\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^{11/2}} \, dx\) [145]

Optimal result

Integrand size = 28, antiderivative size = 160 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{3 \left (d+e x^2\right )^{7/2}}+\frac {x \sqrt {d^2-e^2 x^4}}{8 d \left (d+e x^2\right )^{5/2}}+\frac {19 x \sqrt {d^2-e^2 x^4}}{96 d^2 \left (d+e x^2\right )^{3/2}}+\frac {11 \arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{32 \sqrt {2} d^2 \sqrt {e}} \] Output:

1/3*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(7/2)+1/8*x*(-e^2*x^4+d^2)^(1/2)/d/(e 
*x^2+d)^(5/2)+19/96*x*(-e^2*x^4+d^2)^(1/2)/d^2/(e*x^2+d)^(3/2)+11/64*arcta 
n(2^(1/2)*e^(1/2)*x*(e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2))*2^(1/2)/d^2/e^(1 
/2)
 

Mathematica [A] (verified)

Time = 4.92 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d-e x^2} \left (63 d^2+50 d e x^2+19 e^2 x^4\right )+33 \sqrt {2} \left (d+e x^2\right )^3 \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )\right )}{192 d^2 \sqrt {e} \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}} \] Input:

Integrate[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(11/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d - e*x^2]*(63*d^2 + 50*d*e*x^2 + 1 
9*e^2*x^4) + 33*Sqrt[2]*(d + e*x^2)^3*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d - 
e*x^2]]))/(192*d^2*Sqrt[e]*Sqrt[d - e*x^2]*(d + e*x^2)^(7/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1396, 296, 292, 292, 291, 218}

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.

\(\displaystyle \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 296

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {11 \int \frac {\left (d-e x^2\right )^{3/2}}{\left (e x^2+d\right )^3}dx}{12 d}+\frac {x \left (d-e x^2\right )^{5/2}}{12 d^2 \left (d+e x^2\right )^3}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 292

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {11 \left (\frac {3}{4} \int \frac {\sqrt {d-e x^2}}{\left (e x^2+d\right )^2}dx+\frac {x \left (d-e x^2\right )^{3/2}}{4 d \left (d+e x^2\right )^2}\right )}{12 d}+\frac {x \left (d-e x^2\right )^{5/2}}{12 d^2 \left (d+e x^2\right )^3}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 292

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {11 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx+\frac {x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )}\right )+\frac {x \left (d-e x^2\right )^{3/2}}{4 d \left (d+e x^2\right )^2}\right )}{12 d}+\frac {x \left (d-e x^2\right )^{5/2}}{12 d^2 \left (d+e x^2\right )^3}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {11 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\frac {2 d e x^2}{d-e x^2}+d}d\frac {x}{\sqrt {d-e x^2}}+\frac {x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )}\right )+\frac {x \left (d-e x^2\right )^{3/2}}{4 d \left (d+e x^2\right )^2}\right )}{12 d}+\frac {x \left (d-e x^2\right )^{5/2}}{12 d^2 \left (d+e x^2\right )^3}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {11 \left (\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{2 \sqrt {2} d \sqrt {e}}+\frac {x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )}\right )+\frac {x \left (d-e x^2\right )^{3/2}}{4 d \left (d+e x^2\right )^2}\right )}{12 d}+\frac {x \left (d-e x^2\right )^{5/2}}{12 d^2 \left (d+e x^2\right )^3}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

Int[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(11/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*((x*(d - e*x^2)^(5/2))/(12*d^2*(d + e*x^2)^3) + (11*( 
(x*(d - e*x^2)^(3/2))/(4*d*(d + e*x^2)^2) + (3*((x*Sqrt[d - e*x^2])/(2*d*( 
d + e*x^2)) + ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]]/(2*Sqrt[2]*d*Sqr 
t[e])))/4))/(12*d)))/(Sqrt[d - e*x^2]*Sqrt[d + e*x^2])
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 292

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si 
mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( 
a*(p + 1)))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ 
{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt 
Q[q, 0] && NeQ[p, -1]
 

rule 296

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d))   Int[ 
(a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N 
eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1 
]) && NeQ[p, -1]
 

rule 1396

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x 
_Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d 
+ c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, 
x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* 
e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 1] && EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(595\) vs. \(2(128)=256\).

