3.3.26 \(\int \frac {1}{(c+d x+e x^2) \sqrt {a+b x^4}} \, dx\) [226]

Optimal. Leaf size=1605 \[ -\frac {e^2 \tan ^{-1}\left (\frac {\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} \left (d^2-2 c e\right )} x}{e \left (d+\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (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}\right )}}+\frac {e^2 \tan ^{-1}\left (\frac {\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} \left (d^2-2 c e\right )} x}{e \left (d-\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (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}\right )}}-\frac {e^2 \tanh ^{-1}\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 d^2 e+2 b c^2 e^2+2 a e^4-b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} \sqrt {a+b x^4}}\right )}{\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} \left (d^2-2 c e\right )}}+\frac {e^2 \tanh ^{-1}\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 d^2 e+2 b c^2 e^2+2 a e^4+b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} \sqrt {a+b x^4}}\right )}{\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} \left (d^2-2 c e\right )}}+\frac {\sqrt [4]{b} e \left (d-\sqrt {d^2-4 c e}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {d^2-4 c e} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} e \left (d+\sqrt {d^2-4 c e}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {d^2-4 c e} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}+\frac {e \left (2 \sqrt {a} e^2-\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{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^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {e \left (2 \sqrt {a} e^2-\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{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^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}} \]

[Out]

1/2*b^(1/4)*e*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arct
an(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*(d-(-4*c*e+d^2)^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/
2))^2)^(1/2)/a^(1/4)/(-4*c*e+d^2)^(1/2)/(2*e^2*a^(1/2)+b^(1/2)*(d^2-2*c*e-d*(-4*c*e+d^2)^(1/2)))/(b*x^4+a)^(1/
2)+1/2*e*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(b
^(1/4)*x/a^(1/4))),1/4*(2*e^2*a^(1/2)+b^(1/2)*(d^2-2*c*e-d*(-4*c*e+d^2)^(1/2)))^2/e^2/a^(1/2)/b^(1/2)/(d-(-4*c
*e+d^2)^(1/2))^2,1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*(2*e^2*a^(1/2)-b^(1/2)*(d^2-2*c*e-d*(-4*c*e+d^2)^(1/2)))*(
(b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/b^(1/4)/(d-(-4*c*e+d^2)^(1/2))/(-4*c*e+d^2)^(1/2)/(2*e^2*a^(1
/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*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^
2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2
*b^(1/2))*(d+(-4*c*e+d^2)^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/(-4*c*e+d^2)^(1/2)/(2*e^2*a
^(1/2)+b^(1/2)*(d^2-2*c*e+d*(-4*c*e+d^2)^(1/2)))/(b*x^4+a)^(1/2)-1/2*e*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1
/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/4*(2*e^2*a^(1/2)+b^(1/2)*(d
^2-2*c*e+d*(-4*c*e+d^2)^(1/2)))^2/e^2/a^(1/2)/b^(1/2)/(d+(-4*c*e+d^2)^(1/2))^2,1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/
2))*(2*e^2*a^(1/2)-b^(1/2)*(d^2-2*c*e+d*(-4*c*e+d^2)^(1/2)))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)
/b^(1/4)/(-4*c*e+d^2)^(1/2)/(d+(-4*c*e+d^2)^(1/2))/(2*e^2*a^(1/2)+b^(1/2)*(d^2-2*c*e+d*(-4*c*e+d^2)^(1/2)))/(b
*x^4+a)^(1/2)-1/2*e^2*arctanh(1/4*(4*a*e^2+b*x^2*(d-(-4*c*e+d^2)^(1/2))^2)*2^(1/2)/(b*x^4+a)^(1/2)/(b*d^4-4*b*
c*d^2*e+2*b*c^2*e^2+2*a*e^4-b*d*(-2*c*e+d^2)*(-4*c*e+d^2)^(1/2))^(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*(-2*c*e+d^2)*(-4*c*e+d^2)^(1/2))^(1/2)+1/2*e^2*arctanh(1/4*(4*a*e^2+b*x^2*(d+(
-4*c*e+d^2)^(1/2))^2)*2^(1/2)/(b*x^4+a)^(1/2)/(b*d^4-4*b*c*d^2*e+2*b*c^2*e^2+2*a*e^4+b*d*(-2*c*e+d^2)*(-4*c*e+
d^2)^(1/2))^(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*(-2*c*e+d^2)*(-4*c*e+
d^2)^(1/2))^(1/2)-1/2*e^2*arctan(x*2^(1/2)*(-b*d^4+4*b*c*d^2*e-2*b*c^2*e^2-2*a*e^4-b*d*(-2*c*e+d^2)*(-4*c*e+d^
2)^(1/2))^(1/2)/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*arctan(x*2^(1/2)*(-b*d^4+4*b*c*d
^2*e-2*b*c^2*e^2-2*a*e^4+b*d*(-2*c*e+d^2)*(-4*c*e+d^2)^(1/2))^(1/2)/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)

