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

3.1.4.1 Optimal result

Integrand size = 24, antiderivative size = 822 \[ \int \frac {1}{(d+e x)^2 \sqrt {a+b x^2+c x^4}} \, dx=-\frac {e^3 \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d e \left (2 c d^2+b e^2\right ) \arctan \left (\frac {\sqrt {-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \left (-c d^4-b d^2 e^2-a e^4\right )^{3/2}}-\frac {d e \left (2 c d^2+b e^2\right ) \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b d^2 e^2+a e^4} \sqrt {a+b x^2+c x^4}}\right )}{2 \left (c d^4+b d^2 e^2+a e^4\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (2 c d^2+b e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {a+b x^2+c x^4}} \]

output

-1/2*d*e*(b*e^2+2*c*d^2)*arctan(x*(-a*e^4-b*d^2*e^2-c*d^4)^(1/2)/d/e/(c*x^ 
4+b*x^2+a)^(1/2))/(-a*e^4-b*d^2*e^2-c*d^4)^(3/2)-1/2*d*e*(b*e^2+2*c*d^2)*a 
rctanh(1/2*(b*d^2+2*a*e^2+(b*e^2+2*c*d^2)*x^2)/(a*e^4+b*d^2*e^2+c*d^4)^(1/ 
2)/(c*x^4+b*x^2+a)^(1/2))/(a*e^4+b*d^2*e^2+c*d^4)^(3/2)-e^3*(c*x^4+b*x^2+a 
)^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)/(e*x+d)+e^2*x*c^(1/2)*(c*x^4+b*x^2+a)^(1/2 
)/(a*e^4+b*d^2*e^2+c*d^4)/(a^(1/2)+x^2*c^(1/2))-a^(1/4)*c^(1/4)*e^2*(cos(2 
*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*Elli 
pticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a 
^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/(a*e^4 
+b*d^2*e^2+c*d^4)/(c*x^4+b*x^2+a)^(1/2)+1/2*c^(1/4)*(cos(2*arctan(c^(1/4)* 
x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arct 
an(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/ 
2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/(e^2*a^(1/2)+d 
^2*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)-1/4*(b*e^2+2*c*d^2)*(cos(2*arctan(c^(1/4 
)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*a 
rctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*a^(1/2)+d^2*c^(1/2))^2/d^2/e^2/a^(1/2)/ 
c^(1/2),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))*(a^(1/ 
2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/c^ 
(1/4)/(a*e^4+b*d^2*e^2+c*d^4)/(e^2*a^(1/2)+d^2*c^(1/2))/(c*x^4+b*x^2+a)^(1 
/2)
 

3.1.4.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 17.38 (sec) , antiderivative size = 4019, normalized size of antiderivative = 4.89 \[ \int \frac {1}{(d+e x)^2 \sqrt {a+b x^2+c x^4}} \, dx=\text {Result too large to show} \]

input

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

output

-((e^3*Sqrt[a + b*x^2 + c*x^4])/((c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x))) + 
 (((I/2)*(-b + Sqrt[b^2 - 4*a*c])*e^2*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 
4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*(EllipticE[I*ArcSinh 
[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/ 
(-b + Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqr 
t[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])])) 
/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + ( 
I*c*d^2*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(- 
b + Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^ 
2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])])/(Sqr 
t[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) + (4*c*( 
Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a* 
c]/c]/Sqrt[2])*d^3*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x)^2*S 
qrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c 
]/Sqrt[2]) + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c 
) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sq 
rt[2]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b 
^2 - 4*a*c]/c]/Sqrt[2] + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] 
- Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 
4*a*c]/c]/Sqrt[2]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqr...
 

3.1.4.3 Rubi [A] (verified)

Time = 1.87 (sec) , antiderivative size = 814, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {2264, 25, 2279, 27, 1576, 1154, 219, 2232, 25, 27, 1509, 2226, 27, 1416, 2222}

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.