Time = 0.50 (sec) , antiderivative size = 596, normalized size of antiderivative = 3.72

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{3} \left (-33 \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) e^{3} x^{6} \sqrt {d}+33 \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) e^{3} x^{6} \sqrt {d}-99 \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) d^{\frac {3}{2}} e^{2} x^{4}+99 \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) d^{\frac {3}{2}} e^{2} x^{4}+76 e^{2} \sqrt {-e \,x^{2}+d}\, \sqrt {-d e}\, x^{5}-99 \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) d^{\frac {5}{2}} e \,x^{2}+99 \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) d^{\frac {5}{2}} e \,x^{2}+200 e \sqrt {-e \,x^{2}+d}\, d \sqrt {-d e}\, x^{3}-33 \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) d^{\frac {7}{2}}+33 \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) d^{\frac {7}{2}}+252 \sqrt {-e \,x^{2}+d}\, d^{2} \sqrt {-d e}\, x \right )}{384 d^{2} \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \left (e x -\sqrt {-d e}\right )^{3} \left (e x +\sqrt {-d e}\right )^{3} \sqrt {-d e}}\)	\(596\)

Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

1/384*(-e^2*x^4+d^2)^(1/2)*e^3*(-33*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^ 
2+d)^(1/2)-(-d*e)^(1/2)*x+d)/(e*x-(-d*e)^(1/2)))*e^3*x^6*d^(1/2)+33*2^(1/2 
)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2+d)^(1/2)+(-d*e)^(1/2)*x+d)/(e*x+(-d*e)^( 
1/2)))*e^3*x^6*d^(1/2)-99*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2+d)^(1/2) 
-(-d*e)^(1/2)*x+d)/(e*x-(-d*e)^(1/2)))*d^(3/2)*e^2*x^4+99*2^(1/2)*ln(2*e*( 
2^(1/2)*d^(1/2)*(-e*x^2+d)^(1/2)+(-d*e)^(1/2)*x+d)/(e*x+(-d*e)^(1/2)))*d^( 
3/2)*e^2*x^4+76*e^2*(-e*x^2+d)^(1/2)*(-d*e)^(1/2)*x^5-99*2^(1/2)*ln(2*e*(2 
^(1/2)*d^(1/2)*(-e*x^2+d)^(1/2)-(-d*e)^(1/2)*x+d)/(e*x-(-d*e)^(1/2)))*d^(5 
/2)*e*x^2+99*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2+d)^(1/2)+(-d*e)^(1/2) 
*x+d)/(e*x+(-d*e)^(1/2)))*d^(5/2)*e*x^2+200*e*(-e*x^2+d)^(1/2)*d*(-d*e)^(1 
/2)*x^3-33*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2+d)^(1/2)-(-d*e)^(1/2)*x 
+d)/(e*x-(-d*e)^(1/2)))*d^(7/2)+33*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2 
+d)^(1/2)+(-d*e)^(1/2)*x+d)/(e*x+(-d*e)^(1/2)))*d^(7/2)+252*(-e*x^2+d)^(1/ 
2)*d^2*(-d*e)^(1/2)*x)/d^2/(e*x^2+d)^(1/2)/(-e*x^2+d)^(1/2)/(e*x-(-d*e)^(1 
/2))^3/(e*x+(-d*e)^(1/2))^3/(-d*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.70 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\left [-\frac {33 \, \sqrt {2} {\left (e^{4} x^{8} + 4 \, d e^{3} x^{6} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{2} + d^{4}\right )} \sqrt {-e} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - 4 \, {\left (19 \, e^{3} x^{5} + 50 \, d e^{2} x^{3} + 63 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{384 \, {\left (d^{2} e^{5} x^{8} + 4 \, d^{3} e^{4} x^{6} + 6 \, d^{4} e^{3} x^{4} + 4 \, d^{5} e^{2} x^{2} + d^{6} e\right )}}, -\frac {33 \, \sqrt {2} {\left (e^{4} x^{8} + 4 \, d e^{3} x^{6} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{2} + d^{4}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, {\left (19 \, e^{3} x^{5} + 50 \, d e^{2} x^{3} + 63 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{192 \, {\left (d^{2} e^{5} x^{8} + 4 \, d^{3} e^{4} x^{6} + 6 \, d^{4} e^{3} x^{4} + 4 \, d^{5} e^{2} x^{2} + d^{6} e\right )}}\right ] \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="fricas")
 

