\(\int \frac {1}{(d+e x) (a+b x^2+c x^4)^{3/2}} \, dx\) [270]

Optimal result

Integrand size = 24, antiderivative size = 1010 \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Output:

e*(b*c*d^2+b^2*e^2-2*a*c*e^2+c*(b*e^2+2*c*d^2)*x^2)/(-4*a*c+b^2)/(a*e^4+b* 
d^2*e^2+c*d^4)/(c*x^4+b*x^2+a)^(1/2)+d*x*(c*(-2*a*c+b^2)*d^2+b*(-3*a*c+b^2 
)*e^2+c*(-2*a*c*e^2+b^2*e^2+b*c*d^2)*x^2)/a/(-4*a*c+b^2)/(a*e^4+b*d^2*e^2+ 
c*d^4)/(c*x^4+b*x^2+a)^(1/2)-c^(1/2)*d*(-2*a*c*e^2+b^2*e^2+b*c*d^2)*x*(c*x 
^4+b*x^2+a)^(1/2)/a/(-4*a*c+b^2)/(a*e^4+b*d^2*e^2+c*d^4)/(a^(1/2)+c^(1/2)* 
x^2)+1/2*e^5*arctanh((a*e^4+b*d^2*e^2+c*d^4)^(1/2)*x/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*e^5*arctanh(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)+c^(1/4)*d*(-2*a*c*e^2+b^2*e^2+b*c*d^2)*(a^(1/2) 
+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(si 
n(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/(- 
4*a*c+b^2)/(a*e^4+b*d^2*e^2+c*d^4)/(c*x^4+b*x^2+a)^(1/2)-1/2*c^(3/4)*d*(a^ 
(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*Inverse 
JacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3 
/4)/(b-2*a^(1/2)*c^(1/2))/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)- 
1/4*e^4*(c^(1/2)*d^2-a^(1/2)*e^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/( 
a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))), 
1/4*(c^(1/2)*d^2+a^(1/2)*e^2)^2/a^(1/2)/c^(1/2)/d^2/e^2,1/2*(2-b/a^(1/2)/c 
^(1/2))^(1/2))/a^(1/4)/c^(1/4)/d/(c^(1/2)*d^2+a^(1/2)*e^2)/(a*e^4+b*d^2*e^ 
2+c*d^4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 18.36 (sec) , antiderivative size = 4965, normalized size of antiderivative = 4.92 \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((d + e*x)*(a + b*x^2 + c*x^4)^(3/2)),x]
 

Output:

(-(a*b*c*d^2*e) - a*b^2*e^3 + 2*a^2*c*e^3 - b^2*c*d^3*x + 2*a*c^2*d^3*x - 
b^3*d*e^2*x + 3*a*b*c*d*e^2*x - 2*a*c^2*d^2*e*x^2 - a*b*c*e^3*x^2 - b*c^2* 
d^3*x^3 - b^2*c*d*e^2*x^3 + 2*a*c^2*d*e^2*x^3)/(a*(-b^2 + 4*a*c)*(c*d^4 + 
b*d^2*e^2 + a*e^4)*Sqrt[a + b*x^2 + c*x^4]) + (((I/2)*b*c*(-b + Sqrt[b^2 - 
 4*a*c])*d^3*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 - S 
qrt[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 - Sqrt[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/2)*b^2*(-b + Sqrt[b^2 
- 4*a*c])*d*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 - Sqrt[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*a*c*(-b + Sqrt[b^2 - 
 4*a*c])*d*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 - Sqrt[b^2 - 4*a*c]))]*x]...
 

Rubi [A] (verified)

Time = 3.57 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {2266, 1547, 27, 1576, 1165, 27, 1154, 219, 2206, 27, 1511, 27, 1416, 1509, 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.

