3.4.63 \(\int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) [363]

3.4.63.1 Optimal result

Integrand size = 26, antiderivative size = 240 \[ \int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \]

output

arctan(x*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b-(-4*a*c 
+b^2)^(1/2))^(1/2))*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(-4*a*c+b^2)^(1 
/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-arctan(x*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))) 
^(1/2)/(e*x^2+d)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(2*c*d-e*(b+(-4*a*c+b 
^2)^(1/2)))^(1/2)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
 

3.4.63.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2607\) vs. \(2(240)=480\).

Time = 13.42 (sec) , antiderivative size = 2607, normalized size of antiderivative = 10.86 \[ \int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\text {Result too large to show} \]

input

Integrate[Sqrt[d + e*x^2]/(a + b*x^2 + c*x^4),x]
 

output

(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*S 
qrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[-(Sqrt[2]*Sqrt[(-b + Sqrt[b^2 
 - 4*a*c])/c]) + 2*x] - Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*(2*c*d + (-b + 
Sqrt[b^2 - 4*a*c])*e)*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[Sqrt[2 
]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] + 2*x] - 2*c*Sqrt[(-b + Sqrt[b^2 - 4*a* 
c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[2]*Sqrt[-( 
(b + Sqrt[b^2 - 4*a*c])/c)]) + 2*x] + b*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e 
*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[2]*Sqrt[-((b + Sqr 
t[b^2 - 4*a*c])/c)]) + 2*x] + Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a* 
c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[2]*Sqrt[-( 
(b + Sqrt[b^2 - 4*a*c])/c)]) + 2*x] + 2*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] 
*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[2]*Sqrt[-((b + Sqr 
t[b^2 - 4*a*c])/c)] + 2*x] - b*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2* 
c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a 
*c])/c)] + 2*x] - Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqr 
t[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[2]*Sqrt[-((b + Sqrt[b^2 
- 4*a*c])/c)] + 2*x] + 2*c*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*d*Sqrt[2*d - 
 ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[4*d - 2*Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 
4*a*c])/c]*e*x + 2*Sqrt[4*d + (2*(-b + Sqrt[b^2 - 4*a*c])*e)/c]*Sqrt[d + e 
*x^2]] - b*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*Sqrt[2*d - ((b + Sqrt[b...
 

3.4.63.3 Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1488, 301, 224, 219, 291, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\)

\(\Big \downarrow \) 1488

\(\displaystyle \frac {2 c \int \frac {\sqrt {e x^2+d}}{2 c x^2+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {\sqrt {e x^2+d}}{2 c x^2+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {2 c \left (\frac {\left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \int \frac {1}{\left (2 c x^2+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}dx}{2 c}+\frac {e \int \frac {1}{\sqrt {e x^2+d}}dx}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \int \frac {1}{\left (2 c x^2+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}dx}{2 c}+\frac {e \int \frac {1}{\sqrt {e x^2+d}}dx}{2 c}\right )}{\sqrt {b^2-4 a c}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {2 c \left (\frac {\left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \int \frac {1}{\left (2 c x^2+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}dx}{2 c}+\frac {e \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \int \frac {1}{\left (2 c x^2+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}dx}{2 c}+\frac {e \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 c}\right )}{\sqrt {b^2-4 a c}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 c \left (\frac {\left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \int \frac {1}{\left (2 c x^2+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}dx}{2 c}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \int \frac {1}{\left (2 c x^2+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^2+d}}dx}{2 c}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {2 c \left (\frac {\left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \int \frac {1}{-\frac {\left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right ) x^2}{e x^2+d}+b-\sqrt {b^2-4 a c}}d\frac {x}{\sqrt {e x^2+d}}}{2 c}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \int \frac {1}{-\frac {\left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right ) x^2}{e x^2+d}+b+\sqrt {b^2-4 a c}}d\frac {x}{\sqrt {e x^2+d}}}{2 c}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {2 c \left (\frac {\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 c \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{2 c \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}\)

input

Int[Sqrt[d + e*x^2]/(a + b*x^2 + c*x^4),x]
 

output

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

3.4.63.3.1 Defintions of rubi rules used

rule 218

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

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 291

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

rule 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 1488

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

3.4.63.4 Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {d \sqrt {2}\, \left (\frac {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}\)	\(233\)
pseudoelliptic	\(-\frac {d \sqrt {2}\, \left (\frac {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}\)	\(233\)

input

int((e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)
 

output

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

3.4.63.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 985 vs. \(2 (200) = 400\).

