\(\int \frac {\sqrt {a+c x^2}}{x^2 (d+e x+f x^2)} \, dx\) [56]

Optimal result

Integrand size = 27, antiderivative size = 382 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {\sqrt {a+c x^2}}{d x}-\frac {f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {\sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \]

[Out]

e*arctanh((c*x^2+a)^(1/2)/a^(1/2))*a^(1/2)/d^2-(c*x^2+a)^(1/2)/d/x-1/2*f*arctanh(1/2*(2*a*f-c*x*(e-(-4*d*f+e^2
)^(1/2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2))*(2*c*d^2+a*(e^2-2*d*f+e*
(-4*d*f+e^2)^(1/2)))/d^2*2^(1/2)/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)+1/2*f*a
rctanh(1/2*(2*a*f-c*x*(e+(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/
2)))^(1/2))*(2*c*d^2+a*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))/d^2*2^(1/2)/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+
e*(-4*d*f+e^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6860, 283, 223, 212, 272, 52, 65, 214, 1034, 1094, 1048, 739} \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {f \left (a \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )+2 c d^2\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {f \left (a \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )+2 c d^2\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}-\frac {\sqrt {a+c x^2}}{d x} \]

[In]

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

[Out]

-(Sqrt[a + c*x^2]/(d*x)) - (f*(2*c*d^2 + a*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e
^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2
*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + (f*(2*c*d^2 + a*(e^2 - 2*d*f - e*S
qrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sq
rt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2
- 4*d*f])]) + (Sqrt[a]*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^2

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 214

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 283

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1034

Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[h*(a + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] + Dist[1/(2*f*(p + q + 1)), Int[(a + c*x^2)
^(p - 1)*(d + e*x + f*x^2)^q*Simp[a*h*e*p - a*(h*e - 2*g*f)*(p + q + 1) - 2*h*p*(c*d - a*f)*x - (h*c*e*p + c*(
h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[
p, 0] && NeQ[p + q + 1, 0]

Rule 1048

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(2*c
*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1094

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)
*Sqrt[d + f*x^2]), x], x] /; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {a+c x^2}}{d x^2}-\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {\left (e^2-d f+e f x\right ) \sqrt {a+c x^2}}{d^2 \left (d+e x+f x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\left (e^2-d f+e f x\right ) \sqrt {a+c x^2}}{d+e x+f x^2} \, dx}{d^2}+\frac {\int \frac {\sqrt {a+c x^2}}{x^2} \, dx}{d}-\frac {e \int \frac {\sqrt {a+c x^2}}{x} \, dx}{d^2} \\ & = \frac {e \sqrt {a+c x^2}}{d^2}-\frac {\sqrt {a+c x^2}}{d x}+\frac {c \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d}-\frac {e \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^2}+\frac {\int \frac {a f \left (e^2-d f\right )-e f (c d-a f) x-c d f^2 x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^2 f} \\ & = -\frac {\sqrt {a+c x^2}}{d x}-\frac {c \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d}+\frac {c \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d}-\frac {(a e) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^2}+\frac {\int \frac {c d^2 f^2+a f^2 \left (e^2-d f\right )+\left (c d e f^2-e f^2 (c d-a f)\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^2 f^2} \\ & = -\frac {\sqrt {a+c x^2}}{d x}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{d}-\frac {c \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d}-\frac {(a e) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^2}-\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^2 \sqrt {e^2-4 d f}}+\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^2 \sqrt {e^2-4 d f}} \\ & = -\frac {\sqrt {a+c x^2}}{d x}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}+\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d^2 \sqrt {e^2-4 d f}}-\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d^2 \sqrt {e^2-4 d f}} \\ & = -\frac {\sqrt {a+c x^2}}{d x}-\frac {f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.38 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {d \sqrt {a+c x^2}-\sqrt {a} e x \log (x)+\sqrt {a} e x \log \left (-\sqrt {a}+\sqrt {a+c x^2}\right )+x \text {RootSum}\left [c^2 d+2 \sqrt {a} c e \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {-c^2 d^2 \log (x)-a c e^2 \log (x)+a c d f \log (x)+c^2 d^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+a c e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-a c d f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-2 a^{3/2} e f \log (x) \text {$\#$1}+2 a^{3/2} e f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+c d^2 \log (x) \text {$\#$1}^2+a e^2 \log (x) \text {$\#$1}^2-a d f \log (x) \text {$\#$1}^2-c d^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+a d f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} c e+2 c d \text {$\#$1}-4 a f \text {$\#$1}+3 \sqrt {a} e \text {$\#$1}^2-2 d \text {$\#$1}^3}\&\right ]}{d^2 x} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(775\) vs. \(2(337)=674\).

Time = 0.76 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.03

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+a}}{d x}-\frac {\frac {4 f \sqrt {a}\, e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{\left (-e +\sqrt {-4 d f +e^{2}}\right ) \left (e +\sqrt {-4 d f +e^{2}}\right )}-\frac {\left (f a \sqrt {-4 d f +e^{2}}-c d \sqrt {-4 d f +e^{2}}+a e f +c d e \right ) \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}-\frac {c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 {\left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2} c -\frac {4 c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {-2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{\sqrt {-4 d f +e^{2}}\, \left (-e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}+\frac {\left (f a \sqrt {-4 d f +e^{2}}-c d \sqrt {-4 d f +e^{2}}-a e f -c d e \right ) \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}-\frac {c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 {\left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2} c -\frac {4 c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{\sqrt {-4 d f +e^{2}}\, \left (e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}}{d}\)	\(776\)
default	\(\text {Expression too large to display}\)	\(1415\)

[In]

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

[Out]

-(c*x^2+a)^(1/2)/d/x-1/d*(4*f*a^(1/2)*e/(-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f+e^2)^(1/2))*ln((2*a+2*a^(1/2)*(c*x^
2+a)^(1/2))/x)-(f*a*(-4*d*f+e^2)^(1/2)-c*d*(-4*d*f+e^2)^(1/2)+a*e*f+c*d*e)/(-4*d*f+e^2)^(1/2)/(-e+(-4*d*f+e^2)
^(1/2))*2^(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*ln(((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f
^2-2*c*d*f+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((-(-4*d*f+e^2)
^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c-4*c*(e-(-4*d*f+e^2)^(1/2
))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))/(x-1/2/f*
(-e+(-4*d*f+e^2)^(1/2))))+(f*a*(-4*d*f+e^2)^(1/2)-c*d*(-4*d*f+e^2)^(1/2)-a*e*f-c*d*e)/(-4*d*f+e^2)^(1/2)/(e+(-
4*d*f+e^2)^(1/2))*2^(1/2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*ln((((-4*d*f+e^2)^(1/2)*c
*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*(((-4*d*
f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d*f+e^2)
^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))/(x+1/2
*(e+(-4*d*f+e^2)^(1/2))/f)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2656 vs. \(2 (335) = 670\).

Time = 176.36 (sec) , antiderivative size = 5324, normalized size of antiderivative = 13.94 \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError >> type

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\int \frac {\sqrt {c\,x^2+a}}{x^2\,\left (f\,x^2+e\,x+d\right )} \,d x \]

[In]

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

[Out]

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