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

\(\Big \downarrow \) 2266

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

\(\Big \downarrow \) 1547

\(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} x^4 e^4}{\sqrt {a}}+b e^4+\sqrt {a} \sqrt {c} e^4+c d^2 e^2+\frac {\sqrt {c} \left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) x^2 e^2}{\sqrt {a}}+\frac {\sqrt {c} d^2 \left (c d^2+b e^2\right )}{\sqrt {a}}}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} x^4 e^4}{\sqrt {a}}+b e^4+\sqrt {a} \sqrt {c} e^4+c d^2 e^2+\frac {\sqrt {c} \left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) x^2 e^2}{\sqrt {a}}+\frac {\sqrt {c} d^2 \left (c d^2+b e^2\right )}{\sqrt {a}}}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}}dx\)

\(\Big \downarrow \) 1576

\(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} x^4 e^4}{\sqrt {a}}+b e^4+\sqrt {a} \sqrt {c} e^4+c d^2 e^2+\frac {\sqrt {c} \left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) x^2 e^2}{\sqrt {a}}+\frac {\sqrt {c} d^2 \left (c d^2+b e^2\right )}{\sqrt {a}}}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \int \frac {1}{\left (d^2-e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}}dx^2\)

\(\Big \downarrow \) 1165

\(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} x^4 e^4}{\sqrt {a}}+b e^4+\sqrt {a} \sqrt {c} e^4+c d^2 e^2+\frac {\sqrt {c} \left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) x^2 e^2}{\sqrt {a}}+\frac {\sqrt {c} d^2 \left (c d^2+b e^2\right )}{\sqrt {a}}}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {2 \int -\frac {\left (b^2-4 a c\right ) e^4}{2 \left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{\left (b^2-4 a c\right ) \left (a e^4+b d^2 e^2+c d^4\right )}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} x^4 e^4}{\sqrt {a}}+b e^4+\sqrt {a} \sqrt {c} e^4+c d^2 e^2+\frac {\sqrt {c} \left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) x^2 e^2}{\sqrt {a}}+\frac {\sqrt {c} d^2 \left (c d^2+b e^2\right )}{\sqrt {a}}}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2}{a e^4+b d^2 e^2+c d^4}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} x^4 e^4}{\sqrt {a}}+b e^4+\sqrt {a} \sqrt {c} e^4+c d^2 e^2+\frac {\sqrt {c} \left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) x^2 e^2}{\sqrt {a}}+\frac {\sqrt {c} d^2 \left (c d^2+b e^2\right )}{\sqrt {a}}}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (-\frac {2 e^4 \int \frac {1}{4 \left (c d^4+b e^2 d^2+a e^4\right )-x^4}d\left (-\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{\sqrt {c x^4+b x^2+a}}\right )}{a e^4+b d^2 e^2+c d^4}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle d \left (\frac {\int \frac {\frac {c^{3/2} x^4 e^4}{\sqrt {a}}+b e^4+\sqrt {a} \sqrt {c} e^4+c d^2 e^2+\frac {\sqrt {c} \left (c d^2+b e^2+\sqrt {a} \sqrt {c} e^2\right ) x^2 e^2}{\sqrt {a}}+\frac {\sqrt {c} d^2 \left (c d^2+b e^2\right )}{\sqrt {a}}}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{\left (a e^4+b d^2 e^2+c d^4\right )^{3/2}}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle d \left (\frac {\frac {x \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (c x^2 \left (-2 a c e^2+b^2 e^2+b c d^2\right )-3 a b c e^2-2 a c^2 d^2+b^3 e^2+b^2 c d^2\right )}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {\sqrt {c} \left (\sqrt {c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (b c d^2+b^2 e^2-2 a c e^2\right ) x^2+a \left (2 c^2 d^4+2 \sqrt {a} c^{3/2} e^2 d^2+b c e^2 d^2-b^2 e^4+4 a c e^4+\sqrt {a} b \sqrt {c} e^4\right )\right )}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{\left (a e^4+b d^2 e^2+c d^4\right )^{3/2}}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {\frac {x \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (c x^2 \left (-2 a c e^2+b^2 e^2+b c d^2\right )-3 a b c e^2-2 a c^2 d^2+b^3 e^2+b^2 c d^2\right )}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \int \frac {\sqrt {c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (b c d^2+b^2 e^2-2 a c e^2\right ) x^2+a \left (2 c^2 d^4+2 \sqrt {a} c^{3/2} e^2 d^2+b c e^2 d^2-b^2 e^4+4 a c e^4+\sqrt {a} b \sqrt {c} e^4\right )}{\sqrt {c x^4+b x^2+a}}dx}{a^{3/2} \left (b^2-4 a c\right )}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{\left (a e^4+b d^2 e^2+c d^4\right )^{3/2}}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 1511