________________________________________________________________________________________

Rubi [A]
time = 7.06, antiderivative size = 1605, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6860, 1739, 1231, 226, 1721, 1262, 739, 212} \begin {gather*} -\frac {\text {ArcTan}\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 {ArcTan}\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 {\tanh ^{-1}\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 {\tanh ^{-1}\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}} F\left (2 \text {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}} F\left (2 \text {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}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2};2 \text {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}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2};2 \text {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}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((e^2*ArcTan[(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*ArcTan[(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])*Sqr
t[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])*(S
qrt[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]))*Sqrt[a + b*x^4])
- (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]*Elli
pticF[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]))*Sqrt[a + b*x^4]) + (e*(2*Sqrt[a]*e^2 - Sqrt[b]*(d^2 - 2*c*e - d*Sqrt[d^2 - 4*c*e]))*(S
qrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(2*Sqrt[a]*e^2 + Sqrt[b]*(d^2 - 2
*c*e - d*Sqrt[d^2 - 4*c*e]))^2/(4*Sqrt[a]*Sqrt[b]*e^2*(d - Sqrt[d^2 - 4*c*e])^2), 2*ArcTan[(b^(1/4)*x)/a^(1/4)
], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[d^2 - 4*c*e]*(d - 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]))*Sqrt[a + b*x^4]) - (e*(2*Sqrt[a]*e^2 - Sqrt[b]*(d^2 - 2*c*e + d*Sqrt[d^2 - 4*c*e]))*(Sqr
t[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(2*Sqrt[a]*e^2 + Sqrt[b]*(d^2 - 2*c
*e + d*Sqrt[d^2 - 4*c*e]))^2/(4*Sqrt[a]*Sqrt[b]*e^2*(d + Sqrt[d^2 - 4*c*e])^2), 2*ArcTan[(b^(1/4)*x)/a^(1/4)],
1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[d^2 - 4*c*e]*(d + 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]))*Sqrt[a + b*x^4])

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 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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 1231

