\(\int \frac {(d-e x^2)^{9/2}}{(d^2-e^2 x^4)^{3/2}} \, dx\) [157]

Optimal result

Integrand size = 29, antiderivative size = 153 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {8 d^2 x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}+\frac {19 d x \sqrt {d^2-e^2 x^4}}{8 \sqrt {d-e x^2}}-\frac {e x^3 \sqrt {d^2-e^2 x^4}}{4 \sqrt {d-e x^2}}-\frac {75 d^2 \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{8 \sqrt {e}} \] Output:

8*d^2*x*(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2)+19/8*d*x*(-e^2*x^4+d^2)^(1/2 
)/(-e*x^2+d)^(1/2)-1/4*e*x^3*(-e^2*x^4+d^2)^(1/2)/(-e*x^2+d)^(1/2)-75/8*d^ 
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 = 5.38 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {1}{8} \left (\frac {x \left (83 d^2+17 d e x^2-2 e^2 x^4\right ) \sqrt {d^2-e^2 x^4}}{\sqrt {d-e x^2} \left (d+e x^2\right )}+\frac {75 d^2 \log \left (-d+e x^2\right )}{\sqrt {e}}-\frac {75 d^2 \log \left (d e x-e^2 x^3+\sqrt {e} \sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}\right )}{\sqrt {e}}\right ) \] Input:

Integrate[(d - e*x^2)^(9/2)/(d^2 - e^2*x^4)^(3/2),x]
 

Output:

((x*(83*d^2 + 17*d*e*x^2 - 2*e^2*x^4)*Sqrt[d^2 - e^2*x^4])/(Sqrt[d - e*x^2 
]*(d + e*x^2)) + (75*d^2*Log[-d + e*x^2])/Sqrt[e] - (75*d^2*Log[d*e*x - e^ 
2*x^3 + Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d^2 - e^2*x^4]])/Sqrt[e])/8
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1396, 315, 25, 27, 403, 27, 299, 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.

\(\displaystyle \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int -\frac {d e \left (d-9 e x^2\right ) \left (d-e x^2\right )}{\sqrt {e x^2+d}}dx}{d e}+\frac {2 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}-\frac {\int \frac {d e \left (d-9 e x^2\right ) \left (d-e x^2\right )}{\sqrt {e x^2+d}}dx}{d e}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}-\int \frac {\left (d-9 e x^2\right ) \left (d-e x^2\right )}{\sqrt {e x^2+d}}dx\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {\int \frac {5 d e \left (d-13 e x^2\right )}{\sqrt {e x^2+d}}dx}{4 e}+\frac {2 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}+\frac {1}{4} x \left (d-9 e x^2\right ) \sqrt {d+e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {5}{4} d \int \frac {d-13 e x^2}{\sqrt {e x^2+d}}dx+\frac {2 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}+\frac {1}{4} x \left (d-9 e x^2\right ) \sqrt {d+e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {5}{4} d \left (\frac {15}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {13}{2} x \sqrt {d+e x^2}\right )+\frac {2 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}+\frac {1}{4} x \left (d-9 e x^2\right ) \sqrt {d+e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {5}{4} d \left (\frac {15}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {13}{2} x \sqrt {d+e x^2}\right )+\frac {2 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}+\frac {1}{4} x \left (d-9 e x^2\right ) \sqrt {d+e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {5}{4} d \left (\frac {15 d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}-\frac {13}{2} x \sqrt {d+e x^2}\right )+\frac {2 x \left (d-e x^2\right )^2}{\sqrt {d+e x^2}}+\frac {1}{4} x \left (d-9 e x^2\right ) \sqrt {d+e x^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[(d - e*x^2)^(9/2)/(d^2 - e^2*x^4)^(3/2),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((2*x*(d - e*x^2)^2)/Sqrt[d + e*x^2] + (x 
*(d - 9*e*x^2)*Sqrt[d + e*x^2])/4 - (5*d*((-13*x*Sqrt[d + e*x^2])/2 + (15* 
d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[e])))/4))/Sqrt[d^2 - e^2*x 
^4]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 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 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 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 = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.63

method	result	size
default	\(-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (2 e^{\frac {5}{2}} x^{5}-17 d \,e^{\frac {3}{2}} x^{3}+75 \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right ) d^{2} \sqrt {e \,x^{2}+d}-83 d^{2} x \sqrt {e}\right )}{8 \sqrt {-e \,x^{2}+d}\, \left (e \,x^{2}+d \right ) \sqrt {e}}\)	\(96\)
risch	\(-\frac {x \left (-2 e \,x^{2}+19 d \right ) \sqrt {e \,x^{2}+d}\, \sqrt {\frac {\left (-e \,x^{2}+d \right ) \left (-e^{2} x^{4}+d^{2}\right )}{\left (e \,x^{2}-d \right )^{2}}}\, \left (e \,x^{2}-d \right )}{8 \sqrt {-e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\left (\frac {8 d^{2} x}{\sqrt {e \,x^{2}+d}}-\frac {75 d^{2} \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{8 \sqrt {e}}\right ) \sqrt {\frac {\left (-e \,x^{2}+d \right ) \left (-e^{2} x^{4}+d^{2}\right )}{\left (e \,x^{2}-d \right )^{2}}}\, \left (e \,x^{2}-d \right )}{\sqrt {-e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(200\)