\(\displaystyle d \left (\frac {\frac {x \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (c x^2 \left (-2 a c e^2+b^2 e^2+b c d^2\right )-3 a b c e^2-2 a c^2 d^2+b^3 e^2+b^2 c d^2\right )}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} \sqrt {c} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\sqrt {a} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (-2 a c e^2+b^2 e^2+b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx\right )}{a^{3/2} \left (b^2-4 a c\right )}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{\left (a e^4+b d^2 e^2+c d^4\right )^{3/2}}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {\frac {x \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (c x^2 \left (-2 a c e^2+b^2 e^2+b c d^2\right )-3 a b c e^2-2 a c^2 d^2+b^3 e^2+b^2 c d^2\right )}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\sqrt {a} \sqrt {c} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (a e^4+b d^2 e^2+c d^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (-2 a c e^2+b^2 e^2+b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx\right )}{a^{3/2} \left (b^2-4 a c\right )}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{\left (a e^4+b d^2 e^2+c d^4\right )^{3/2}}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 1416

\(\displaystyle d \left (\frac {\frac {x \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (c x^2 \left (-2 a c e^2+b^2 e^2+b c d^2\right )-3 a b c e^2-2 a c^2 d^2+b^3 e^2+b^2 c d^2\right )}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^4+b d^2 e^2+c d^4\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt {a+b x^2+c x^4}}-\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (-2 a c e^2+b^2 e^2+b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx\right )}{a^{3/2} \left (b^2-4 a c\right )}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^6 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{\left (a e^4+b d^2 e^2+c d^4\right )^{3/2}}-\frac {2 \left (-2 a c e^2+b^2 e^2+c x^2 \left (b e^2+2 c d^2\right )+b c d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^4+b d^2 e^2+c d^4\right )}\right )\)

\(\Big \downarrow \) 1509

\(\displaystyle d \left (\frac {\int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx e^6}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right )}+\frac {\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) x \left (e^2 b^3+c d^2 b^2-3 a c e^2 b-2 a c^2 d^2+c \left (b c d^2+b^2 e^2-2 a c e^2\right ) x^2\right )}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {c x^4+b x^2+a}}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \sqrt [4]{c} \left (c d^4+b e^2 d^2+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt {c x^4+b x^2+a}}-\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (b c d^2+b^2 e^2-2 a c e^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {c x^4+b x^2+a}}-\frac {x \sqrt {c x^4+b x^2+a}}{\sqrt {c} x^2+\sqrt {a}}\right )\right )}{a^{3/2} \left (b^2-4 a c\right )}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right )}{\left (c d^4+b e^2 d^2+a e^4\right )^{3/2}}-\frac {2 \left (b c d^2+b^2 e^2-2 a c e^2+c \left (2 c d^2+b e^2\right ) x^2\right )}{\left (b^2-4 a c\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \sqrt {c x^4+b x^2+a}}\right )\)

\(\Big \downarrow \) 2222

\(\displaystyle d \left (\frac {\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^6}{\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right )}+\frac {\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) x \left (e^2 b^3+c d^2 b^2-3 a c e^2 b-2 a c^2 d^2+c \left (b c d^2+b^2 e^2-2 a c e^2\right ) x^2\right )}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {c x^4+b x^2+a}}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \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 {c x^4+b x^2+a}}-\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (b c d^2+b^2 e^2-2 a c e^2\right ) \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 )\right )}{a^{3/2} \left (b^2-4 a c\right )}}{\left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right )}\right )-\frac {1}{2} e \left (\frac {e^4 \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 )}{\left (c d^4+b e^2 d^2+a e^4\right )^{3/2}}-\frac {2 \left (b c d^2+b^2 e^2-2 a c e^2+c \left (2 c d^2+b e^2\right ) x^2\right )}{\left (b^2-4 a c\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \sqrt {c x^4+b x^2+a}}\right )\)

