\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^{3/2}} \, dx\) [141]

Optimal result

Integrand size = 28, antiderivative size = 113 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {3 d x \sqrt {d^2-e^2 x^4}}{8 \sqrt {d+e x^2}}+\frac {x \left (d^2-e^2 x^4\right )^{3/2}}{4 \left (d+e x^2\right )^{3/2}}+\frac {3 d^2 \arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{8 \sqrt {e}} \] Output:

3/8*d*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+1/4*x*(-e^2*x^4+d^2)^(3/2)/(e 
*x^2+d)^(3/2)+3/8*d^2*arctan(e^(1/2)*x*(e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2 
))/e^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {1}{8} \left (\frac {x \left (5 d-2 e x^2\right ) \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}+\frac {3 i d^2 \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}\right ) \] Input:

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

Output:

((x*(5*d - 2*e*x^2)*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2] + ((3*I)*d^2*Log[ 
(-2*I)*Sqrt[e]*x + (2*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2]])/Sqrt[e])/8
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1396, 211, 211, 224, 216}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*((x*(d - e*x^2)^(3/2))/4 + (3*d*((x*Sqrt[d - e*x^2])/ 
2 + (d*ArcTan[(Sqrt[e]*x)/Sqrt[d - e*x^2]])/(2*Sqrt[e])))/4))/(Sqrt[d - e* 
x^2]*Sqrt[d + e*x^2])
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

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

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 [A] (verified)

Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (-2 e^{\frac {3}{2}} x^{3} \sqrt {-e \,x^{2}+d}+5 \sqrt {e}\, \sqrt {-e \,x^{2}+d}\, d x +3 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d^{2}\right )}{8 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}}\)	\(96\)
risch	\(\frac {x \left (-2 e \,x^{2}+5 d \right ) \sqrt {-e \,x^{2}+d}\, \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{8 \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {3 d^{2} \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{8 \sqrt {e}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(143\)

Input:

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

Output:

1/8*(-e^2*x^4+d^2)^(1/2)*(-2*e^(3/2)*x^3*(-e*x^2+d)^(1/2)+5*e^(1/2)*(-e*x^ 
2+d)^(1/2)*d*x+3*arctan(e^(1/2)*x/(-e*x^2+d)^(1/2))*d^2)/(e*x^2+d)^(1/2)/( 
-e*x^2+d)^(1/2)/e^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.24 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (d^{2} e x^{2} + d^{3}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right ) + 2 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (2 \, e^{2} x^{3} - 5 \, d e x\right )} \sqrt {e x^{2} + d}}{16 \, {\left (e^{2} x^{2} + d e\right )}}, -\frac {3 \, {\left (d^{2} e x^{2} + d^{3}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) + \sqrt {-e^{2} x^{4} + d^{2}} {\left (2 \, e^{2} x^{3} - 5 \, d e x\right )} \sqrt {e x^{2} + d}}{8 \, {\left (e^{2} x^{2} + d e\right )}}\right ] \] Input:

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

Output:

[-1/16*(3*(d^2*e*x^2 + d^3)*sqrt(-e)*log(-(2*e^2*x^4 + d*e*x^2 - 2*sqrt(-e 
^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(-e)*x - d^2)/(e*x^2 + d)) + 2*sqrt(-e^2 
*x^4 + d^2)*(2*e^2*x^3 - 5*d*e*x)*sqrt(e*x^2 + d))/(e^2*x^2 + d*e), -1/8*( 
3*(d^2*e*x^2 + d^3)*sqrt(e)*arctan(sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sq 
rt(e)*x/(e^2*x^4 - d^2)) + sqrt(-e^2*x^4 + d^2)*(2*e^2*x^3 - 5*d*e*x)*sqrt 
(e*x^2 + d))/(e^2*x^2 + d*e)]
 

Sympy [F]

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

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

Output:

Integral((-(-d + e*x**2)*(d + e*x**2))**(3/2)/(d + e*x**2)**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x - 7*int((sqrt(d + e*x**2)*sq 
rt(d**2 - e**2*x**4)*x**2)/(d**2 - e**2*x**4),x)*d*e + 5*int((sqrt(d + e*x 
**2)*sqrt(d**2 - e**2*x**4))/(d**2 - e**2*x**4),x)*d**2)/4