\(\int \frac {(d+c x^2) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx\) [2939]

Optimal result

Integrand size = 40, antiderivative size = 348 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx=-\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {1}{2} a d \text {RootSum}\left [b^4 e-2 b^2 e \text {$\#$1}^4+4 a^2 f \text {$\#$1}^4+e \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 e \text {$\#$1}-2 a^2 f \text {$\#$1}-e \text {$\#$1}^5}\&\right ]-\frac {a c f \text {RootSum}\left [b^4 e-2 b^2 e \text {$\#$1}^4+4 a^2 f \text {$\#$1}^4+e \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 e \text {$\#$1}-2 a^2 f \text {$\#$1}-e \text {$\#$1}^5}\&\right ]}{2 e} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1370\) vs. \(2(348)=696\).

Time = 2.46 (sec) , antiderivative size = 1370, normalized size of antiderivative = 3.94, number of steps used = 33, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {6857, 2142, 14, 2144, 1642, 840, 1183, 648, 632, 210, 642} \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx=-\frac {c b^2}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \arctan \left (\frac {\sqrt {-e} \left (\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}}-\sqrt {2} \sqrt [4]{-e} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}\right )}{\sqrt {2} (-e)^{9/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \arctan \left (\frac {\sqrt {-e} \left (\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}}+\sqrt {2} \sqrt [4]{-e} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}\right )}{\sqrt {2} (-e)^{9/4} \sqrt {f}}-\frac {\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} (d e-c f) \arctan \left (\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}-\sqrt {2} (-e)^{3/4} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {-e} \sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}}}\right )}{\sqrt {2} (-e)^{7/4} \sqrt {f}}+\frac {\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} (d e-c f) \arctan \left (\frac {\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}} (-e)^{3/4}+\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{\sqrt {-e} \sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}}}\right )}{\sqrt {2} (-e)^{7/4} \sqrt {f}}+\frac {\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} (d e-c f) \log \left (\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-e}\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} (d e-c f) \log \left (\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {f} a+\sqrt {-b^2} \sqrt {-e}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} \sqrt [4]{-e}\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left ((-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-e)^{3/4}\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left ((-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {-b^2} (-e)^{3/4}\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}} \]

[In]

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

[Out]

-((b^2*c)/(a*e*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(3*a*e) + (Sqrt[Sqrt[
-b^2]*(-e)^(3/2) + a*e*Sqrt[f]]*(d*e - c*f)*ArcTan[(Sqrt[-e]*(Sqrt[Sqrt[-b^2]*Sqrt[-e] + a*Sqrt[f]] - Sqrt[2]*
(-e)^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]))/Sqrt[Sqrt[-b^2]*(-e)^(3/2) + a*e*Sqrt[f]]])/(Sqrt[2]*(-e)^(9/4)*S
qrt[f]) - (Sqrt[Sqrt[-b^2]*(-e)^(3/2) + a*e*Sqrt[f]]*(d*e - c*f)*ArcTan[(Sqrt[-e]*(Sqrt[Sqrt[-b^2]*Sqrt[-e] +
a*Sqrt[f]] + Sqrt[2]*(-e)^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]))/Sqrt[Sqrt[-b^2]*(-e)^(3/2) + a*e*Sqrt[f]]])/
(Sqrt[2]*(-e)^(9/4)*Sqrt[f]) - (Sqrt[Sqrt[-b^2]*Sqrt[-e] + a*Sqrt[f]]*(d*e - c*f)*ArcTan[(Sqrt[Sqrt[-b^2]*(-e)
^(3/2) + a*e*Sqrt[f]] - Sqrt[2]*(-e)^(3/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(Sqrt[-e]*Sqrt[Sqrt[-b^2]*Sqrt[-e]
+ a*Sqrt[f]])])/(Sqrt[2]*(-e)^(7/4)*Sqrt[f]) + (Sqrt[Sqrt[-b^2]*Sqrt[-e] + a*Sqrt[f]]*(d*e - c*f)*ArcTan[(Sqr
t[Sqrt[-b^2]*(-e)^(3/2) + a*e*Sqrt[f]] + Sqrt[2]*(-e)^(3/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(Sqrt[-e]*Sqrt[Sq
rt[-b^2]*Sqrt[-e] + a*Sqrt[f]])])/(Sqrt[2]*(-e)^(7/4)*Sqrt[f]) + (Sqrt[Sqrt[-b^2]*Sqrt[-e] + a*Sqrt[f]]*(d*e -
c*f)*Log[Sqrt[-b^2]*(-e)^(1/4) - Sqrt[2]*Sqrt[Sqrt[-b^2]*Sqrt[-e] + a*Sqrt[f]]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]
] + (-e)^(1/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(2*Sqrt[2]*(-e)^(7/4)*Sqrt[f]) - (Sqrt[Sqrt[-b^2]*Sqrt[-e] + a*Sq
rt[f]]*(d*e - c*f)*Log[Sqrt[-b^2]*(-e)^(1/4) + Sqrt[2]*Sqrt[Sqrt[-b^2]*Sqrt[-e] + a*Sqrt[f]]*Sqrt[a*x + Sqrt[b
^2 + a^2*x^2]] + (-e)^(1/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(2*Sqrt[2]*(-e)^(7/4)*Sqrt[f]) - (Sqrt[Sqrt[-b^2]*(-
e)^(3/2) + a*e*Sqrt[f]]*(d*e - c*f)*Log[Sqrt[-b^2]*(-e)^(3/4) - Sqrt[2]*Sqrt[Sqrt[-b^2]*(-e)^(3/2) + a*e*Sqrt[
f]]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + (-e)^(3/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(2*Sqrt[2]*(-e)^(9/4)*Sqrt[f])
+ (Sqrt[Sqrt[-b^2]*(-e)^(3/2) + a*e*Sqrt[f]]*(d*e - c*f)*Log[Sqrt[-b^2]*(-e)^(3/4) + Sqrt[2]*Sqrt[Sqrt[-b^2]*(
-e)^(3/2) + a*e*Sqrt[f]]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + (-e)^(3/4)*(a*x + Sqrt[b^2 + a^2*x^2])])/(2*Sqrt[2]
*(-e)^(9/4)*Sqrt[f])

