\(\int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx\) [6]

Optimal result

Integrand size = 36, antiderivative size = 107 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx=-\frac {B \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{e^{3/2}}+\frac {(B d+A e) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{\sqrt {2} d e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 5.78 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.43 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx=\frac {\frac {\sqrt {2} (B d+A e) \sqrt {d^2-e^2 x^4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d \sqrt {d-e x^2} \sqrt {d+e x^2}}+2 B \left (\log \left (-d+e x^2\right )-\log \left (d e x-e^2 x^3+\sqrt {e} \sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}\right )\right )}{2 e^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1396, 398, 224, 219, 291, 221}

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.

\(\displaystyle \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 221

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

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 291

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

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(88)=176\).

Time = 0.20 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.20

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (A \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}+\sqrt {d e}\, x +d \right )}{e x -\sqrt {d e}}\right ) \sqrt {2}\, \sqrt {d}\, e^{\frac {3}{2}}-A \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}-\sqrt {d e}\, x +d \right )}{e x +\sqrt {d e}}\right ) \sqrt {2}\, \sqrt {d}\, e^{\frac {3}{2}}+B \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}+\sqrt {d e}\, x +d \right )}{e x -\sqrt {d e}}\right ) \sqrt {2}\, d^{\frac {3}{2}} \sqrt {e}-B \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}-\sqrt {d e}\, x +d \right )}{e x +\sqrt {d e}}\right ) \sqrt {2}\, d^{\frac {3}{2}} \sqrt {e}+2 A \sqrt {d e}\, \ln \left (\frac {\sqrt {e}\, \sqrt {-\frac {\left (-e x +\sqrt {-d e}\right ) \left (e x +\sqrt {-d e}\right )}{e}}+e x}{\sqrt {e}}\right ) e -2 A \sqrt {d e}\, \ln \left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {e}+e x}{\sqrt {e}}\right ) e -2 B \sqrt {d e}\, \ln \left (\frac {\sqrt {e}\, \sqrt {-\frac {\left (-e x +\sqrt {-d e}\right ) \left (e x +\sqrt {-d e}\right )}{e}}+e x}{\sqrt {e}}\right ) d -2 B \sqrt {d e}\, \ln \left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {e}+e x}{\sqrt {e}}\right ) d \right )}{2 \sqrt {-e \,x^{2}+d}\, \sqrt {e \,x^{2}+d}\, \left (-\sqrt {-d e}+\sqrt {d e}\right ) \left (\sqrt {-d e}+\sqrt {d e}\right ) \sqrt {e}\, \sqrt {d e}}\)	\(449\)

Input:

int((B*x^2+A)/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2),x,method=_RETURNVERBOS 
E)
 

Output:

1/2/(-e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(1/2)*(A*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^ 
2+d)^(1/2)+(d*e)^(1/2)*x+d)/(e*x-(d*e)^(1/2)))*2^(1/2)*d^(1/2)*e^(3/2)-A*l 
n(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/(e*x+(d*e)^(1/2))) 
*2^(1/2)*d^(1/2)*e^(3/2)+B*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)+(d*e)^( 
1/2)*x+d)/(e*x-(d*e)^(1/2)))*2^(1/2)*d^(3/2)*e^(1/2)-B*ln(2*e*(2^(1/2)*d^( 
1/2)*(e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/(e*x+(d*e)^(1/2)))*2^(1/2)*d^(3/2)*e 
^(1/2)+2*A*(d*e)^(1/2)*ln((e^(1/2)*(-1/e*(-e*x+(-d*e)^(1/2))*(e*x+(-d*e)^( 
1/2)))^(1/2)+e*x)/e^(1/2))*e-2*A*(d*e)^(1/2)*ln(((e*x^2+d)^(1/2)*e^(1/2)+e 
*x)/e^(1/2))*e-2*B*(d*e)^(1/2)*ln((e^(1/2)*(-1/e*(-e*x+(-d*e)^(1/2))*(e*x+ 
(-d*e)^(1/2)))^(1/2)+e*x)/e^(1/2))*d-2*B*(d*e)^(1/2)*ln(((e*x^2+d)^(1/2)*e 
^(1/2)+e*x)/e^(1/2))*d)/(e*x^2+d)^(1/2)/(-(-d*e)^(1/2)+(d*e)^(1/2))/((-d*e 
)^(1/2)+(d*e)^(1/2))/e^(1/2)/(d*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.84 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx=\left [\frac {2 \, B d \sqrt {e} \log \left (\frac {2 \, e^{2} x^{4} - d e x^{2} + 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e x^{2} - d}\right ) + \sqrt {2} {\left (B d + A e\right )} \sqrt {e} \log \left (-\frac {3 \, e^{2} x^{4} - 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )}{4 \, d e^{2}}, -\frac {2 \, B d \sqrt {-e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) - \sqrt {2} {\left (B d + A e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right )}{2 \, d e^{2}}\right ] \] Input:

integrate((B*x^2+A)/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2),x, algorithm="fr 
icas")
 

Output:

[1/4*(2*B*d*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)) + sqrt(2)*(B*d + A*e)*sqrt(e)*l 
og(-(3*e^2*x^4 - 2*d*e*x^2 - 2*sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + 
d)*sqrt(e)*x - d^2)/(e^2*x^4 - 2*d*e*x^2 + d^2)))/(d*e^2), -1/2*(2*B*d*sqr 
t(-e)*arctan(sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(-e)*x/(e^2*x^4 - d 
^2)) - sqrt(2)*(B*d + A*e)*sqrt(-e)*arctan(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sq 
rt(-e*x^2 + d)*sqrt(-e)*x/(e^2*x^4 - d^2)))/(d*e^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}} \,d x } \] Input:

integrate((B*x^2+A)/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2),x, algorithm="ma 
xima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx=\frac {B \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{e^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (B d + A e\right )} \log \left (\frac {{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{4 \, e^{\frac {3}{2}} {\left | d \right |}} \] Input:

integrate((B*x^2+A)/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2),x, algorithm="gi 
ac")
 

Output:

B*log(abs(-sqrt(e)*x + sqrt(e*x^2 + d)))/e^(3/2) + 1/4*sqrt(2)*(B*d + A*e) 
*log(abs(2*(sqrt(e)*x - sqrt(e*x^2 + d))^2 - 4*sqrt(2)*abs(d) - 6*d)/abs(2 
*(sqrt(e)*x - sqrt(e*x^2 + d))^2 + 4*sqrt(2)*abs(d) - 6*d))/(e^(3/2)*abs(d 
))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {d^2-e^2\,x^4}\,\sqrt {d-e\,x^2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.79 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}} \, dx=\frac {\sqrt {e}\, \left (-\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a e -\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d +\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a e +\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d +\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a e +\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d -\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a e -\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d -4 \,\mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d \right )}{4 d \,e^{2}} \] Input:

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

Output:

(sqrt(e)*( - sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + s 
qrt(e)*x)/sqrt(d))*a*e - sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - 
 sqrt(d) + sqrt(e)*x)/sqrt(d))*b*d + sqrt(2)*log((sqrt(d + e*x**2) - sqrt( 
d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*a*e + sqrt(2)*log((sqrt(d + e*x 
**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*b*d + sqrt(2)*log(( 
sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*a*e + s 
qrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt 
(d))*b*d - sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + sqr 
t(e)*x)/sqrt(d))*a*e - sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + s 
qrt(d) + sqrt(e)*x)/sqrt(d))*b*d - 4*log((sqrt(d + e*x**2) + sqrt(e)*x)/sq 
rt(d))*b*d))/(4*d*e**2)