\(\int \frac {\sqrt {a+c x^2}}{d+e x+f x^2} \, dx\) [54]

Optimal result

Integrand size = 24, antiderivative size = 298 \[ \int \frac {\sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{f}-\frac {\sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f \sqrt {e^2-4 d f}}+\frac {\sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f \sqrt {e^2-4 d f}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1005, 223, 212, 1048, 739} \[ \int \frac {\sqrt {a+c x^2}}{d+e x+f x^2} \, dx=-\frac {\sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )} \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} f \sqrt {e^2-4 d f}}+\frac {\sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )} \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} f \sqrt {e^2-4 d f}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{f} \]

[In]

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

[Out]

(Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/f - (Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*ArcT
anh[(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]*f*Sqrt[e^2 - 4*d*f]) + (Sqrt[2*a*f^2 + c*(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]*f*Sqrt[e^2 - 4*d*f])

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 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 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 1005

Int[Sqrt[(a_) + (c_.)*(x_)^2]/((d_) + (e_.)*(x_) + (f_.)*(x_)^2), x_Symbol] :> Dist[c/f, Int[1/Sqrt[a + c*x^2]
, x], x] - Dist[1/f, Int[(c*d - a*f + c*e*x)/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)), x], x] /; FreeQ[{a, c, d, e,
f}, x] && NeQ[e^2 - 4*d*f, 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]

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

Mathematica [C] (verified)

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

Time = 0.29 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\frac {-\sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )+\text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {a c e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 c^{3/2} d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{f} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1211\) vs. \(2(259)=518\).

Time = 0.74 (sec) , antiderivative size = 1212, normalized size of antiderivative = 4.07

method	result	size
default	\(\text {Expression too large to display}\)	\(1212\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (257) = 514\).

Time = 138.40 (sec) , antiderivative size = 2384, normalized size of antiderivative = 8.00 \[ \int \frac {\sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*(sqrt(2)*f*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 + (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*
f^2 - 4*d*f^3))*log((4*c^2*d*e*x - 2*a*c*e^2 + sqrt(2)*(c*e^3 - 4*c*d*e*f - (e^3*f^2 - 4*d*e*f^3)*sqrt(c^2*e^2
/(e^2*f^4 - 4*d*f^5)))*sqrt(c*x^2 + a)*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 + (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2
*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3)) + 2*(a*e^2*f^2 - 4*a*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/x) - sqr
t(2)*f*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 + (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d
*f^3))*log((4*c^2*d*e*x - 2*a*c*e^2 - sqrt(2)*(c*e^3 - 4*c*d*e*f - (e^3*f^2 - 4*d*e*f^3)*sqrt(c^2*e^2/(e^2*f^4
- 4*d*f^5)))*sqrt(c*x^2 + a)*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 + (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*
d*f^5)))/(e^2*f^2 - 4*d*f^3)) + 2*(a*e^2*f^2 - 4*a*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/x) + sqrt(2)*f*sq
rt((c*e^2 - 2*c*d*f + 2*a*f^2 - (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3))*lo
g((4*c^2*d*e*x - 2*a*c*e^2 + sqrt(2)*(c*e^3 - 4*c*d*e*f + (e^3*f^2 - 4*d*e*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^
5)))*sqrt(c*x^2 + a)*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 - (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/
(e^2*f^2 - 4*d*f^3)) - 2*(a*e^2*f^2 - 4*a*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/x) - sqrt(2)*f*sqrt((c*e^2
- 2*c*d*f + 2*a*f^2 - (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3))*log((4*c^2*
d*e*x - 2*a*c*e^2 - sqrt(2)*(c*e^3 - 4*c*d*e*f + (e^3*f^2 - 4*d*e*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))*sqrt
(c*x^2 + a)*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 - (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*f^2
- 4*d*f^3)) - 2*(a*e^2*f^2 - 4*a*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/x) + 2*sqrt(c)*log(-2*c*x^2 - 2*sqr
t(c*x^2 + a)*sqrt(c)*x - a))/f, 1/4*(sqrt(2)*f*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 + (e^2*f^2 - 4*d*f^3)*sqrt(c^2*
e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3))*log((4*c^2*d*e*x - 2*a*c*e^2 + sqrt(2)*(c*e^3 - 4*c*d*e*f - (e^
3*f^2 - 4*d*e*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))*sqrt(c*x^2 + a)*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 + (e^2*f
^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3)) + 2*(a*e^2*f^2 - 4*a*d*f^3)*sqrt(c^2*e^2
/(e^2*f^4 - 4*d*f^5)))/x) - sqrt(2)*f*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 + (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*
f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3))*log((4*c^2*d*e*x - 2*a*c*e^2 - sqrt(2)*(c*e^3 - 4*c*d*e*f - (e^3*f^2 - 4
*d*e*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))*sqrt(c*x^2 + a)*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 + (e^2*f^2 - 4*d*
f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3)) + 2*(a*e^2*f^2 - 4*a*d*f^3)*sqrt(c^2*e^2/(e^2*f^4
- 4*d*f^5)))/x) + sqrt(2)*f*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 - (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d
*f^5)))/(e^2*f^2 - 4*d*f^3))*log((4*c^2*d*e*x - 2*a*c*e^2 + sqrt(2)*(c*e^3 - 4*c*d*e*f + (e^3*f^2 - 4*d*e*f^3)
*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))*sqrt(c*x^2 + a)*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 - (e^2*f^2 - 4*d*f^3)*sqrt
(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3)) - 2*(a*e^2*f^2 - 4*a*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^
5)))/x) - sqrt(2)*f*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 - (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/(
e^2*f^2 - 4*d*f^3))*log((4*c^2*d*e*x - 2*a*c*e^2 - sqrt(2)*(c*e^3 - 4*c*d*e*f + (e^3*f^2 - 4*d*e*f^3)*sqrt(c^2
*e^2/(e^2*f^4 - 4*d*f^5)))*sqrt(c*x^2 + a)*sqrt((c*e^2 - 2*c*d*f + 2*a*f^2 - (e^2*f^2 - 4*d*f^3)*sqrt(c^2*e^2/
(e^2*f^4 - 4*d*f^5)))/(e^2*f^2 - 4*d*f^3)) - 2*(a*e^2*f^2 - 4*a*d*f^3)*sqrt(c^2*e^2/(e^2*f^4 - 4*d*f^5)))/x) -
4*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/f]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-e^2>0)', see `assume?` f
or more deta

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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