Input:

int((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8*(-e^2*x^4+d^2)^(1/2)*(2*e^(5/2)*x^5-17*d*e^(3/2)*x^3+75*ln(x*e^(1/2)+ 
(e*x^2+d)^(1/2))*d^2*(e*x^2+d)^(1/2)-83*d^2*x*e^(1/2))/(-e*x^2+d)^(1/2)/(e 
*x^2+d)/e^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.93 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\left [\frac {75 \, {\left (d^{2} e^{2} x^{4} - d^{4}\right )} \sqrt {e} \log \left (\frac {2 \, e^{2} x^{4} - d e x^{2} + 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e x^{2} - d}\right ) + 2 \, {\left (2 \, e^{3} x^{5} - 17 \, d e^{2} x^{3} - 83 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{16 \, {\left (e^{3} x^{4} - d^{2} e\right )}}, -\frac {75 \, {\left (d^{2} e^{2} x^{4} - d^{4}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) - {\left (2 \, e^{3} x^{5} - 17 \, d e^{2} x^{3} - 83 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{8 \, {\left (e^{3} x^{4} - d^{2} e\right )}}\right ] \] Input:

integrate((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
 

Output:

[1/16*(75*(d^2*e^2*x^4 - 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*(2*e^3*x^ 
5 - 17*d*e^2*x^3 - 83*d^2*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(e^3 
*x^4 - d^2*e), -1/8*(75*(d^2*e^2*x^4 - 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)) - (2*e^3*x^5 - 17*d*e^ 
2*x^3 - 83*d^2*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(e^3*x^4 - d^2* 
e)]
 

Sympy [F]

\[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {\left (d - e x^{2}\right )^{\frac {9}{2}}}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((-e*x**2+d)**(9/2)/(-e**2*x**4+d**2)**(3/2),x)
 

Output:

Integral((d - e*x**2)**(9/2)/(-(-d + e*x**2)*(d + e*x**2))**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {{\left (-e x^{2} + d\right )}^{\frac {9}{2}}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-e*x^2 + d)^(9/2)/(-e^2*x^4 + d^2)^(3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.41 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {75 \, d^{2} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{8 \, \sqrt {e}} - \frac {{\left ({\left (2 \, e^{2} x^{2} - 17 \, d e\right )} x^{2} - 83 \, d^{2}\right )} x}{8 \, \sqrt {e x^{2} + d}} \] Input:

integrate((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
 

Output:

75/8*d^2*log(abs(-sqrt(e)*x + sqrt(e*x^2 + d)))/sqrt(e) - 1/8*((2*e^2*x^2 
- 17*d*e)*x^2 - 83*d^2)*x/sqrt(e*x^2 + d)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {{\left (d-e\,x^2\right )}^{9/2}}{{\left (d^2-e^2\,x^4\right )}^{3/2}} \,d x \] Input:

int((d - e*x^2)^(9/2)/(d^2 - e^2*x^4)^(3/2),x)
 

Output:

int((d - e*x^2)^(9/2)/(d^2 - e^2*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d-e x^2\right )^{9/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {83 \sqrt {e \,x^{2}+d}\, d^{2} e x +17 \sqrt {e \,x^{2}+d}\, d \,e^{2} x^{3}-2 \sqrt {e \,x^{2}+d}\, e^{3} x^{5}-75 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{3}-75 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2} e \,x^{2}+69 \sqrt {e}\, d^{3}+69 \sqrt {e}\, d^{2} e \,x^{2}}{8 e \left (e \,x^{2}+d \right )} \] Input:

int((-e*x^2+d)^(9/2)/(-e^2*x^4+d^2)^(3/2),x)
 

Output:

(83*sqrt(d + e*x**2)*d**2*e*x + 17*sqrt(d + e*x**2)*d*e**2*x**3 - 2*sqrt(d 
 + e*x**2)*e**3*x**5 - 75*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt( 
d))*d**3 - 75*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d**2*e*x 
**2 + 69*sqrt(e)*d**3 + 69*sqrt(e)*d**2*e*x**2)/(8*e*(d + e*x**2))