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

\(\Big \downarrow \) 2264

\(\displaystyle -\frac {\int -\frac {c e^3 x^3+c d e^2 x^2-c d^2 e x+d \left (c d^2+b e^2\right )}{(d+e x) \sqrt {c x^4+b x^2+a}}dx}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {c e^3 x^3+c d e^2 x^2-c d^2 e x+d \left (c d^2+b e^2\right )}{(d+e x) \sqrt {c x^4+b x^2+a}}dx}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 2279

\(\displaystyle \frac {\int \frac {-c e^4 x^4+2 c d^2 e^2 x^2+d^2 \left (c d^2+b e^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx+\int \frac {\left (-c e d^3-e \left (c d^2+b e^2\right ) d\right ) x}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-c e^4 x^4+2 c d^2 e^2 x^2+d^2 \left (c d^2+b e^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx-d e \left (b e^2+2 c d^2\right ) \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 1576

\(\displaystyle \frac {\int \frac {-c e^4 x^4+2 c d^2 e^2 x^2+d^2 \left (c d^2+b e^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx-\frac {1}{2} d e \left (b e^2+2 c d^2\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\int \frac {-c e^4 x^4+2 c d^2 e^2 x^2+d^2 \left (c d^2+b e^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx+d e \left (b e^2+2 c d^2\right ) \int \frac {1}{4 \left (c d^4+b e^2 d^2+a e^4\right )-x^4}d\left (-\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{\sqrt {c x^4+b x^2+a}}\right )}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\int \frac {-c e^4 x^4+2 c d^2 e^2 x^2+d^2 \left (c d^2+b e^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx-\frac {d e \left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 2232

\(\displaystyle \frac {-\frac {\int -\frac {c e^2 \left (\left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) d^2+\sqrt {c} e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) x^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{c e^2}-\sqrt {a} \sqrt {c} e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx-\frac {d e \left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {c e^2 \left (\left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) d^2+\sqrt {c} e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) x^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{c e^2}-\sqrt {a} \sqrt {c} e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx-\frac {d e \left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) d^2+\sqrt {c} e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx-\sqrt {c} e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx-\frac {d e \left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\int \frac {\left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) d^2+\sqrt {c} e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx-\sqrt {c} e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )-\frac {d e \left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 2226

\(\displaystyle \frac {\frac {\sqrt {a} d^2 e^2 \left (b e^2+2 c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt {c} \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}-\sqrt {c} e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )-\frac {d e \left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {d^2 e^2 \left (b e^2+2 c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt {c} \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}-\sqrt {c} e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )-\frac {d e \left (b e^2+2 c d^2\right ) \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}}{a e^4+b d^2 e^2+c d^4}-\frac {e^3 \sqrt {a+b x^2+c x^4}}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}\)

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 2222

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

input

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

output

-((e^3*Sqrt[a + b*x^2 + c*x^4])/((c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x))) + 
 (-1/2*(d*e*(2*c*d^2 + b*e^2)*ArcTanh[(b*d^2 + 2*a*e^2 + (2*c*d^2 + b*e^2) 
*x^2)/(2*Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c 
*d^4 + b*d^2*e^2 + a*e^4] - Sqrt[c]*e^2*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sq 
rt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + 
 c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)] 
, (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])) + (c^(1 
/4)*(c*d^4 + b*d^2*e^2 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + 
c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 
 (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt 
[a + b*x^2 + c*x^4]) + (d^2*e^2*(2*c*d^2 + b*e^2)*(((Sqrt[c]*d^2 + Sqrt[a] 
*e^2)*ArcTanh[(Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]*x)/(d*e*Sqrt[a + b*x^2 + c* 
x^4])])/(2*d*e*Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]) + ((Sqrt[a]/d^2 - Sqrt[c]/ 
e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x 
^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2 
), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4) 
*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/(Sqrt[c]*d^2 + Sqrt[a]*e^2))/(c*d^4 + 
b*d^2*e^2 + a*e^4)
 

