrt [4]{-c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {-b^2} (-c)^{5/4}}+\frac {\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}-\sqrt {2} (-c)^{3/4} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {-c} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}}\right )}{\sqrt {-b^2} (-c)^{3/4}}-\frac {\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}} (-c)^{3/4}+\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{\sqrt {-c} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}}}\right )}{\sqrt {-b^2} (-c)^{3/4}}+\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \log \left (\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-c}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {d} \log \left (\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {d} a+\sqrt {-b^2} \sqrt {-c}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-c}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \log \left ((-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-c)^{3/4}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \log \left ((-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-c)^{3/4}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-d + c*x^2)/((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

-1/3*b^2/(a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/a - (Sqrt[2]*Sqrt[Sqrt[-b^2]*
(-c)^(3/2) + a*c*Sqrt[d]]*Sqrt[d]*ArcTan[(Sqrt[-c]*(Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]] - Sqrt[2]*(-c)^(1/4)
*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]))/Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]])/(Sqrt[-b^2]*(-c)^(5/4)) + (Sqrt
[2]*Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]*Sqrt[d]*ArcTan[(Sqrt[-c]*(Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]
+ Sqrt[2]*(-c)^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]))/Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]])/(Sqrt[-b^2]
*(-c)^(5/4)) + (Sqrt[2]*Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*Sqrt[d]*ArcTan[(Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c
*Sqrt[d]] - Sqrt[2]*(-c)^(3/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(Sqrt[-c]*Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]
])])/(Sqrt[-b^2]*(-c)^(3/4)) - (Sqrt[2]*Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*Sqrt[d]*ArcTan[(Sqrt[Sqrt[-b^2]*
(-c)^(3/2) + a*c*Sqrt[d]] + Sqrt[2]*(-c)^(3/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(Sqrt[-c]*Sqrt[Sqrt[-b^2]*Sqrt
[-c] + a*Sqrt[d]])])/(Sqrt[-b^2]*(-c)^(3/4)) + (Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*Sqrt[d]*Log[Sqrt[-b^2]*(
-c)^(1/4) - Sqrt[2]*Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + (-c)^(1/4)*(a*x +
Sqrt[b^2 + a^2*x^2])])/(Sqrt[2]*Sqrt[-b^2]*(-c)^(3/4)) - (Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*Sqrt[d]*Log[Sq
rt[-b^2]*(-c)^(1/4) + Sqrt[2]*Sqrt[Sqrt[-b^2]*Sqrt[-c] + a*Sqrt[d]]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + (-c)^(1/
4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(Sqrt[2]*Sqrt[-b^2]*(-c)^(3/4)) - (Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]*
Sqrt[d]*Log[Sqrt[-b^2]*(-c)^(3/4) - Sqrt[2]*Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]*Sqrt[a*x + Sqrt[b^2 + a^
2*x^2]] + (-c)^(3/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(Sqrt[2]*Sqrt[-b^2]*(-c)^(5/4)) + (Sqrt[Sqrt[-b^2]*(-c)^(3/
2) + a*c*Sqrt[d]]*Sqrt[d]*Log[Sqrt[-b^2]*(-c)^(3/4) + Sqrt[2]*Sqrt[Sqrt[-b^2]*(-c)^(3/2) + a*c*Sqrt[d]]*Sqrt[a
*x + Sqrt[b^2 + a^2*x^2]] + (-c)^(3/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(Sqrt[2]*Sqrt[-b^2]*(-c)^(5/4))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 840

