Integrand size = 21, antiderivative size = 473 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {e^2 x \sqrt {d+e x^2}}{4 a c}+\frac {x \left (d+e x^2\right )^{5/2}}{4 a \left (a+c x^4\right )}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (4 c d^2 e+2 a e^3-\left (3 c d^2+a e^2\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} c^{3/2} \sqrt {c d^2+a e^2}}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (4 c d^2 e+2 a e^3-\left (3 c d^2+a e^2\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} c^{3/2} \sqrt {c d^2+a e^2}} \] Output:
-1/4*e^2*x*(e*x^2+d)^(1/2)/a/c+1/4*x*(e*x^2+d)^(5/2)/a/(c*x^4+a)+1/16*(a^( 1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(4*c*d^2*e+2*a*e^3-(a*e^2+3*c*d^2)*(e-(a *e^2+c*d^2)^(1/2)/a^(1/2)))*arctan(2^(1/2)*a^(1/4)*c^(1/2)*(a^(1/2)*e+(a*e ^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2) ^(1/2))-c*d*x^2))*2^(1/2)/a^(5/4)/c^(3/2)/(a*e^2+c*d^2)^(1/2)+1/16*(-a^(1/ 2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(4*c*d^2*e+2*a*e^3-(a*e^2+3*c*d^2)*(e+(a*e ^2+c*d^2)^(1/2)/a^(1/2)))*arctanh(2^(1/2)*a^(1/4)*c^(1/2)*(-a^(1/2)*e+(a*e ^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e-(a*e^2+c*d^2) ^(1/2))-c*d*x^2))*2^(1/2)/a^(5/4)/c^(3/2)/(a*e^2+c*d^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.57 (sec) , antiderivative size = 708, normalized size of antiderivative = 1.50 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx=\frac {x \sqrt {d+e x^2} \left (c d^2-a e^2+2 c d e x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {49 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-16 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+10 c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{2 c^2}+\frac {e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {2 c^2 d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-97 a c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+32 a^2 e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+2 c^2 d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-20 a c d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 c^2 d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a c e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{4 a c^2} \] Input:
Integrate[(d + e*x^2)^(5/2)/(a + c*x^4)^2,x]
Output:
(x*Sqrt[d + e*x^2]*(c*d^2 - a*e^2 + 2*c*d*e*x^2))/(4*a*c*(a + c*x^4)) + (e ^(7/2)*RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 + 16*a*e^2*#1^2 - 4*c*d*# 1^3 + c*#1^4 & , (49*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 16*a*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 10*c *d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*Log[d + 2*e* x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2)/(c*d^3 - 3*c*d^2*#1 - 8*a*e^ 2*#1 + 3*c*d*#1^2 - c*#1^3) & ])/(2*c^2) + (e^(3/2)*RootSum[c*d^4 - 4*c*d^ 3*#1 + 6*c*d^2*#1^2 + 16*a*e^2*#1^2 - 4*c*d*#1^3 + c*#1^4 & , (2*c^2*d^4*L og[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 97*a*c*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 32*a^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 2*c^2*d^3*Log[d + 2*e*x^2 - 2*Sqrt[ e]*x*Sqrt[d + e*x^2] - #1]*#1 - 20*a*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x *Sqrt[d + e*x^2] - #1]*#1 + 2*c^2*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - a*c*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 2] - #1]*#1^2)/(c*d^3 - 3*c*d^2*#1 - 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ] )/(4*a*c^2)
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2}dx\) |
Input:
Int[(d + e*x^2)^(5/2)/(a + c*x^4)^2,x]
Output:
$Aborted
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(870\) vs. \(2(385)=770\).
