\(\int \frac {1}{(c+d x+e x^2) \sqrt {a+b x^4}} \, dx\) [281]

Optimal result

Integrand size = 24, antiderivative size = 1613 \[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx =\text {Too large to display} \] Output:

-1/2*e^2*arctanh(2^(1/2)*(2*a*e^4+b*(d^4-4*c*d^2*e+2*c^2*e^2+d^3*(-4*c*e+d 
^2)^(1/2)-2*c*d*e*(-4*c*e+d^2)^(1/2)))^(1/2)*x/e/(d+(-4*c*e+d^2)^(1/2))/(b 
*x^4+a)^(1/2))*2^(1/2)/(-4*c*e+d^2)^(1/2)/(2*a*e^4+b*(d^4-4*c*d^2*e+2*c^2* 
e^2+d^3*(-4*c*e+d^2)^(1/2)-2*c*d*e*(-4*c*e+d^2)^(1/2)))^(1/2)+1/2*e^2*arct 
anh(1/2*2^(1/2)*(8*a*e^4+2*b*(2*d^4-8*c*d^2*e+4*c^2*e^2-2*d^3*(-4*c*e+d^2) 
^(1/2)+4*c*d*e*(-4*c*e+d^2)^(1/2)))^(1/2)*x/e/(d-(-4*c*e+d^2)^(1/2))/(b*x^ 
4+a)^(1/2))*2^(1/2)/(-4*c*e+d^2)^(1/2)/(2*a*e^4+b*(d^4-4*c*d^2*e+2*c^2*e^2 
-d^3*(-4*c*e+d^2)^(1/2)+2*c*d*e*(-4*c*e+d^2)^(1/2)))^(1/2)-1/2*e^2*arctanh 
(1/4*(4*a*e^2+b*(d-(-4*c*e+d^2)^(1/2))^2*x^2)*2^(1/2)/(b*d^4-4*b*c*d^2*e+2 
*b*c^2*e^2+2*a*e^4-b*d*(-4*c*e+d^2)^(1/2)*(-2*c*e+d^2))^(1/2)/(b*x^4+a)^(1 
/2))*2^(1/2)/(-4*c*e+d^2)^(1/2)/(b*d^4-4*b*c*d^2*e+2*b*c^2*e^2+2*a*e^4-b*d 
*(-4*c*e+d^2)^(1/2)*(-2*c*e+d^2))^(1/2)+1/2*e^2*arctanh(1/4*(4*a*e^2+b*(d+ 
(-4*c*e+d^2)^(1/2))^2*x^2)*2^(1/2)/(b*d^4-4*b*c*d^2*e+2*b*c^2*e^2+2*a*e^4+ 
b*d*(-4*c*e+d^2)^(1/2)*(-2*c*e+d^2))^(1/2)/(b*x^4+a)^(1/2))*2^(1/2)/(-4*c* 
e+d^2)^(1/2)/(b*d^4-4*b*c*d^2*e+2*b*c^2*e^2+2*a*e^4+b*d*(-4*c*e+d^2)^(1/2) 
*(-2*c*e+d^2))^(1/2)+1/2*b^(1/4)*e*(d-(-4*c*e+d^2)^(1/2))*(a^(1/2)+b^(1/2) 
*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b 
^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/(-4*c*e+d^2)^(1/2)/(2*a^(1/2)*e^2+b 
^(1/2)*(d^2-2*c*e-d*(-4*c*e+d^2)^(1/2)))/(b*x^4+a)^(1/2)-1/2*b^(1/4)*e*(d+ 
(-4*c*e+d^2)^(1/2))*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x...
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 11.69 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=-\frac {i \sqrt {1+\frac {b x^4}{a}} \left (\left (-d^2+\sqrt {d^4-4 c d^2 e}\right ) \operatorname {EllipticPi}\left (\frac {2 i \sqrt {a} e^2}{\sqrt {b} \left (d^2-2 c e+\sqrt {d^4-4 c d^2 e}\right )},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right ),-1\right )+\left (d^2+\sqrt {d^4-4 c d^2 e}\right ) \operatorname {EllipticPi}\left (-\frac {2 i \sqrt {a} e^2}{\sqrt {b} \left (-d^2+2 c e+\sqrt {d^4-4 c d^2 e}\right )},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right ),-1\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} c \sqrt {d^4-4 c d^2 e} \sqrt {a+b x^4}}+\sqrt {b} d \text {RootSum}\left [a^2 e^2-2 a \sqrt {b} d^2 \text {$\#$1}+4 a \sqrt {b} c e \text {$\#$1}+4 b c^2 \text {$\#$1}^2-2 a e^2 \text {$\#$1}^2+2 \sqrt {b} d^2 \text {$\#$1}^3-4 \sqrt {b} c e \text {$\#$1}^3+e^2 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {b} x^2+\sqrt {a+b x^4}-\text {$\#$1}\right ) \text {$\#$1}}{-a \sqrt {b} d^2+2 a \sqrt {b} c e+4 b c^2 \text {$\#$1}-2 a e^2 \text {$\#$1}+3 \sqrt {b} d^2 \text {$\#$1}^2-6 \sqrt {b} c e \text {$\#$1}^2+2 e^2 \text {$\#$1}^3}\&\right ] \] Input:

Integrate[1/((c + d*x + e*x^2)*Sqrt[a + b*x^4]),x]
 

Output:

((-1/2*I)*Sqrt[1 + (b*x^4)/a]*((-d^2 + Sqrt[d^4 - 4*c*d^2*e])*EllipticPi[( 
(2*I)*Sqrt[a]*e^2)/(Sqrt[b]*(d^2 - 2*c*e + Sqrt[d^4 - 4*c*d^2*e])), I*ArcS 
inh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + (d^2 + Sqrt[d^4 - 4*c*d^2*e])*Elli 
pticPi[((-2*I)*Sqrt[a]*e^2)/(Sqrt[b]*(-d^2 + 2*c*e + Sqrt[d^4 - 4*c*d^2*e] 
)), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1]))/(Sqrt[(I*Sqrt[b])/Sqrt[a 
]]*c*Sqrt[d^4 - 4*c*d^2*e]*Sqrt[a + b*x^4]) + Sqrt[b]*d*RootSum[a^2*e^2 - 
2*a*Sqrt[b]*d^2*#1 + 4*a*Sqrt[b]*c*e*#1 + 4*b*c^2*#1^2 - 2*a*e^2*#1^2 + 2* 
Sqrt[b]*d^2*#1^3 - 4*Sqrt[b]*c*e*#1^3 + e^2*#1^4 & , (Log[-(Sqrt[b]*x^2) + 
 Sqrt[a + b*x^4] - #1]*#1)/(-(a*Sqrt[b]*d^2) + 2*a*Sqrt[b]*c*e + 4*b*c^2*# 
1 - 2*a*e^2*#1 + 3*Sqrt[b]*d^2*#1^2 - 6*Sqrt[b]*c*e*#1^2 + 2*e^2*#1^3) & ]
 

Rubi [A] (warning: unable to verify)

Time = 20.67 (sec) , antiderivative size = 1590, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {7279, 7239, 7279, 2009}

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.

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {b d^4-4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} x}{e \left (d-\sqrt {d^2-4 c e}\right ) \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {2 a e^4+b \left (d^4-\sqrt {d^2-4 c e} d^3-4 c e d^2+2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {b d^4-4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} x}{e \left (d+\sqrt {d^2-4 c e}\right ) \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {2 a e^4+b \left (d^4+\sqrt {d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )}}-\frac {\text {arctanh}\left (\frac {4 a e^2+b \left (d-\sqrt {d^2-4 c e}\right )^2 x^2}{2 \sqrt {2} \sqrt {b d^4-4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {b d^4-4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2}}+\frac {\text {arctanh}\left (\frac {4 a e^2+b \left (d+\sqrt {d^2-4 c e}\right )^2 x^2}{2 \sqrt {2} \sqrt {b d^4-4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {b d^4-4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2}}+\frac {\sqrt [4]{b} \left (d-\sqrt {d^2-4 c e}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt {d^2-4 c e} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}-\frac {\sqrt [4]{b} \left (d+\sqrt {d^2-4 c e}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt {d^2-4 c e} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}+\frac {\left (2 \sqrt {a} e^2-\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \left (4 e^2+\frac {\sqrt {b} \left (d-\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right )^2}{16 \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {d^2-4 c e} \left (d-\sqrt {d^2-4 c e}\right ) \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}-\frac {\left (2 \sqrt {a} e^2-\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \left (4 e^2+\frac {\sqrt {b} \left (d+\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right )^2}{16 \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {d^2-4 c e} \left (d+\sqrt {d^2-4 c e}\right ) \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}\)

Input:

Int[1/((c + d*x + e*x^2)*Sqrt[a + b*x^4]),x]
 

Output:

(e^2*ArcTanh[(Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 - b 
*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]*x)/(e*(d - Sqrt[d^2 - 4*c*e])*Sqrt[a + 
 b*x^4])])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[2*a*e^4 + b*(d^4 - 4*c*d^2*e + 
2*c^2*e^2 - d^3*Sqrt[d^2 - 4*c*e] + 2*c*d*e*Sqrt[d^2 - 4*c*e])]) - (e^2*Ar 
cTanh[(Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 + b*d*Sqrt 
[d^2 - 4*c*e]*(d^2 - 2*c*e)]*x)/(e*(d + Sqrt[d^2 - 4*c*e])*Sqrt[a + b*x^4] 
)])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[2*a*e^4 + b*(d^4 - 4*c*d^2*e + 2*c^2*e 
^2 + d^3*Sqrt[d^2 - 4*c*e] - 2*c*d*e*Sqrt[d^2 - 4*c*e])]) - (e^2*ArcTanh[( 
4*a*e^2 + b*(d - Sqrt[d^2 - 4*c*e])^2*x^2)/(2*Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d 
^2*e + 2*b*c^2*e^2 + 2*a*e^4 - b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]*Sqrt[a 
 + b*x^4])])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2 
*e^2 + 2*a*e^4 - b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]) + (e^2*ArcTanh[(4*a 
*e^2 + b*(d + Sqrt[d^2 - 4*c*e])^2*x^2)/(2*Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d^2* 
e + 2*b*c^2*e^2 + 2*a*e^4 + b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]*Sqrt[a + 
b*x^4])])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^ 
2 + 2*a*e^4 + b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]) + (b^(1/4)*e*(d - Sqrt 
[d^2 - 4*c*e])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b] 
*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*Sqrt[d^ 
2 - 4*c*e]*(2*Sqrt[a]*e^2 + Sqrt[b]*(d^2 - 2*c*e - d*Sqrt[d^2 - 4*c*e]))*S 
qrt[a + b*x^4]) - (b^(1/4)*e*(d + Sqrt[d^2 - 4*c*e])*(Sqrt[a] + Sqrt[b]...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7279

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] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.77 (sec) , antiderivative size = 1153, normalized size of antiderivative = 0.71

method	result	size
default	\(\text {Expression too large to display}\)	\(1153\)
elliptic	\(\text {Expression too large to display}\)	\(1153\)

Input:

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

Output:

-1/2/(-4*c*e+d^2)^(1/2)/(1/2*b/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2* 
b/e^3*d^2*c+b/e^3*c*d*(-4*c*e+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)*arctanh(1/2/(1 
/2*b/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*d^2*c+b/e^3*c*d*(-4* 
c*e+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d^2-1/2/(1/2*b 
/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*d^2*c+b/e^3*c*d*(-4*c*e+ 
d^2)^(1/2)+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d*(-4*c*e+d^2)^(1/ 
2)-1/(1/2*b/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*d^2*c+b/e^3*c 
*d*(-4*c*e+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e*c+1/(1/2* 
b/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*d^2*c+b/e^3*c*d*(-4*c*e 
+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*a)-2/(-4*c*e+d^2)^(1/2)/(I* 
b^(1/2)/a^(1/2))^(1/2)*e/(-d+(-4*c*e+d^2)^(1/2))*(1-I*b^(1/2)/a^(1/2)*x^2) 
^(1/2)*(1+I*b^(1/2)/a^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticPi(x*(I*b^( 
1/2)/a^(1/2))^(1/2),-4*I/b^(1/2)*a^(1/2)*e^2/(-d+(-4*c*e+d^2)^(1/2))^2,(-I 
*b^(1/2)/a^(1/2))^(1/2)/(I*b^(1/2)/a^(1/2))^(1/2))+1/2/(-4*c*e+d^2)^(1/2)/ 
(1/2*b/e^4*d^4+1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*d^2*c-b/e^3*c*d*(- 
4*c*e+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)*arctanh(1/2/(1/2*b/e^4*d^4+1/2*b/e^4*d 
^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*d^2*c-b/e^3*c*d*(-4*c*e+d^2)^(1/2)+b/e^2*c^2 
+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d^2+1/2/(1/2*b/e^4*d^4+1/2*b/e^4*d^3*( 
-4*c*e+d^2)^(1/2)-2*b/e^3*d^2*c-b/e^3*c*d*(-4*c*e+d^2)^(1/2)+b/e^2*c^2+a)^ 
(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d*(-4*c*e+d^2)^(1/2)-1/(1/2*b/e^4*d^4+1...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int \frac {1}{\sqrt {a + b x^{4}} \left (c + d x + e x^{2}\right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a} {\left (e x^{2} + d x + c\right )}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a} {\left (e x^{2} + d x + c\right )}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int \frac {1}{\sqrt {b\,x^4+a}\,\left (e\,x^2+d\,x+c\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int \frac {\sqrt {b \,x^{4}+a}}{b e \,x^{6}+b d \,x^{5}+b c \,x^{4}+a e \,x^{2}+a d x +a c}d x \] Input:

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

Output:

int(sqrt(a + b*x**4)/(a*c + a*d*x + a*e*x**2 + b*c*x**4 + b*d*x**5 + b*e*x 
**6),x)