3.1.4.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1154

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

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

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

rule 1576

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

rule 2222

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

rule 2226

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

rule 2232

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x 
, 2], C = Coeff[P4x, x, 4]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + b*x^2 
 + c*x^4], x], x] + Simp[1/(c*e)   Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - 
 a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b 
, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] 
&&  !GtQ[b^2 - 4*a*c, 0]
 

rule 2264

Int[((d_) + (e_.)*(x_))^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Sy 
mbol] :> Simp[e^3*(d + e*x)^(q + 1)*(Sqrt[a + b*x^2 + c*x^4]/((q + 1)*(c*d^ 
4 + b*d^2*e^2 + a*e^4))), x] + Simp[1/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4)) 
   Int[((d + e*x)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[d*(q + 1)*(c*d^2 + 
b*e^2) - e*(c*d^2*(q + 1) + b*e^2*(q + 2))*x + c*d*e^2*(q + 1)*x^2 - c*e^3* 
(q + 3)*x^3, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c*d^4 + b*d^2*e 
^2 + a*e^4, 0] && ILtQ[q, -1]
 

rule 2279

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x 
_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x 
, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e 
^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4 
)/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e}, 
x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4 
, 0]
 

3.1.4.4 Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 771, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {e^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (e^{4} a +e^{2} d^{2} b +d^{4} c \right ) \left (e x +d \right )}-\frac {c \,d^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \left (e^{4} a +e^{2} d^{2} b +d^{4} c \right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {e^{2} c a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (e^{4} a +e^{2} d^{2} b +d^{4} c \right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {d \left (b \,e^{2}+2 c \,d^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \Pi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{\left (e^{4} a +e^{2} d^{2} b +d^{4} c \right ) e}\)	\(771\)
elliptic	\(-\frac {e^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (e^{4} a +e^{2} d^{2} b +d^{4} c \right ) \left (e x +d \right )}-\frac {c \,d^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \left (e^{4} a +e^{2} d^{2} b +d^{4} c \right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {e^{2} c a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (e^{4} a +e^{2} d^{2} b +d^{4} c \right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {d \left (b \,e^{2}+2 c \,d^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \Pi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{\left (e^{4} a +e^{2} d^{2} b +d^{4} c \right ) e}\)	\(771\)

input

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

output

-e^3*(c*x^4+b*x^2+a)^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)/(e*x+d)-1/4*c*d^2/(a*e^ 
4+b*d^2*e^2+c*d^4)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4* 
a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x 
^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2 
),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/2*e^2*c/(a*e^4+b*d^2*e^ 
2+c*d^4)*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2) 
^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2 
+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2 
)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE 
(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^ 
2)^(1/2))/a/c)^(1/2)))+d*(b*e^2+2*c*d^2)/(a*e^4+b*d^2*e^2+c*d^4)/e*(-1/2/( 
c/e^4*d^4+b/e^2*d^2+a)^(1/2)*arctanh(1/2*(2*c*x^2/e^2*d^2+b/e^2*d^2+b*x^2+ 
2*a)/(c/e^4*d^4+b/e^2*d^2+a)^(1/2)/(c*x^4+b*x^2+a)^(1/2))+2^(1/2)/((-b+(-4 
*a*c+b^2)^(1/2))/a)^(1/2)*e/d*(1-1/2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)* 
(1+1/2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticP 
i(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),2/(-b+(-4*a*c+b^2)^(1/2) 
)*a*e^2/d^2,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2 
)^(1/2))/a)^(1/2)))
 

3.1.4.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.1.4.6 Sympy [F]

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

input

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

output

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

3.1.4.7 Maxima [F]

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

input

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

output

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

3.1.4.8 Giac [F]

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

input

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

output

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

3.1.4.9 Mupad [F(-1)]

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

input

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

output

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