Output:

[-1/384*(33*sqrt(2)*(e^4*x^8 + 4*d*e^3*x^6 + 6*d^2*e^2*x^4 + 4*d^3*e*x^2 + 
 d^4)*sqrt(-e)*log(-(3*e^2*x^4 + 2*d*e*x^2 - 2*sqrt(2)*sqrt(-e^2*x^4 + d^2 
)*sqrt(e*x^2 + d)*sqrt(-e)*x - d^2)/(e^2*x^4 + 2*d*e*x^2 + d^2)) - 4*(19*e 
^3*x^5 + 50*d*e^2*x^3 + 63*d^2*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/ 
(d^2*e^5*x^8 + 4*d^3*e^4*x^6 + 6*d^4*e^3*x^4 + 4*d^5*e^2*x^2 + d^6*e), -1/ 
192*(33*sqrt(2)*(e^4*x^8 + 4*d*e^3*x^6 + 6*d^2*e^2*x^4 + 4*d^3*e*x^2 + d^4 
)*sqrt(e)*arctan(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(e)*x/(e 
^2*x^4 - d^2)) - 2*(19*e^3*x^5 + 50*d*e^2*x^3 + 63*d^2*e*x)*sqrt(-e^2*x^4 
+ d^2)*sqrt(e*x^2 + d))/(d^2*e^5*x^8 + 4*d^3*e^4*x^6 + 6*d^4*e^3*x^4 + 4*d 
^5*e^2*x^2 + d^6*e)]
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((-e**2*x**4+d**2)**(3/2)/(e*x**2+d)**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="maxima")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^(11/2), x)
 

Giac [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="giac")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{11/2}} \,d x \] Input:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(11/2),x)
 

Output:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(11/2), x)
 

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {13 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, d^{2} x +7 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, d e \,x^{3}+2 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e^{2} x^{5}+44 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{7} e +176 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{6} e^{2} x^{2}+264 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{5} e^{3} x^{4}+176 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{4} e^{4} x^{6}+44 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{6} x^{12}-4 d \,e^{5} x^{10}-5 d^{2} e^{4} x^{8}+5 d^{4} e^{2} x^{4}+4 d^{5} e \,x^{2}+d^{6}}d x \right ) d^{3} e^{5} x^{8}}{13 d^{2} \left (e^{4} x^{8}+4 d \,e^{3} x^{6}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{2}+d^{4}\right )} \] Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(11/2),x)
 

Output:

(13*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*d**2*x + 7*sqrt(d + e*x**2)*sq 
rt(d**2 - e**2*x**4)*d*e*x**3 + 2*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)* 
e**2*x**5 + 44*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**6 + 
4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 4*d*e**5*x**10 - e** 
6*x**12),x)*d**7*e + 176*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2 
)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 4*d*e**5*x 
**10 - e**6*x**12),x)*d**6*e**2*x**2 + 264*int((sqrt(d + e*x**2)*sqrt(d**2 
 - e**2*x**4)*x**2)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2*x**4 - 5*d**2*e**4 
*x**8 - 4*d*e**5*x**10 - e**6*x**12),x)*d**5*e**3*x**4 + 176*int((sqrt(d + 
 e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**6 + 4*d**5*e*x**2 + 5*d**4*e**2* 
x**4 - 5*d**2*e**4*x**8 - 4*d*e**5*x**10 - e**6*x**12),x)*d**4*e**4*x**6 + 
 44*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**6 + 4*d**5*e*x* 
*2 + 5*d**4*e**2*x**4 - 5*d**2*e**4*x**8 - 4*d*e**5*x**10 - e**6*x**12),x) 
*d**3*e**5*x**8)/(13*d**2*(d**4 + 4*d**3*e*x**2 + 6*d**2*e**2*x**4 + 4*d*e 
**3*x**6 + e**4*x**8))