\(\int \frac {(d-e x^2)^{7/2}}{(d^2-e^2 x^4)^{3/2}} \, dx\) [158]

Optimal result

Integrand size = 29, antiderivative size = 112 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {4 d x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}+\frac {x \sqrt {d^2-e^2 x^4}}{2 \sqrt {d-e x^2}}-\frac {7 d \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{2 \sqrt {e}} \] Output:

4*d*x*(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2)+1/2*x*(-e^2*x^4+d^2)^(1/2)/(-e 
*x^2+d)^(1/2)-7/2*d*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 = 5.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {x \left (9 d+e x^2\right ) \sqrt {d^2-e^2 x^4}}{\sqrt {d-e x^2} \left (d+e x^2\right )}+\frac {7 d \log \left (-d+e x^2\right )}{\sqrt {e}}-\frac {7 d \log \left (d e x-e^2 x^3+\sqrt {e} \sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}\right )}{\sqrt {e}}\right ) \] Input:

Integrate[(d - e*x^2)^(7/2)/(d^2 - e^2*x^4)^(3/2),x]
 

Output:

((x*(9*d + e*x^2)*Sqrt[d^2 - e^2*x^4])/(Sqrt[d - e*x^2]*(d + e*x^2)) + (7* 
d*Log[-d + e*x^2])/Sqrt[e] - (7*d*Log[d*e*x - e^2*x^3 + Sqrt[e]*Sqrt[d - e 
*x^2]*Sqrt[d^2 - e^2*x^4]])/Sqrt[e])/2
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1396, 315, 25, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {7 d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {5}{2} x \sqrt {d+e x^2}+\frac {2 x \left (d-e x^2\right )}{\sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[(d - e*x^2)^(7/2)/(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))/Sqrt[d + e*x^2] + (5*x 
*Sqrt[d + e*x^2])/2 - (7*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[e 
])))/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 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 = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.74

method	result	size
default	\(-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (-e^{\frac {3}{2}} x^{3}+7 \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right ) d \sqrt {e \,x^{2}+d}-9 \sqrt {e}\, d x \right )}{2 \sqrt {-e \,x^{2}+d}\, \left (e \,x^{2}+d \right ) \sqrt {e}}\)	\(83\)
risch	\(-\frac {x \sqrt {e \,x^{2}+d}\, \sqrt {\frac {\left (-e \,x^{2}+d \right ) \left (-e^{2} x^{4}+d^{2}\right )}{\left (e \,x^{2}-d \right )^{2}}}\, \left (e \,x^{2}-d \right )}{2 \sqrt {-e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\left (\frac {4 d x}{\sqrt {e \,x^{2}+d}}-\frac {7 d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right ) \sqrt {\frac {\left (-e \,x^{2}+d \right ) \left (-e^{2} x^{4}+d^{2}\right )}{\left (e \,x^{2}-d \right )^{2}}}\, \left (e \,x^{2}-d \right )}{\sqrt {-e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(186\)

Input:

int((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(-e^2*x^4+d^2)^(1/2)*(-e^(3/2)*x^3+7*ln(x*e^(1/2)+(e*x^2+d)^(1/2))*d* 
(e*x^2+d)^(1/2)-9*e^(1/2)*d*x)/(-e*x^2+d)^(1/2)/(e*x^2+d)/e^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.38 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\left [\frac {7 \, {\left (d e^{2} x^{4} - d^{3}\right )} \sqrt {e} \log \left (\frac {2 \, e^{2} x^{4} - d e x^{2} + 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e x^{2} - d}\right ) - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (e^{2} x^{3} + 9 \, d e x\right )} \sqrt {-e x^{2} + d}}{4 \, {\left (e^{3} x^{4} - d^{2} e\right )}}, -\frac {7 \, {\left (d e^{2} x^{4} - d^{3}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) + \sqrt {-e^{2} x^{4} + d^{2}} {\left (e^{2} x^{3} + 9 \, d e x\right )} \sqrt {-e x^{2} + d}}{2 \, {\left (e^{3} x^{4} - d^{2} e\right )}}\right ] \] Input:

integrate((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
 

Output:

[1/4*(7*(d*e^2*x^4 - 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)*(e^2*x^3 + 9*d*e*x)*sqrt(-e*x^2 + d))/(e^3*x^4 - d^2*e), -1/2*(7*( 
d*e^2*x^4 - d^3)*sqrt(-e)*arctan(sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqr 
t(-e)*x/(e^2*x^4 - d^2)) + sqrt(-e^2*x^4 + d^2)*(e^2*x^3 + 9*d*e*x)*sqrt(- 
e*x^2 + d))/(e^3*x^4 - d^2*e)]
 

Sympy [F]

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

integrate((-e*x**2+d)**(7/2)/(-e**2*x**4+d**2)**(3/2),x)
 

Output:

Integral((d - e*x**2)**(7/2)/(-(-d + e*x**2)*(d + e*x**2))**(3/2), x)
 

Maxima [F]

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

integrate((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-e*x^2 + d)^(7/2)/(-e^2*x^4 + d^2)^(3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.41 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {{\left (e x^{2} + 9 \, d\right )} x}{2 \, \sqrt {e x^{2} + d}} + \frac {7 \, d \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{2 \, \sqrt {e}} \] Input:

integrate((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
 

Output:

1/2*(e*x^2 + 9*d)*x/sqrt(e*x^2 + d) + 7/2*d*log(abs(-sqrt(e)*x + sqrt(e*x^ 
2 + d)))/sqrt(e)
 

Mupad [F(-1)]

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

int((d - e*x^2)^(7/2)/(d^2 - e^2*x^4)^(3/2),x)
 

Output:

int((d - e*x^2)^(7/2)/(d^2 - e^2*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {36 \sqrt {e \,x^{2}+d}\, d e x +4 \sqrt {e \,x^{2}+d}\, e^{2} x^{3}-28 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}-28 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d e \,x^{2}+33 \sqrt {e}\, d^{2}+33 \sqrt {e}\, d e \,x^{2}}{8 e \left (e \,x^{2}+d \right )} \] Input:

int((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(3/2),x)
 

Output:

(36*sqrt(d + e*x**2)*d*e*x + 4*sqrt(d + e*x**2)*e**2*x**3 - 28*sqrt(e)*log 
((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d**2 - 28*sqrt(e)*log((sqrt(d + e 
*x**2) + sqrt(e)*x)/sqrt(d))*d*e*x**2 + 33*sqrt(e)*d**2 + 33*sqrt(e)*d*e*x 
**2)/(8*e*(d + e*x**2))