Time = 0.83 (sec) , antiderivative size = 985, normalized size of antiderivative = 4.10 \[ \int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b d - 2 \, a e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} d \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {e x^{2} + d} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x \sqrt {-\frac {b d - 2 \, a e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - 2 \, a d^{2} + {\left (b d^{2} - 4 \, a d e\right )} x^{2}}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b d - 2 \, a e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} d \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {e x^{2} + d} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x \sqrt {-\frac {b d - 2 \, a e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - 2 \, a d^{2} + {\left (b d^{2} - 4 \, a d e\right )} x^{2}}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b d - 2 \, a e - {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} d \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {e x^{2} + d} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x \sqrt {-\frac {b d - 2 \, a e - {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} + 2 \, a d^{2} - {\left (b d^{2} - 4 \, a d e\right )} x^{2}}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b d - 2 \, a e - {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} d \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {e x^{2} + d} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x \sqrt {-\frac {b d - 2 \, a e - {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} + 2 \, a d^{2} - {\left (b d^{2} - 4 \, a d e\right )} x^{2}}{x^{2}}\right ) \]

input

integrate((e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")
 

output

1/4*sqrt(1/2)*sqrt(-(b*d - 2*a*e + (a*b^2 - 4*a^2*c)*sqrt(d^2/(a^2*b^2 - 4 
*a^3*c)))/(a*b^2 - 4*a^2*c))*log(-((a*b^2 - 4*a^2*c)*d*sqrt(d^2/(a^2*b^2 - 
 4*a^3*c))*x^2 + 4*sqrt(1/2)*(a^2*b^2 - 4*a^3*c)*sqrt(e*x^2 + d)*sqrt(d^2/ 
(a^2*b^2 - 4*a^3*c))*x*sqrt(-(b*d - 2*a*e + (a*b^2 - 4*a^2*c)*sqrt(d^2/(a^ 
2*b^2 - 4*a^3*c)))/(a*b^2 - 4*a^2*c)) - 2*a*d^2 + (b*d^2 - 4*a*d*e)*x^2)/x 
^2) - 1/4*sqrt(1/2)*sqrt(-(b*d - 2*a*e + (a*b^2 - 4*a^2*c)*sqrt(d^2/(a^2*b 
^2 - 4*a^3*c)))/(a*b^2 - 4*a^2*c))*log(-((a*b^2 - 4*a^2*c)*d*sqrt(d^2/(a^2 
*b^2 - 4*a^3*c))*x^2 - 4*sqrt(1/2)*(a^2*b^2 - 4*a^3*c)*sqrt(e*x^2 + d)*sqr 
t(d^2/(a^2*b^2 - 4*a^3*c))*x*sqrt(-(b*d - 2*a*e + (a*b^2 - 4*a^2*c)*sqrt(d 
^2/(a^2*b^2 - 4*a^3*c)))/(a*b^2 - 4*a^2*c)) - 2*a*d^2 + (b*d^2 - 4*a*d*e)* 
x^2)/x^2) + 1/4*sqrt(1/2)*sqrt(-(b*d - 2*a*e - (a*b^2 - 4*a^2*c)*sqrt(d^2/ 
(a^2*b^2 - 4*a^3*c)))/(a*b^2 - 4*a^2*c))*log(((a*b^2 - 4*a^2*c)*d*sqrt(d^2 
/(a^2*b^2 - 4*a^3*c))*x^2 + 4*sqrt(1/2)*(a^2*b^2 - 4*a^3*c)*sqrt(e*x^2 + d 
)*sqrt(d^2/(a^2*b^2 - 4*a^3*c))*x*sqrt(-(b*d - 2*a*e - (a*b^2 - 4*a^2*c)*s 
qrt(d^2/(a^2*b^2 - 4*a^3*c)))/(a*b^2 - 4*a^2*c)) + 2*a*d^2 - (b*d^2 - 4*a* 
d*e)*x^2)/x^2) - 1/4*sqrt(1/2)*sqrt(-(b*d - 2*a*e - (a*b^2 - 4*a^2*c)*sqrt 
(d^2/(a^2*b^2 - 4*a^3*c)))/(a*b^2 - 4*a^2*c))*log(((a*b^2 - 4*a^2*c)*d*sqr 
t(d^2/(a^2*b^2 - 4*a^3*c))*x^2 - 4*sqrt(1/2)*(a^2*b^2 - 4*a^3*c)*sqrt(e*x^ 
2 + d)*sqrt(d^2/(a^2*b^2 - 4*a^3*c))*x*sqrt(-(b*d - 2*a*e - (a*b^2 - 4*a^2 
*c)*sqrt(d^2/(a^2*b^2 - 4*a^3*c)))/(a*b^2 - 4*a^2*c)) + 2*a*d^2 - (b*d^...
 

3.4.63.6 Sympy [F]

\[ \int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int \frac {\sqrt {d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \]

input

integrate((e*x**2+d)**(1/2)/(c*x**4+b*x**2+a),x)
 

output

Integral(sqrt(d + e*x**2)/(a + b*x**2 + c*x**4), x)
 

3.4.63.7 Maxima [F]

\[ \int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{c x^{4} + b x^{2} + a} \,d x } \]

input

integrate((e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")
 

output

integrate(sqrt(e*x^2 + d)/(c*x^4 + b*x^2 + a), x)
 

3.4.63.8 Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\text {Timed out} \]

input

integrate((e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="giac")
 

output

Timed out
 

3.4.63.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int \frac {\sqrt {e\,x^2+d}}{c\,x^4+b\,x^2+a} \,d x \]

input

int((d + e*x^2)^(1/2)/(a + b*x^2 + c*x^4),x)
 

output

int((d + e*x^2)^(1/2)/(a + b*x^2 + c*x^4), x)