Integrand size = 33, antiderivative size = 752 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx=-\frac {e (B c d+A c e-3 a C e) x \sqrt {d+e x^2}}{4 a c^2}-\frac {B e x \left (d+e x^2\right )^{3/2}}{4 a c}+\frac {x \left (A c-a C+B c x^2\right ) \left (d+e x^2\right )^{5/2}}{4 a c \left (a+c x^4\right )}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (\sqrt {a} c^2 d^3 (B d+A e)+a^{5/2} e^3 (7 C d+4 B e)+3 A c^2 d^3 \sqrt {c d^2+a e^2}-a^2 e^2 (13 C d+4 B e) \sqrt {c d^2+a e^2}+a c d \sqrt {c d^2+a e^2} \left (C d^2+e (2 B d+A e)\right )+a^{3/2} c d e \left (7 C d^2+e (5 B d+A e)\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{7/4} c^{5/2} d \sqrt {c d^2+a e^2}}+\frac {e^{3/2} (5 C d+2 B e) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2}-\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (\left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (A c d \left (3 c d^2+a e^2\right )+a \left (c d^2 (C d+2 B e)-a e^2 (13 C d+4 B e)\right )\right )-d \left (B c d \left (c d^2+7 a e^2\right )+2 e \left (a C \left (4 c d^2-3 a e^2\right )+A c \left (2 c d^2+a e^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} c^{5/2} d \sqrt {c d^2+a e^2}} \] Output:
-1/4*e*(A*c*e+B*c*d-3*C*a*e)*x*(e*x^2+d)^(1/2)/a/c^2-1/4*B*e*x*(e*x^2+d)^( 3/2)/a/c+1/4*x*(B*c*x^2+A*c-C*a)*(e*x^2+d)^(5/2)/a/c/(c*x^4+a)+1/16*(a^(1/ 2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(a^(1/2)*c^2*d^3*(A*e+B*d)+a^(5/2)*e^3*(4* B*e+7*C*d)+3*A*c^2*d^3*(a*e^2+c*d^2)^(1/2)-a^2*e^2*(4*B*e+13*C*d)*(a*e^2+c *d^2)^(1/2)+a*c*d*(a*e^2+c*d^2)^(1/2)*(C*d^2+e*(A*e+2*B*d))+a^(3/2)*c*d*e* (7*C*d^2+e*(A*e+5*B*d)))*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^(7/4)/c^(5/2)/d/(a*e^2+c*d^2)^(1/2)+1/2*e^(3/2)*( 2*B*e+5*C*d)*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/c^2-1/16*(-a^(1/2)*e+(a*e^ 2+c*d^2)^(1/2))^(1/2)*((e+(a*e^2+c*d^2)^(1/2)/a^(1/2))*(A*c*d*(a*e^2+3*c*d ^2)+a*(c*d^2*(2*B*e+C*d)-a*e^2*(4*B*e+13*C*d)))-d*(B*c*d*(7*a*e^2+c*d^2)+2 *e*(a*C*(-3*a*e^2+4*c*d^2)+A*c*(a*e^2+2*c*d^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^(5/2)/d/(a*e^2 +c*d^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.80 (sec) , antiderivative size = 2013, normalized size of antiderivative = 2.68 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx=\text {Result too large to show} \] Input:
Integrate[((d + e*x^2)^(5/2)*(A + B*x^2 + C*x^4))/(a + c*x^4)^2,x]
Output:
((2*c*x*Sqrt[d + e*x^2]*(3*a^2*C*e^2 + B*c^2*d^2*x^2 + A*c*(-(a*e^2) + c*d *(d + 2*e*x^2)) - a*c*(B*e*(2*d + e*x^2) + C*(d^2 + 2*d*e*x^2 - 2*e^2*x^4) )))/(a*(a + c*x^4)) - 4*c*e^(3/2)*(5*C*d + 2*B*e)*Log[-(Sqrt[e]*x) + Sqrt[ d + e*x^2]] + 4*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 & , (3*c^2*C*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e] *x*Sqrt[d + e*x^2] - #1] + 19*B*c^2*d^3*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sq rt[d + e*x^2] - #1] + 49*A*c^2*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[ d + e*x^2] - #1] - 50*a*c*C*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 48*a*B*c*d*e^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 2] - #1] - 16*a*A*c*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1 ] + 16*a^2*C*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2*c ^2*C*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 6*B*c^2* d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 10*A*c^2*d* e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 20*a*c*C*d*e^ 2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 8*a*B*c*e^3*Log [d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 3*c^2*C*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 3*B*c^2*d*e*Log[d + 2*e *x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + A*c^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 2*a*c*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...