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 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*} \text {integral}& = \int \left (\frac {c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{e}+\frac {(d e-c f) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{e \left (f+e x^2\right )}\right ) \, dx \\ & = \frac {c \int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx}{e}+\frac {(d e-c f) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx}{e} \\ & = \frac {c \text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a e}+\frac {(d e-c f) \int \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {f} \left (\sqrt {f}-\sqrt {-e} x\right )}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {f} \left (\sqrt {f}+\sqrt {-e} x\right )}\right ) \, dx}{e} \\ & = \frac {c \text {Subst}\left (\int \left (\frac {b^2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a e}+\frac {(d e-c f) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {f}-\sqrt {-e} x} \, dx}{2 e \sqrt {f}}+\frac {(d e-c f) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {f}+\sqrt {-e} x} \, dx}{2 e \sqrt {f}} \\ & = -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {(d e-c f) \text {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (b^2 \sqrt {-e}+2 a \sqrt {f} x-\sqrt {-e} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 e \sqrt {f}}+\frac {(d e-c f) \text {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (-b^2 \sqrt {-e}+2 a \sqrt {f} x+\sqrt {-e} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 e \sqrt {f}} \\ & = -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {(d e-c f) \text {Subst}\left (\int \left (-\frac {1}{\sqrt {-e} \sqrt {x}}+\frac {2 \left (b^2 e-a \sqrt {-e} \sqrt {f} x\right )}{e \sqrt {x} \left (b^2 \sqrt {-e}+2 a \sqrt {f} x-\sqrt {-e} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 e \sqrt {f}}+\frac {(d e-c f) \text {Subst}\left (\int \left (\frac {1}{\sqrt {-e} \sqrt {x}}+\frac {2 \left (b^2 e+a \sqrt {-e} \sqrt {f} x\right )}{e \sqrt {x} \left (-b^2 \sqrt {-e}+2 a \sqrt {f} x+\sqrt {-e} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 e \sqrt {f}} \\ & = -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {(d e-c f) \text {Subst}\left (\int \frac {b^2 e-a \sqrt {-e} \sqrt {f} x}{\sqrt {x} \left (b^2 \sqrt {-e}+2 a \sqrt {f} x-\sqrt {-e} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{e^2 \sqrt {f}}+\frac {(d e-c f) \text {Subst}\left (\int \frac {b^2 e+a \sqrt {-e} \sqrt {f} x}{\sqrt {x} \left (-b^2 \sqrt {-e}+2 a \sqrt {f} x+\sqrt {-e} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{e^2 \sqrt {f}} \\ & = -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {(2 (d e-c f)) \text {Subst}\left (\int \frac {b^2 e-a \sqrt {-e} \sqrt {f} x^2}{b^2 \sqrt {-e}+2 a \sqrt {f} x^2-\sqrt {-e} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}+\frac {(2 (d e-c f)) \text {Subst}\left (\int \frac {b^2 e+a \sqrt {-e} \sqrt {f} x^2}{-b^2 \sqrt {-e}+2 a \sqrt {f} x^2+\sqrt {-e} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}} \\ & = -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}-\frac {(d e-c f) \text {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 e \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}-\left (b^2 e+a \sqrt {-b^2} \sqrt {-e} \sqrt {f}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-e)^{9/4} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {f}}-\frac {(d e-c f) \text {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 e \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+\left (b^2 e+a \sqrt {-b^2} \sqrt {-e} \sqrt {f}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-e)^{9/4} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {f}}+\frac {(d e-c f) \text {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 e \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}-\left (b^2 e-a \sqrt {-b^2} \sqrt {-e} \sqrt {f}\right ) x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-e)^{7/4} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {f}}+\frac {(d e-c f) \text {Subst}\left (\int \frac {\frac {\sqrt {2} b^2 e \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+\left (b^2 e-a \sqrt {-b^2} \sqrt {-e} \sqrt {f}\right ) x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {2} \sqrt {-b^2} (-e)^{7/4} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {f}} \\ & = -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}-a \sqrt {f}\right ) (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 e^2 \sqrt {f}}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}-a \sqrt {f}\right ) (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 e^2 \sqrt {f}}+\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f)\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\left (\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} x}{\sqrt [4]{-e}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}\right ) (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 e^2 \sqrt {f}}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}\right ) (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 e^2 \sqrt {f}}-\frac {\left (\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f)\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+2 