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

Rule 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1183

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

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2142

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

Rule 2144

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

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-d+c x^2}{\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {2 d}{\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=-\left ((2 d) \int \frac {1}{\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\right )+\int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {b^2+x^2}{x^{5/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-(2 d) \int \left (\frac {1}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {1}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=\frac {\text {Subst}\left (\int \left (\frac {b^2}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-\sqrt {d} \int \frac {1}{\left (\sqrt {d}-\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx-\sqrt {d} \int \frac {1}{\left (\sqrt {d}+\sqrt {-c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\sqrt {d} \text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {d} \text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\sqrt {d} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {-c} x^{3/2}}+\frac {2 \left (b^2 c-a \sqrt {-c} \sqrt {d} x\right )}{c x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {d} \text {Subst}\left (\int \left (\frac {1}{\sqrt {-c} x^{3/2}}+\frac {2 \left (b^2 c+a \sqrt {-c} \sqrt {d} x\right )}{c x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 c-a \sqrt {-c} \sqrt {d} x}{x^{3/2} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c}-\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 c+a \sqrt {-c} \sqrt {d} x}{x^{3/2} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {-a b^2 c \sqrt {d}-b^2 (-c)^{3/2} x}{\sqrt {x} \left (b^2 \sqrt {-c}+2 a \sqrt {d} x-\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 (-c)^{3/2}}-\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {-a b^2 c \sqrt {d}+b^2 (-c)^{3/2} x}{\sqrt {x} \left (-b^2 \sqrt {-c}+2 a \sqrt {d} x+\sqrt {-c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 (-c)^{3/2}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\left (4 \sqrt {d}\right ) \text {Subst}\left (\int \frac {-a b^2 c \sqrt {d}-b^2 (-c)^{3/2} x^2}{b^2 \sqrt {-c}+2 a \sqrt {d} x^2-\sqrt {-c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 (-c)^{3/2}}-\frac {\left (4 \sqrt {d}\right ) \text {Subst}\left (\int \frac {-a b^2 c \sqrt {d}+b^2 (-c)^{3/2} x^2}{-b^2 \sqrt {-c}+2 a \sqrt {d} x^2+\sqrt {-c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 (-c)^{3/2}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}{\sqrt [4]{-c}}-\left (b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\left (-b^2\right )^{3/2} (-c)^{7/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}}{\sqrt [4]{-c}}+\left (b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\left (-b^2\right )^{3/2} (-c)^{7/4} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}{(-c)^{3/4}}-\left (-b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\left (-b^2\right )^{3/2} (-c)^{5/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}+\frac {\left (\sqrt {2} \sqrt {d}\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} a b^2 c \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}}{(-c)^{3/4}}+\left (-b^2 \sqrt {-b^2} (-c)^{3/2}-a b^2 c \sqrt {d}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\left (-b^2\right )^{3/2} (-c)^{5/4} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4}}-\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4}}+\frac {\left (\left (b^2 \sqrt {-c}-a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}+\frac {\left (\left (b^2 \sqrt {-c}-a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}-\frac {\left (\left (b^2 \sqrt {-c}+a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}-\frac {\left (\left (b^2 \sqrt {-c}+a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} x}{\sqrt [4]{-c}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}-\frac {\left (\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4}}+\frac {\left (\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} x}{(-c)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4}}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d} \log \left (\sqrt {-b^2} \sqrt [4]{-c}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d} \log \left (\sqrt {-b^2} \sqrt [4]{-c}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \log \left (\sqrt {-b^2} (-c)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \log \left (\sqrt {-b^2} (-c)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4}}-\frac {\left (2 \left (b^2 \sqrt {-c}-a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {d}}{\sqrt {-c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}-\frac {\left (2 \left (b^2 \sqrt {-c}-a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {d}}{\sqrt {-c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}+\frac {\left (2 \left (b^2 \sqrt {-c}+a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} c+a \sqrt {-c} \sqrt {d}\right )}{c}-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}+\frac {\left (2 \left (b^2 \sqrt {-c}+a \sqrt {-b^2} \sqrt {d}\right ) \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} c+a \sqrt {-c} \sqrt {d}\right )}{c}-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \tan ^{-1}\left (\frac {(-c)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {-b^2} (-c)^{5/4}}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{-c} \left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}\right )}{\sqrt {-b^2} (-c)^{3/4}}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \tan ^{-1}\left (\frac {(-c)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}{\sqrt [4]{-c}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}\right )}{\sqrt {-b^2} (-c)^{5/4}}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{-c} \left (\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}}}{(-c)^{3/4}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}}}\right )}{\sqrt {-b^2} (-c)^{3/4}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d} \log \left (\sqrt {-b^2} \sqrt [4]{-c}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {d} \log \left (\sqrt {-b^2} \sqrt [4]{-c}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-c}+a \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-c} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{3/4}}-\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \log \left (\sqrt {-b^2} (-c)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4}}+\frac {\sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {d} \log \left (\sqrt {-b^2} (-c)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-c)^{3/2}+a c \sqrt {d}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-c)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{\sqrt {2} \sqrt {-b^2} (-c)^{5/4}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 195, normalized size = 0.98 \begin {gather*} \frac {2 \left (b^2+3 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}-a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-b^2 c \text {$\#$1}^3+2 a^2 d \text {$\#$1}^3+c \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-d + c*x^2)/((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

(2*(b^2 + 3*a*x*(a*x + Sqrt[b^2 + a^2*x^2])))/(3*a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) - a*d*RootSum[b^4*c - 2*
b^2*c*#1^4 + 4*a^2*d*#1^4 + c*#1^8 & , (b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] + Log[Sqrt[a*x + Sqrt[b^
2 + a^2*x^2]] - #1]*#1^4)/(-(b^2*c*#1^3) + 2*a^2*d*#1^3 + c*#1^7) & ]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2-d)/(c*x^2+d)/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x)