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2257 |
| \(\displaystyle \int \left (\frac {\left (d+e x^2\right )^{5/2} \left (-a C+A c+B c x^2\right )}{c \left (a+c x^4\right )^2}+\frac {C \left (d+e x^2\right )^{5/2}}{c \left (a+c x^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(A c-a C) \int \frac {\left (e x^2+d\right )^{5/2}}{\left (c x^4+a\right )^2}dx}{c}+B \int \frac {x^2 \left (e x^2+d\right )^{5/2}}{\left (c x^4+a\right )^2}dx-\frac {C \left (-3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+\sqrt {-a} a e^3+c^{3/2} d^3\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} c^{5/2} \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {C \sqrt {e} \left (-20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{16 \sqrt {-a} c^{5/2}}+\frac {C \sqrt {e} \left (20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{16 \sqrt {-a} c^{5/2}}-\frac {C \left (3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+(-a)^{3/2} e^3+c^{3/2} d^3\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} c^{5/2} \sqrt {\sqrt {-a} e+\sqrt {c} d}}+\frac {C e x \sqrt {d+e x^2} \left (4 \sqrt {-a} e+7 \sqrt {c} d\right )}{16 \sqrt {-a} c^2}+\frac {C e x \sqrt {d+e x^2} \left (7 \sqrt {-a} \sqrt {c} d+4 a e\right )}{16 a c^2}\) |
Input:
Int[((d + e*x^2)^(5/2)*(A + B*x^2 + C*x^4))/(a + c*x^4)^2,x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a , c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(1732\) vs. \(2(644)=1288\).
Time = 2.80 (sec) , antiderivative size = 1733, normalized size of antiderivative = 2.30
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1733\) |
| default | \(\text {Expression too large to display}\) | \(13770\) |
| risch | \(\text {Expression too large to display}\) | \(37570\) |
Input:
int((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*(1/4*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*e^(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)*(((-9/20*c^3*d^3*x^4 *(A*e+B*d)*a^(5/2)-(a*B*(c*x^4+a)*e^4+1/4*d*(c*x^4+a)*(A*c+7*C*a)*e^3+41/2 0*B*c*d^2*(c*x^4+a)*e^2+9/20*c*d^3*(7*C*a+c*(7*C*x^4+A))*e+9/20*B*c^2*d^4) *a^(7/2))*(a*e^2+c*d^2)^(1/2)+1/4*(-4*a^2*B*e^3+a*d*(A*c-13*C*a)*e^2+2*B*a *c*d^2*e+(3*A*c^2+C*a*c)*d^3)*(a*e^2+9/5*c*d^2)*(c*x^4+a)*a^2)*(a*(a*e^2+c *d^2))^(1/2)+1/4*(a*e^2+9/5*c*d^2)*(c*x^4+a)*((A*c*d*e+4*B*a*e^2+B*c*d^2+7 *C*a*d*e)*a^(7/2)*(a*e^2+c*d^2)^(1/2)-(-4*a^2*B*e^3+a*d*(A*c-13*C*a)*e^2+2 *B*a*c*d^2*e+(3*A*c^2+C*a*c)*d^3)*a^3)*e)*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+x^2*(a*e^2+c*d^2)^(1/2))/x ^2)-1/4*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*e^(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)*(((-9/20*c^3*d^3*x^4*( A*e+B*d)*a^(5/2)-(a*B*(c*x^4+a)*e^4+1/4*d*(c*x^4+a)*(A*c+7*C*a)*e^3+41/20* B*c*d^2*(c*x^4+a)*e^2+9/20*c*d^3*(7*C*a+c*(7*C*x^4+A))*e+9/20*B*c^2*d^4)*a ^(7/2))*(a*e^2+c*d^2)^(1/2)+1/4*(-4*a^2*B*e^3+a*d*(A*c-13*C*a)*e^2+2*B*a*c *d^2*e+(3*A*c^2+C*a*c)*d^3)*(a*e^2+9/5*c*d^2)*(c*x^4+a)*a^2)*(a*(a*e^2+c*d ^2))^(1/2)+1/4*(a*e^2+9/5*c*d^2)*(c*x^4+a)*((A*c*d*e+4*B*a*e^2+B*c*d^2+7*C *a*d*e)*a^(7/2)*(a*e^2+c*d^2)^(1/2)-(-4*a^2*B*e^3+a*d*(A*c-13*C*a)*e^2+2*B *a*c*d^2*e+(3*A*c^2+C*a*c)*d^3)*a^3)*e)*ln((a^(1/2)*(e*x^2+d)+x^2*(a*e^2+c *d^2)^(1/2)+(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x)/...