Time = 1.17 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.84
| method | result | size |
| pseudoelliptic | \(-\frac {\frac {d^{2} \left (\left (a^{\frac {9}{2}}+c \,x^{4} a^{\frac {7}{2}}\right ) e \sqrt {a \,e^{2}+c \,d^{2}}+a^{3} \left (c \,x^{4}+a \right ) \left (a \,e^{2}+3 c \,d^{2}\right )\right ) c \arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )}{2}-\frac {d^{2} \left (\left (a^{\frac {9}{2}}+c \,x^{4} a^{\frac {7}{2}}\right ) e \sqrt {a \,e^{2}+c \,d^{2}}+a^{3} \left (c \,x^{4}+a \right ) \left (a \,e^{2}+3 c \,d^{2}\right )\right ) c \arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )}{2}+\left (\frac {\left (\left (-e \left (c \,x^{4} a^{\frac {5}{2}}+a^{\frac {7}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+a^{2} \left (c \,x^{4}+a \right ) \left (a \,e^{2}+3 c \,d^{2}\right )\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+e \left (\left (a^{\frac {9}{2}}+c \,x^{4} a^{\frac {7}{2}}\right ) e \sqrt {a \,e^{2}+c \,d^{2}}-a^{3} \left (c \,x^{4}+a \right ) \left (a \,e^{2}+3 c \,d^{2}\right )\right )\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )-\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{8}-\frac {\left (\left (-e \left (c \,x^{4} a^{\frac {5}{2}}+a^{\frac {7}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+a^{2} \left (c \,x^{4}+a \right ) \left (a \,e^{2}+3 c \,d^{2}\right )\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+e \left (\left (a^{\frac {9}{2}}+c \,x^{4} a^{\frac {7}{2}}\right ) e \sqrt {a \,e^{2}+c \,d^{2}}-a^{3} \left (c \,x^{4}+a \right ) \left (a \,e^{2}+3 c \,d^{2}\right )\right )\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{8}+\left (-c d \left (2 e \,x^{2}+d \right ) a^{\frac {7}{2}}+a^{\frac {9}{2}} e^{2}\right ) x d c \sqrt {e \,x^{2}+d}\right ) \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}{4 a^{\frac {9}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, d \,c^{2} \left (c \,x^{4}+a \right )}\) | \(871\) |
| default | \(\text {Expression too large to display}\) | \(13518\) |
Input:
int((e*x^2+d)^(5/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/4/a^(9/2)/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a* e)^(1/2)*(1/2*d^2*((a^(9/2)+c*x^4*a^(7/2))*e*(a*e^2+c*d^2)^(1/2)+a^3*(c*x^ 4+a)*(a*e^2+3*c*d^2))*c*arctan((2*a^(1/2)*(e*x^2+d)^(1/2)+(2*(a*(a*e^2+c*d ^2))^(1/2)+2*a*e)^(1/2)*x)/x/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c* d^2))^(1/2)-2*a*e)^(1/2))-1/2*d^2*((a^(9/2)+c*x^4*a^(7/2))*e*(a*e^2+c*d^2) ^(1/2)+a^3*(c*x^4+a)*(a*e^2+3*c*d^2))*c*arctan(((2*(a*(a*e^2+c*d^2))^(1/2) +2*a*e)^(1/2)*x-2*a^(1/2)*(e*x^2+d)^(1/2))/x/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2 )-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2))+(1/8*((-e*(c*x^4*a^(5/2)+a^(7/2) )*(a*e^2+c*d^2)^(1/2)+a^2*(c*x^4+a)*(a*e^2+3*c*d^2))*(a*(a*e^2+c*d^2))^(1/ 2)+e*((a^(9/2)+c*x^4*a^(7/2))*e*(a*e^2+c*d^2)^(1/2)-a^3*(c*x^4+a)*(a*e^2+3 *c*d^2)))*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*ln((a^(1/2)*(e*x^2+d)-(e *x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2 )*x^2)/x^2)-1/8*((-e*(c*x^4*a^(5/2)+a^(7/2))*(a*e^2+c*d^2)^(1/2)+a^2*(c*x^ 4+a)*(a*e^2+3*c*d^2))*(a*(a*e^2+c*d^2))^(1/2)+e*((a^(9/2)+c*x^4*a^(7/2))*e *(a*e^2+c*d^2)^(1/2)-a^3*(c*x^4+a)*(a*e^2+3*c*d^2)))*(2*(a*(a*e^2+c*d^2))^ (1/2)+2*a*e)^(1/2)*ln((a^(1/2)*(e*x^2+d)+(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^ 2))^(1/2)+2*a*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)+(-c*d*(2*e*x^2+d)*a ^(7/2)+a^(9/2)*e^2)*x*d*c*(e*x^2+d)^(1/2))*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)- 2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2))/d/c^2/(c*x^4+a)
Leaf count of result is larger than twice the leaf count of optimal. 1951 vs. \(2 (387) = 774\).
