\(\int \frac {(d^2-e^2 x^4)^{3/2}}{\sqrt {d+e x^2}} \, dx\) [140]

Optimal result

Integrand size = 28, antiderivative size = 156 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\frac {9 d^2 x \sqrt {d^2-e^2 x^4}}{16 \sqrt {d+e x^2}}+\frac {d e x^3 \sqrt {d^2-e^2 x^4}}{24 \sqrt {d+e x^2}}-\frac {e^2 x^5 \sqrt {d^2-e^2 x^4}}{6 \sqrt {d+e x^2}}+\frac {7 d^3 \arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{16 \sqrt {e}} \] Output:

9/16*d^2*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+1/24*d*e*x^3*(-e^2*x^4+d^2 
)^(1/2)/(e*x^2+d)^(1/2)-1/6*e^2*x^5*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+7 
/16*d^3*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 = 2.69 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\frac {1}{48} \left (\frac {x \left (27 d^2+2 d e x^2-8 e^2 x^4\right ) \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}+\frac {21 i d^3 \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}\right ) \] Input:

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

Output:

((x*(27*d^2 + 2*d*e*x^2 - 8*e^2*x^4)*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2] 
+ ((21*I)*d^3*Log[(-2*I)*Sqrt[e]*x + (2*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^ 
2]])/Sqrt[e])/48
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1396, 299, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 299

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7}{6} d \int \left (d-e x^2\right )^{3/2}dx-\frac {1}{6} x \left (d-e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7}{6} d \left (\frac {3}{4} d \int \sqrt {d-e x^2}dx+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {1}{6} x \left (d-e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7}{6} d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\sqrt {d-e x^2}}dx+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {1}{6} x \left (d-e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7}{6} d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\frac {e x^2}{d-e x^2}+1}d\frac {x}{\sqrt {d-e x^2}}+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {1}{6} x \left (d-e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {7}{6} d \left (\frac {3}{4} d \left (\frac {d \arctan \left (\frac {\sqrt {e} x}{\sqrt {d-e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {1}{6} x \left (d-e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*(-1/6*(x*(d - e*x^2)^(5/2)) + (7*d*((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))/6))/(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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 = 117, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (-8 e^{\frac {5}{2}} x^{5} \sqrt {-e \,x^{2}+d}+2 d \,e^{\frac {3}{2}} x^{3} \sqrt {-e \,x^{2}+d}+27 \sqrt {-e \,x^{2}+d}\, \sqrt {e}\, d^{2} x +21 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d^{3}\right )}{48 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}}\)	\(117\)
risch	\(\frac {x \left (-8 e^{2} x^{4}+2 d e \,x^{2}+27 d^{2}\right ) \sqrt {-e \,x^{2}+d}\, \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{48 \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {7 d^{3} \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{16 \sqrt {e}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(154\)

Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/48*(-e^2*x^4+d^2)^(1/2)*(-8*e^(5/2)*x^5*(-e*x^2+d)^(1/2)+2*d*e^(3/2)*x^3 
*(-e*x^2+d)^(1/2)+27*(-e*x^2+d)^(1/2)*e^(1/2)*d^2*x+21*arctan(e^(1/2)*x/(- 
e*x^2+d)^(1/2))*d^3)/(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 = 275, normalized size of antiderivative = 1.76 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\left [-\frac {21 \, {\left (d^{3} e x^{2} + d^{4}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right ) + 2 \, {\left (8 \, e^{3} x^{5} - 2 \, d e^{2} x^{3} - 27 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{96 \, {\left (e^{2} x^{2} + d e\right )}}, -\frac {21 \, {\left (d^{3} e x^{2} + d^{4}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) + {\left (8 \, e^{3} x^{5} - 2 \, d e^{2} x^{3} - 27 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{48 \, {\left (e^{2} x^{2} + d e\right )}}\right ] \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="fricas")
 

Output:

[-1/96*(21*(d^3*e*x^2 + d^4)*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*(8*e^3*x 
^5 - 2*d*e^2*x^3 - 27*d^2*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/(e^2* 
x^2 + d*e), -1/48*(21*(d^3*e*x^2 + d^4)*sqrt(e)*arctan(sqrt(-e^2*x^4 + d^2 
)*sqrt(e*x^2 + d)*sqrt(e)*x/(e^2*x^4 - d^2)) + (8*e^3*x^5 - 2*d*e^2*x^3 - 
27*d^2*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/(e^2*x^2 + d*e)]
 

Sympy [F]

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

integrate((-e**2*x**4+d**2)**(3/2)/(e*x**2+d)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.44 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=-\frac {7 \, d^{3} \log \left ({\left | -\sqrt {-e} x + \sqrt {-e x^{2} + d} \right |}\right )}{16 \, \sqrt {-e}} - \frac {1}{48} \, {\left (2 \, {\left (4 \, e^{2} x^{2} - d e\right )} x^{2} - 27 \, d^{2}\right )} \sqrt {-e x^{2} + d} x \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

-7/16*d^3*log(abs(-sqrt(-e)*x + sqrt(-e*x^2 + d)))/sqrt(-e) - 1/48*(2*(4*e 
^2*x^2 - d*e)*x^2 - 27*d^2)*sqrt(-e*x^2 + d)*x
 

Mupad [F(-1)]

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

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(1/2),x)
 

Output:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\frac {5 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, d x}{24}-\frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e \,x^{3}}{6}-\frac {17 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{2} e}{24}+\frac {19 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{3}}{24} \] Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(1/2),x)
 

Output:

(5*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*d*x - 4*sqrt(d + e*x**2)*sqrt(d 
**2 - e**2*x**4)*e*x**3 - 17*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)* 
x**2)/(d**2 - e**2*x**4),x)*d**2*e + 19*int((sqrt(d + e*x**2)*sqrt(d**2 - 
e**2*x**4))/(d**2 - e**2*x**4),x)*d**3)/24