\(\int \frac {\sqrt {d^2-e^2 x^4}}{(d+e x^2)^{7/2}} \, dx\) [136]

Optimal result

Integrand size = 28, antiderivative size = 128 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{4 d \left (d+e x^2\right )^{5/2}}+\frac {5 x \sqrt {d^2-e^2 x^4}}{16 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 \arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{16 \sqrt {2} d^2 \sqrt {e}} \] Output:

1/4*x*(-e^2*x^4+d^2)^(1/2)/d/(e*x^2+d)^(5/2)+5/16*x*(-e^2*x^4+d^2)^(1/2)/d 
^2/(e*x^2+d)^(3/2)+7/32*arctan(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 = 3.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d-e x^2} \left (9 d+5 e x^2\right )+7 \sqrt {2} \left (d+e x^2\right )^2 \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )\right )}{32 d^2 \sqrt {e} \sqrt {d-e x^2} \left (d+e x^2\right )^{5/2}} \] Input:

Integrate[Sqrt[d^2 - e^2*x^4]/(d + e*x^2)^(7/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d - e*x^2]*(9*d + 5*e*x^2) + 7*Sqrt 
[2]*(d + e*x^2)^2*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]]))/(32*d^2*Sq 
rt[e]*Sqrt[d - e*x^2]*(d + e*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1396, 296, 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.

\(\displaystyle \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 296

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7 \int \frac {\sqrt {d-e x^2}}{\left (e x^2+d\right )^2}dx}{8 d}+\frac {x \left (d-e x^2\right )^{3/2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 292

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7 \left (\frac {1}{2} \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx+\frac {x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )}\right )}{8 d}+\frac {x \left (d-e x^2\right )^{3/2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7 \left (\frac {1}{2} \int \frac {1}{\frac {2 d e x^2}{d-e x^2}+d}d\frac {x}{\sqrt {d-e x^2}}+\frac {x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )}\right )}{8 d}+\frac {x \left (d-e x^2\right )^{3/2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7 \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{2 \sqrt {2} d \sqrt {e}}+\frac {x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )}\right )}{8 d}+\frac {x \left (d-e x^2\right )^{3/2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

Int[Sqrt[d^2 - e^2*x^4]/(d + e*x^2)^(7/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*((x*(d - e*x^2)^(3/2))/(8*d^2*(d + e*x^2)^2) + (7*((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*Sqrt[e])))/(8*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. \(451\) vs. \(2(102)=204\).

Time = 0.46 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.53

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{2} \left (-7 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, e^{2} x^{4} \sqrt {d}+7 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, e^{2} x^{4} \sqrt {d}-14 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {3}{2}} e \,x^{2}+14 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {3}{2}} e \,x^{2}+20 \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, e \,x^{3}-7 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {5}{2}}+7 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, d^{\frac {5}{2}}+36 d \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x \right )}{64 d^{2} \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \left (e x -\sqrt {-d e}\right )^{2} \left (e x +\sqrt {-d e}\right )^{2} \sqrt {-d e}}\)	\(452\)

Input:

int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/64*(-e^2*x^4+d^2)^(1/2)*e^2*(-7*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)))*2^(1/2)*e^2*x^4*d^(1/2)+7*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)))*2^(1 
/2)*e^2*x^4*d^(1/2)-14*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)))*2^(1/2)*d^(3/2)*e*x^2+14*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)))*2^(1/2)*d^(3/2)* 
e*x^2+20*(-d*e)^(1/2)*(-e*x^2+d)^(1/2)*e*x^3-7*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)))*2^(1/2)*d^(5/2)+7*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))) 
*2^(1/2)*d^(5/2)+36*d*(-d*e)^(1/2)*(-e*x^2+d)^(1/2)*x)/d^2/(e*x^2+d)^(1/2) 
/(-e*x^2+d)^(1/2)/(e*x-(-d*e)^(1/2))^2/(e*x+(-d*e)^(1/2))^2/(-d*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.86 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\left [-\frac {7 \, \sqrt {2} {\left (e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}\right )} \sqrt {-e} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - 4 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (5 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {e x^{2} + d}}{64 \, {\left (d^{2} e^{4} x^{6} + 3 \, d^{3} e^{3} x^{4} + 3 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}, -\frac {7 \, \sqrt {2} {\left (e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (5 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {e x^{2} + d}}{32 \, {\left (d^{2} e^{4} x^{6} + 3 \, d^{3} e^{3} x^{4} + 3 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}\right ] \] Input:

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="fricas")
 

Output:

[-1/64*(7*sqrt(2)*(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3)*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*sqrt(-e^2*x^4 + d^2)*(5 
*e^2*x^3 + 9*d*e*x)*sqrt(e*x^2 + d))/(d^2*e^4*x^6 + 3*d^3*e^3*x^4 + 3*d^4* 
e^2*x^2 + d^5*e), -1/32*(7*sqrt(2)*(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + 
d^3)*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*sqrt(-e^2*x^4 + d^2)*(5*e^2*x^3 + 9*d*e*x)*sqrt(e*x^ 
2 + d))/(d^2*e^4*x^6 + 3*d^3*e^3*x^4 + 3*d^4*e^2*x^2 + d^5*e)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )}}{\left (d + e x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((-e**2*x**4+d**2)**(1/2)/(e*x**2+d)**(7/2),x)
 

Output:

Integral(sqrt(-(-d + e*x**2)*(d + e*x**2))/(d + e*x**2)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {-e^{2} x^{4} + d^{2}}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-e^2*x^4 + d^2)/(e*x^2 + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {-e^{2} x^{4} + d^{2}}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-e^2*x^4 + d^2)/(e*x^2 + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {d^2-e^2\,x^4}}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:

int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2)^(7/2),x)
 

Output:

int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {5 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, d x +2 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e \,x^{3}+14 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{5} x^{10}-3 d \,e^{4} x^{8}-2 d^{2} e^{3} x^{6}+2 d^{3} e^{2} x^{4}+3 d^{4} e \,x^{2}+d^{5}}d x \right ) d^{5} e +42 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{5} x^{10}-3 d \,e^{4} x^{8}-2 d^{2} e^{3} x^{6}+2 d^{3} e^{2} x^{4}+3 d^{4} e \,x^{2}+d^{5}}d x \right ) d^{4} e^{2} x^{2}+42 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{5} x^{10}-3 d \,e^{4} x^{8}-2 d^{2} e^{3} x^{6}+2 d^{3} e^{2} x^{4}+3 d^{4} e \,x^{2}+d^{5}}d x \right ) d^{3} e^{3} x^{4}+14 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{5} x^{10}-3 d \,e^{4} x^{8}-2 d^{2} e^{3} x^{6}+2 d^{3} e^{2} x^{4}+3 d^{4} e \,x^{2}+d^{5}}d x \right ) d^{2} e^{4} x^{6}}{5 d^{2} \left (e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}\right )} \] Input:

int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(7/2),x)
 

Output:

(5*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*d*x + 2*sqrt(d + e*x**2)*sqrt(d 
**2 - e**2*x**4)*e*x**3 + 14*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)* 
x**2)/(d**5 + 3*d**4*e*x**2 + 2*d**3*e**2*x**4 - 2*d**2*e**3*x**6 - 3*d*e* 
*4*x**8 - e**5*x**10),x)*d**5*e + 42*int((sqrt(d + e*x**2)*sqrt(d**2 - e** 
2*x**4)*x**2)/(d**5 + 3*d**4*e*x**2 + 2*d**3*e**2*x**4 - 2*d**2*e**3*x**6 
- 3*d*e**4*x**8 - e**5*x**10),x)*d**4*e**2*x**2 + 42*int((sqrt(d + e*x**2) 
*sqrt(d**2 - e**2*x**4)*x**2)/(d**5 + 3*d**4*e*x**2 + 2*d**3*e**2*x**4 - 2 
*d**2*e**3*x**6 - 3*d*e**4*x**8 - e**5*x**10),x)*d**3*e**3*x**4 + 14*int(( 
sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**2)/(d**5 + 3*d**4*e*x**2 + 2*d* 
*3*e**2*x**4 - 2*d**2*e**3*x**6 - 3*d*e**4*x**8 - e**5*x**10),x)*d**2*e**4 
*x**6)/(5*d**2*(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6))