Time = 10.82 (sec) , antiderivative size = 1951, normalized size of antiderivative = 4.12 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(5/2)/(c*x^4+a)^2,x, algorithm="fricas")
Output:
1/32*((a*c^2*x^4 + a^2*c)*sqrt(-(15*c^2*d^4*e + 15*a*c*d^2*e^3 + 4*a^2*e^5 + a^3*c^3*sqrt(-(81*c^2*d^10 + 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3) ))/(a^3*c^3))*log((81*c^3*d^10 + 162*a*c^2*d^8*e^2 + 101*a^2*c*d^6*e^4 + 2 0*a^3*d^4*e^6 + (9*a^3*c^4*d^5 + 13*a^4*c^3*d^3*e^2 + 4*a^5*c^2*d*e^4)*x^2 *sqrt(-(81*c^2*d^10 + 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)) + 2*(81* c^3*d^9*e + 162*a*c^2*d^7*e^3 + 101*a^2*c*d^5*e^5 + 20*a^3*d^3*e^7)*x^2 + 2*sqrt(e*x^2 + d)*((3*a^5*c^4*d^2 + 2*a^6*c^3*e^2)*x*sqrt(-(81*c^2*d^10 + 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)) - (9*a^2*c^3*d^6*e + 5*a^3*c^2 *d^4*e^3)*x)*sqrt(-(15*c^2*d^4*e + 15*a*c*d^2*e^3 + 4*a^2*e^5 + a^3*c^3*sq rt(-(81*c^2*d^10 + 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)))/(a^3*c^3)) )/x^2) - (a*c^2*x^4 + a^2*c)*sqrt(-(15*c^2*d^4*e + 15*a*c*d^2*e^3 + 4*a^2* e^5 + a^3*c^3*sqrt(-(81*c^2*d^10 + 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c ^3)))/(a^3*c^3))*log((81*c^3*d^10 + 162*a*c^2*d^8*e^2 + 101*a^2*c*d^6*e^4 + 20*a^3*d^4*e^6 + (9*a^3*c^4*d^5 + 13*a^4*c^3*d^3*e^2 + 4*a^5*c^2*d*e^4)* x^2*sqrt(-(81*c^2*d^10 + 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)) + 2*( 81*c^3*d^9*e + 162*a*c^2*d^7*e^3 + 101*a^2*c*d^5*e^5 + 20*a^3*d^3*e^7)*x^2 - 2*sqrt(e*x^2 + d)*((3*a^5*c^4*d^2 + 2*a^6*c^3*e^2)*x*sqrt(-(81*c^2*d^10 + 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)) - (9*a^2*c^3*d^6*e + 5*a^3* c^2*d^4*e^3)*x)*sqrt(-(15*c^2*d^4*e + 15*a*c*d^2*e^3 + 4*a^2*e^5 + a^3*c^3 *sqrt(-(81*c^2*d^10 + 90*a*c*d^8*e^2 + 25*a^2*d^6*e^4)/(a^7*c^3)))/(a^3...
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {5}{2}}}{\left (a + c x^{4}\right )^{2}}\, dx \] Input:
integrate((e*x**2+d)**(5/2)/(c*x**4+a)**2,x)
Output:
Integral((d + e*x**2)**(5/2)/(a + c*x**4)**2, x)
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (c x^{4} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(5/2)/(c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(5/2)/(c*x^4 + a)^2, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(5/2)/(c*x^4+a)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{{\left (c\,x^4+a\right )}^2} \,d x \] Input:
int((d + e*x^2)^(5/2)/(a + c*x^4)^2,x)
Output:
int((d + e*x^2)^(5/2)/(a + c*x^4)^2, x)
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^2} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) d^{2}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) e^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) d e \] Input:
int((e*x^2+d)^(5/2)/(c*x^4+a)^2,x)
Output:
int(sqrt(d + e*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*d**2 + int((sqrt(d + e*x**2)*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*e**2 + 2*int((sqrt(d + e*x**2)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*d*e