\(\int \frac {(d-e x^2)^{9/2}}{(d^2-e^2 x^4)^{5/2}} \, dx\) [166]

Optimal result

Integrand size = 29, antiderivative size = 110 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {4 d x \left (d-e x^2\right )^{3/2}}{3 \left (d^2-e^2 x^4\right )^{3/2}}-\frac {4 x \sqrt {d-e x^2}}{3 \sqrt {d^2-e^2 x^4}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{\sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 5.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=-\frac {4 e x^3 \sqrt {d^2-e^2 x^4}}{3 \sqrt {d-e x^2} \left (d+e x^2\right )^2}-\frac {\log \left (-d+e x^2\right )}{\sqrt {e}}+\frac {\log \left (d e x-e^2 x^3+\sqrt {e} \sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}\right )}{\sqrt {e}} \] Input:

Integrate[(d - e*x^2)^(9/2)/(d^2 - e^2*x^4)^(5/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1396, 315, 27, 298, 224, 219}

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.

\(\displaystyle \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

Int[(d - e*x^2)^(9/2)/(d^2 - e^2*x^4)^(5/2),x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[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 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

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 = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (3 \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right ) e \,x^{2} \sqrt {e \,x^{2}+d}-4 e^{\frac {3}{2}} x^{3}+3 \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right ) d \sqrt {e \,x^{2}+d}\right )}{3 \sqrt {-e \,x^{2}+d}\, \left (e \,x^{2}+d \right )^{2} \sqrt {e}}\)	\(107\)

Input:

int((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.95 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\left [\frac {8 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} e^{2} x^{3} + 3 \, {\left (e^{3} x^{6} + d e^{2} x^{4} - d^{2} e x^{2} - d^{3}\right )} \sqrt {e} \log \left (\frac {2 \, e^{2} x^{4} - d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e x^{2} - d}\right )}{6 \, {\left (e^{4} x^{6} + d e^{3} x^{4} - d^{2} e^{2} x^{2} - d^{3} e\right )}}, \frac {4 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} e^{2} x^{3} + 3 \, {\left (e^{3} x^{6} + d e^{2} x^{4} - d^{2} e x^{2} - d^{3}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right )}{3 \, {\left (e^{4} x^{6} + d e^{3} x^{4} - d^{2} e^{2} x^{2} - d^{3} e\right )}}\right ] \] Input:

integrate((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

[1/6*(8*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*e^2*x^3 + 3*(e^3*x^6 + d*e^2 
*x^4 - 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)))/(e^4*x^6 + d*e^3*x 
^4 - d^2*e^2*x^2 - d^3*e), 1/3*(4*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*e^ 
2*x^3 + 3*(e^3*x^6 + d*e^2*x^4 - d^2*e*x^2 - d^3)*sqrt(-e)*arctan(sqrt(-e^ 
2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(-e)*x/(e^2*x^4 - d^2)))/(e^4*x^6 + d*e^ 
3*x^4 - d^2*e^2*x^2 - d^3*e)]
 

Sympy [F]

\[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (d - e x^{2}\right )^{\frac {9}{2}}}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((-e*x**2+d)**(9/2)/(-e**2*x**4+d**2)**(5/2),x)
 

Output:

Integral((d - e*x**2)**(9/2)/(-(-d + e*x**2)*(d + e*x**2))**(5/2), x)
 

Maxima [F]

\[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (-e x^{2} + d\right )}^{\frac {9}{2}}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-e*x^2 + d)^(9/2)/(-e^2*x^4 + d^2)^(5/2), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.35 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=-\frac {4 \, e x^{3}}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} - \frac {\log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{\sqrt {e}} \] Input:

integrate((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
 

Output:

-4/3*e*x^3/(e*x^2 + d)^(3/2) - log(abs(-sqrt(e)*x + sqrt(e*x^2 + d)))/sqrt 
(e)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d-e\,x^2\right )}^{9/2}}{{\left (d^2-e^2\,x^4\right )}^{5/2}} \,d x \] Input:

int((d - e*x^2)^(9/2)/(d^2 - e^2*x^4)^(5/2),x)
 

Output:

int((d - e*x^2)^(9/2)/(d^2 - e^2*x^4)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.37 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {-4 \sqrt {e \,x^{2}+d}\, e^{2} x^{3}+3 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}+6 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d e \,x^{2}+3 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}-4 \sqrt {e}\, d^{2}-8 \sqrt {e}\, d e \,x^{2}-4 \sqrt {e}\, e^{2} x^{4}}{3 e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(5/2),x)
 

Output:

( - 4*sqrt(d + e*x**2)*e**2*x**3 + 3*sqrt(e)*log((sqrt(d + e*x**2) + sqrt( 
e)*x)/sqrt(d))*d**2 + 6*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d) 
)*d*e*x**2 + 3*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*e**2*x* 
*4 - 4*sqrt(e)*d**2 - 8*sqrt(e)*d*e*x**2 - 4*sqrt(e)*e**2*x**4)/(3*e*(d**2 
 + 2*d*e*x**2 + e**2*x**4))