x}{\sqrt {-b^2}-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\left (\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+2 x}{\sqrt {-b^2}+\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} x}{(-e)^{3/4}}+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}} \\ & = -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} \sqrt [4]{-e}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} \sqrt [4]{-e}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} (-e)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} (-e)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}-a \sqrt {f}\right ) (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} e+a \sqrt {-e} \sqrt {f}\right )}{e}-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}+\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}-a \sqrt {f}\right ) (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \left (\sqrt {-b^2} e+a \sqrt {-e} \sqrt {f}\right )}{e}-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}\right ) (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {f}}{\sqrt {-e}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}}-\frac {\left (\left (\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}\right ) (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{-2 \left (\sqrt {-b^2}+\frac {a \sqrt {f}}{\sqrt {-e}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{e^2 \sqrt {f}} \\ & = -\frac {b^2 c}{a e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a e}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \arctan \left (\frac {(-e)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}\right )}{\sqrt {2} (-e)^{9/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \arctan \left (\frac {\sqrt [4]{-e} \left (\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}-\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}\right )}{\sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \arctan \left (\frac {(-e)^{3/4} \left (\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}{\sqrt [4]{-e}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}\right )}{\sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \arctan \left (\frac {\sqrt [4]{-e} \left (\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}}}{(-e)^{3/4}}+\sqrt {2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}}}\right )}{\sqrt {2} (-e)^{7/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} \sqrt [4]{-e}-\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} \sqrt [4]{-e}+\sqrt {2} \sqrt {\sqrt {-b^2} \sqrt {-e}+a \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt [4]{-e} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{7/4} \sqrt {f}}-\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} (-e)^{3/4}-\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}}+\frac {\sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} (d e-c f) \log \left (\sqrt {-b^2} (-e)^{3/4}+\sqrt {2} \sqrt {\sqrt {-b^2} (-e)^{3/2}+a e \sqrt {f}} \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(-e)^{3/4} \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{2 \sqrt {2} (-e)^{9/4} \sqrt {f}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.59 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx=\frac {\frac {4 c \left (-b^2+a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+3 a (d e-c f) \text {RootSum}\left [b^4 e-2 b^2 e \text {$\#$1}^4+4 a^2 f \text {$\#$1}^4+e \text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-b^2 e \text {$\#$1}+2 a^2 f \text {$\#$1}+e \text {$\#$1}^5}\&\right ]}{6 e} \]

[In]

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

[Out]

((4*c*(-b^2 + a*x*(a*x + Sqrt[b^2 + a^2*x^2])))/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) + 3*a*(d*e - c*f)*RootSum[
b^4*e - 2*b^2*e*#1^4 + 4*a^2*f*#1^4 + e*#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*e*#1) + 2*a^2*f*#1 + e*#1^5) & ])/(6*e)

Maple [N/A] (verified)

Not integrable

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.10

\[\int \frac {\left (c \,x^{2}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}{e \,x^{2}+f}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 2.04 (sec) , antiderivative size = 12728, normalized size of antiderivative = 36.57 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 1.93 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.10 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx=\int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} + d\right )}{e x^{2} + f}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx=\int { \frac {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{e x^{2} + f} \,d x } \]

[In]

integrate((c*x^2+d)*(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)/(e*x^2+f),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx=\int { \frac {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{e x^{2} + f} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 7.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{f+e x^2} \, dx=\int \frac {\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (c\,x^2+d\right )}{e\,x^2+f} \,d x \]

[In]

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

[Out]

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