[Out]

int((c*x^2-d)/(c*x^2+d)/(a*x+(a^2*x^2+b^2)^(1/2))^(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((c*x^2-d)/(c*x^2+d)/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 - d)/((c*x^2 + d)*sqrt(a*x + sqrt(a^2*x^2 + b^2))), x)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.51, size = 1950, normalized size = 9.75 \begin {gather*} \frac {12 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (\sqrt {a^{3} d^{6} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} d^{6} - {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} b^{2} c d^{5} - 2 \, a^{4} d^{6}\right )} \sqrt {-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}}} {\left (b^{4} c^{4} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} c d^{3}\right )} \sqrt {-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}} - {\left (a b^{4} c^{4} d^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{3} c d^{6}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}}{a^{2} d^{7}}\right ) - 12 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{3} d^{6} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} d^{6} + {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} b^{2} c d^{5} + 2 \, a^{4} d^{6}\right )} \sqrt {\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}}} {\left (b^{4} c^{4} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} c d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {3}{4}} - {\left (a b^{4} c^{4} d^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{3} c d^{6}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {3}{4}}}{a^{2} d^{7}}\right ) - 3 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} + 8 \, {\left (b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 3 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} - 8 \, {\left (b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 3 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} + 8 \, {\left (b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) - 3 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} - 8 \, {\left (b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {-\frac {a^{2} b^{2} c d^{5} - a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) - 2 \, {\left (a^{2} x^{2} - \sqrt {a^{2} x^{2} + b^{2}} a x - b^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{3 \, a b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(12*a*b^2*(-(2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4
)*arctan((sqrt(a^3*d^6*x + sqrt(a^2*x^2 + b^2)*a^2*d^6 - (2*a^2*b^4*c^3*d^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b
^8*c^6)) + a^2*b^2*c*d^5 - 2*a^4*d^6)*sqrt(-(2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - b^2*c*d^2
+ 2*a^2*d^3)/(b^4*c^3)))*(b^4*c^4*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - a^2*c*d^3)*sqrt(-(2*b^4*c^3*sqr
t(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3)) - (a*b^4*c^4*d^3*sqrt(-(a^2*b^2*c*
d^5 - a^4*d^6)/(b^8*c^6)) - a^3*c*d^6)*sqrt(a*x + sqrt(a^2*x^2 + b^2))*sqrt(-(2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 -
a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3)))*(-(2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^
6)) - b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)/(a^2*d^7)) - 12*a*b^2*((2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6
)/(b^8*c^6)) + b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)*arctan((sqrt(a^3*d^6*x + sqrt(a^2*x^2 + b^2)*a^2*d^6 +
(2*a^2*b^4*c^3*d^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - a^2*b^2*c*d^5 + 2*a^4*d^6)*sqrt((2*b^4*c^3*sqr
t(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3)))*(b^4*c^4*sqrt(-(a^2*b^2*c*d^5 - a
^4*d^6)/(b^8*c^6)) + a^2*c*d^3)*((2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 - 2*a^2*d^3
)/(b^4*c^3))^(3/4) - (a*b^4*c^4*d^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + a^3*c*d^6)*sqrt(a*x + sqrt(a^
2*x^2 + b^2))*((2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(3/4)
)/(a^2*d^7)) - 3*a*b^2*((2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^
3))^(1/4)*log(8*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*d^3 + 8*(b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6))
+ a^2*d^3)*((2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)) +
3*a*b^2*((2*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)*log(
8*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*d^3 - 8*(b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + a^2*d^3)*((2
*b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)) + 3*a*b^2*(-(2*
b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)*log(8*sqrt(a*x +
sqrt(a^2*x^2 + b^2))*a*d^3 + 8*(b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - a^2*d^3)*(-(2*b^4*c^3*sqr
t(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)) - 3*a*b^2*(-(2*b^4*c^3*sqrt
(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)*log(8*sqrt(a*x + sqrt(a^2*x^2
+ b^2))*a*d^3 - 8*(b^4*c^3*sqrt(-(a^2*b^2*c*d^5 - a^4*d^6)/(b^8*c^6)) - a^2*d^3)*(-(2*b^4*c^3*sqrt(-(a^2*b^2*
c*d^5 - a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)) - 2*(a^2*x^2 - sqrt(a^2*x^2 + b^2)*a*x
- b^2)*sqrt(a*x + sqrt(a^2*x^2 + b^2)))/(a*b^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{2} - d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} + d\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2-d)/(c*x**2+d)/(a*x+(a**2*x**2+b**2)**(1/2))**(1/2),x)

[Out]

Integral((c*x**2 - d)/(sqrt(a*x + sqrt(a**2*x**2 + b**2))*(c*x**2 + d)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 - d)/((c*x^2 + d)*sqrt(a*x + sqrt(a^2*x^2 + b^2))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(d - c*x^2)/((a*x + (b^2 + a^2*x^2)^(1/2))^(1/2)*(d + c*x^2)),x)

[Out]

int(-(d - c*x^2)/((a*x + (b^2 + a^2*x^2)^(1/2))^(1/2)*(d + c*x^2)), x)

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