Input:

Int[1/((d + e*x)*(a + b*x^2 + c*x^4)^(3/2)),x]
 

Output:

-1/2*(e*((-2*(b*c*d^2 + b^2*e^2 - 2*a*c*e^2 + c*(2*c*d^2 + b*e^2)*x^2))/(( 
b^2 - 4*a*c)*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[a + b*x^2 + c*x^4]) + (e^4*A 
rcTanh[(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])])/(c*d^4 + b*d^2*e^2 + a*e^4)^(3/2))) + 
 d*((((Sqrt[c]*d^2 + Sqrt[a]*e^2)*x*(b^2*c*d^2 - 2*a*c^2*d^2 + b^3*e^2 - 3 
*a*b*c*e^2 + c*(b*c*d^2 + b^2*e^2 - 2*a*c*e^2)*x^2))/(a^(3/2)*(b^2 - 4*a*c 
)*Sqrt[a + b*x^2 + c*x^4]) - (Sqrt[c]*(-((Sqrt[c]*d^2 + Sqrt[a]*e^2)*(b*c* 
d^2 + b^2*e^2 - 2*a*c*e^2)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[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/(Sqr 
t[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4]))) + (a^(1/4)*(b + 2*S 
qrt[a]*Sqrt[c])*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*Sqrt[a + b*x^2 + c*x 
^4])))/(a^(3/2)*(b^2 - 4*a*c)))/(((Sqrt[c]*d^2)/Sqrt[a] + e^2)*(c*d^4 + b* 
d^2*e^2 + a*e^4)) + (e^6*(((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...
 

Defintions of rubi rules used

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 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, 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 1511

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

rule 1547

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_.) + (e_.)*(x_)^2), x_Sym 
bol] :> Simp[-(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(e^(2*p)*(Rt[c/a, 2]*d - e) 
)   Int[(1 + Rt[c/a, 2]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] 
+ Simp[(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(Rt[c/a, 2]*d - e)   Int[(a + b*x^ 
2 + c*x^4)^p*ExpandToSum[((Rt[c/a, 2]*d - e)*(c*d^2 - b*d*e + a*e^2)^(-p - 
1/2) + ((1 + Rt[c/a, 2]*x^2)*(a + b*x^2 + c*x^4)^(-p - 1/2))/e^(2*p))/(d + 
e*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N 
eQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0] && NeQ[c*d^2 - a*e^2, 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 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

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 2266

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbo 
l] :> Simp[d   Int[(a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2), x], x] - Simp[e 
 Int[x*((a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2)), x], x] /; FreeQ[{a, b, c, d 
, e}, x] && IntegerQ[p + 1/2]
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 1111, normalized size of antiderivative = 1.10

method	result	size
default	\(\text {Expression too large to display}\)	\(1111\)
elliptic	\(\text {Expression too large to display}\)	\(1111\)

Input:

int(1/(e*x+d)/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(e*x+d)/(c*x**4+b*x**2+a)**(3/2),x)
 

Output:

Integral(1/((d + e*x)*(a + b*x**2 + c*x**4)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*(e*x + d)), x)
 

Giac [F]

\[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*(e*x + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int(1/((d + e*x)*(a + b*x^2 + c*x^4)^(3/2)),x)
 

Output:

int(1/((d + e*x)*(a + b*x^2 + c*x^4)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (e x +d \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:

int(1/(e*x+d)/(c*x^4+b*x^2+a)^(3/2),x)
 

Output:

int(1/(e*x+d)/(c*x^4+b*x^2+a)^(3/2),x)