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

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1739

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

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*} \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx &=\int \left (\frac {2 e}{\sqrt {d^2-4 c e} \left (d-\sqrt {d^2-4 c e}+2 e x\right ) \sqrt {a+b x^4}}-\frac {2 e}{\sqrt {d^2-4 c e} \left (d+\sqrt {d^2-4 c e}+2 e x\right ) \sqrt {a+b x^4}}\right ) \, dx\\ &=\frac {(2 e) \int \frac {1}{\left (d-\sqrt {d^2-4 c e}+2 e x\right ) \sqrt {a+b x^4}} \, dx}{\sqrt {d^2-4 c e}}-\frac {(2 e) \int \frac {1}{\left (d+\sqrt {d^2-4 c e}+2 e x\right ) \sqrt {a+b x^4}} \, dx}{\sqrt {d^2-4 c e}}\\ &=-\frac {\left (4 e^2\right ) \int \frac {x}{\left (\left (d-\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx}{\sqrt {d^2-4 c e}}+\frac {\left (4 e^2\right ) \int \frac {x}{\left (\left (d+\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx}{\sqrt {d^2-4 c e}}-\left (2 e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1}{\left (\left (d-\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx-\left (2 e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1}{\left (\left (d+\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx\\ &=-\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (\left (d-\sqrt {d^2-4 c e}\right )^2-4 e^2 x\right ) \sqrt {a+b x^2}} \, dx,x,x^2\right )}{\sqrt {d^2-4 c e}}+\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (\left (d+\sqrt {d^2-4 c e}\right )^2-4 e^2 x\right ) \sqrt {a+b x^2}} \, dx,x,x^2\right )}{\sqrt {d^2-4 c e}}-\frac {\left (\sqrt {b} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )}-\frac {\left (4 \sqrt {a} e^3 \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\left (\left (d-\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx}{2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )}-\frac {\left (\sqrt {b} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )}-\frac {\left (4 \sqrt {a} e^3 \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\left (\left (d+\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx}{2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )}\\ &=-\frac {e^2 \tan ^{-1}\left (\frac {\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} \left (d^2-2 c e\right )} x}{e \left (d+\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (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}\right )}}+\frac {e^2 \tan ^{-1}\left (\frac {\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} \left (d^2-2 c e\right )} x}{e \left (d-\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (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}\right )}}-\frac {\sqrt [4]{b} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{a} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (4 e^2-\frac {\sqrt {b} \left (d-\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (d-\sqrt {d^2-4 c e}\right )^2 \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{a} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (4 e^2-\frac {\sqrt {b} \left (d+\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (d+\sqrt {d^2-4 c e}\right )^2 \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}+\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {1}{16 a e^4+b \left (d-\sqrt {d^2-4 c e}\right )^4-x^2} \, dx,x,\frac {-4 a e^2-b \left (d-\sqrt {d^2-4 c e}\right )^2 x^2}{\sqrt {a+b x^4}}\right )}{\sqrt {d^2-4 c e}}-\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {1}{16 a e^4+b \left (d+\sqrt {d^2-4 c e}\right )^4-x^2} \, dx,x,\frac {-4 a e^2-b \left (d+\sqrt {d^2-4 c e}\right )^2 x^2}{\sqrt {a+b x^4}}\right )}{\sqrt {d^2-4 c e}}\\ &=-\frac {e^2 \tan ^{-1}\left (\frac {\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} \left (d^2-2 c e\right )} x}{e \left (d+\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (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}\right )}}+\frac {e^2 \tan ^{-1}\left (\frac {\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} \left (d^2-2 c e\right )} x}{e \left (d-\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (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}\right )}}-\frac {e^2 \tanh ^{-1}\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 d^2 e+2 b c^2 e^2+2 a e^4-b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} \sqrt {a+b x^4}}\right )}{\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} \left (d^2-2 c e\right )}}+\frac {e^2 \tanh ^{-1}\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 d^2 e+2 b c^2 e^2+2 a e^4+b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} \sqrt {a+b x^4}}\right )}{\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} \left (d^2-2 c e\right )}}-\frac {\sqrt [4]{b} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{a} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (4 e^2-\frac {\sqrt {b} \left (d-\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (d-\sqrt {d^2-4 c e}\right )^2 \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{a} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (4 e^2-\frac {\sqrt {b} \left (d+\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (d+\sqrt {d^2-4 c e}\right )^2 \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 10.97, size = 455, normalized size = 0.28 \begin {gather*} -\frac {i \sqrt {1+\frac {b x^4}{a}} \left (\left (-d^2+\sqrt {d^4-4 c d^2 e}\right ) \Pi \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 )};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )+\left (d^2+\sqrt {d^4-4 c d^2 e}\right ) \Pi \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 )};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-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*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + (d^2 + Sqrt[d^4 - 4*c*d^2*e])*El
lipticPi[((-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])/Sqr
t[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) & ]

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Maple [C] Result contains complex when optimal does not.
time = 0.32, size = 1153, normalized size = 0.72 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x^{4}} \left (c + d x + e x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\sqrt {b\,x^4+a}\,\left (e\,x^2+d\,x+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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