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(c*x^4+a)^2,x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(5/2)*(C*x**4+B*x**2+A)/(c*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (c x^{4} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(c*x^4+a)^2,x, algorithm="maxima ")
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^(5/2)/(c*x^4 + a)^2, x)
Time = 0.19 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(c*x^4+a)^2,x, algorithm="giac")
Output:
1/2*sqrt(e*x^2 + d)*C*e^2*x/c^2 - 1/4*(5*C*d*e^(3/2) + 2*B*e^(5/2))*log((s qrt(e)*x - sqrt(e*x^2 + d))^2)/c^2 - 1/2*((sqrt(e)*x - sqrt(e*x^2 + d))^6* B*c^2*d^3*sqrt(e) - 4*(sqrt(e)*x - sqrt(e*x^2 + d))^6*C*a*c*d^2*e^(3/2) + 4*(sqrt(e)*x - sqrt(e*x^2 + d))^6*A*c^2*d^2*e^(3/2) - 5*(sqrt(e)*x - sqrt( e*x^2 + d))^6*B*a*c*d*e^(5/2) + 2*(sqrt(e)*x - sqrt(e*x^2 + d))^6*C*a^2*e^ (7/2) - 2*(sqrt(e)*x - sqrt(e*x^2 + d))^6*A*a*c*e^(7/2) - 3*(sqrt(e)*x - s qrt(e*x^2 + d))^4*B*c^2*d^4*sqrt(e) + 6*(sqrt(e)*x - sqrt(e*x^2 + d))^4*C* a*c*d^3*e^(3/2) - 6*(sqrt(e)*x - sqrt(e*x^2 + d))^4*A*c^2*d^3*e^(3/2) - 5* (sqrt(e)*x - sqrt(e*x^2 + d))^4*B*a*c*d^2*e^(5/2) + 16*(sqrt(e)*x - sqrt(e *x^2 + d))^4*C*a^2*d*e^(7/2) - 16*(sqrt(e)*x - sqrt(e*x^2 + d))^4*A*a*c*d* e^(7/2) + 8*(sqrt(e)*x - sqrt(e*x^2 + d))^4*B*a^2*e^(9/2) + 3*(sqrt(e)*x - sqrt(e*x^2 + d))^2*B*c^2*d^5*sqrt(e) - 4*(sqrt(e)*x - sqrt(e*x^2 + d))^2* C*a*c*d^4*e^(3/2) + 4*(sqrt(e)*x - sqrt(e*x^2 + d))^2*A*c^2*d^4*e^(3/2) + (sqrt(e)*x - sqrt(e*x^2 + d))^2*B*a*c*d^3*e^(5/2) - 2*(sqrt(e)*x - sqrt(e* x^2 + d))^2*C*a^2*d^2*e^(7/2) + 2*(sqrt(e)*x - sqrt(e*x^2 + d))^2*A*a*c*d^ 2*e^(7/2) - B*c^2*d^6*sqrt(e) + 2*C*a*c*d^5*e^(3/2) - 2*A*c^2*d^5*e^(3/2) + B*a*c*d^4*e^(5/2))/(((sqrt(e)*x - sqrt(e*x^2 + d))^8*c - 4*(sqrt(e)*x - sqrt(e*x^2 + d))^6*c*d + 6*(sqrt(e)*x - sqrt(e*x^2 + d))^4*c*d^2 + 16*(sqr t(e)*x - sqrt(e*x^2 + d))^4*a*e^2 - 4*(sqrt(e)*x - sqrt(e*x^2 + d))^2*c*d^ 3 + c*d^4)*a*c^2)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}\,\left (C\,x^4+B\,x^2+A\right )}{{\left (c\,x^4+a\right )}^2} \,d x \] Input:
int(((d + e*x^2)^(5/2)*(A + B*x^2 + C*x^4))/(a + c*x^4)^2,x)
Output:
int(((d + e*x^2)^(5/2)*(A + B*x^2 + C*x^4))/(a + c*x^4)^2, x)
\[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,d^{2}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{8}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) c \,e^{2}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{6}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) b \,e^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{6}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) c d e +\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,e^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) b d e +\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) c \,d^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a d e +\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) b \,d^{2} \] Input:
int((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(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)*a*d**2 + int((sqrt (d + e*x**2)*x**8)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*c*e**2 + int((sqrt(d + e*x**2)*x**6)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*b*e**2 + 2*int((sqrt(d + e*x**2)*x**6)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*c*d*e + int((sqrt(d + e*x**2)*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*e**2 + 2*int((sqrt(d + e*x**2)*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*b*d*e + int((sqrt(d + e*x **2)*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*c*d**2 + 2*int((sqrt(d + e*x **2)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*d*e + int((sqrt(d + e*x**2 )*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*b*d**2