\(\int \frac {(d+e x^2)^{3/2}}{d^2-e^2 x^4} \, dx\) [27]

Optimal result

Integrand size = 26, antiderivative size = 62 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{d^2-e^2 x^4} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{d^2-e^2 x^4} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {d-e x^2+\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}\right )+\log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{\sqrt {e}} \] Input:

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

Output:

(Sqrt[2]*ArcTanh[(d - e*x^2 + Sqrt[e]*x*Sqrt[d + e*x^2])/(Sqrt[2]*d)] + Lo 
g[-(Sqrt[e]*x) + Sqrt[d + e*x^2]])/Sqrt[e]
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1388, 301, 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.

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {2}}{2 x \sqrt {e}}\right )-\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )}{\sqrt {e}}\)	\(50\)
default	\(\text {Expression too large to display}\)	\(1356\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 199, normalized size of antiderivative = 3.21 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{d^2-e^2 x^4} \, dx=\left [\frac {\sqrt {2} \sqrt {e} \log \left (\frac {17 \, e^{2} x^{4} + 14 \, d e x^{2} + d^{2} + \frac {4 \, \sqrt {2} {\left (3 \, e^{2} x^{3} + d e x\right )} \sqrt {e x^{2} + d}}{\sqrt {e}}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 2 \, \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right )}{4 \, e}, -\frac {\sqrt {2} e \sqrt {-\frac {1}{e}} \arctan \left (\frac {\sqrt {2} {\left (3 \, e x^{2} + d\right )} \sqrt {e x^{2} + d} \sqrt {-\frac {1}{e}}}{4 \, {\left (e x^{3} + d x\right )}}\right ) - 2 \, \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{2 \, e}\right ] \] Input:

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

Output:

[1/4*(sqrt(2)*sqrt(e)*log((17*e^2*x^4 + 14*d*e*x^2 + d^2 + 4*sqrt(2)*(3*e^ 
2*x^3 + d*e*x)*sqrt(e*x^2 + d)/sqrt(e))/(e^2*x^4 - 2*d*e*x^2 + d^2)) + 2*s 
qrt(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d))/e, -1/2*(sqrt(2)*e 
*sqrt(-1/e)*arctan(1/4*sqrt(2)*(3*e*x^2 + d)*sqrt(e*x^2 + d)*sqrt(-1/e)/(e 
*x^3 + d*x)) - 2*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/e]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (46) = 92\).

Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.76 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{d^2-e^2 x^4} \, dx=\frac {\sqrt {2} d \log \left (\frac {{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{2 \, \sqrt {e} {\left | d \right |}} + \frac {\log \left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2}\right )}{2 \, \sqrt {e}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.47 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{d^2-e^2 x^4} \, dx=\frac {\sqrt {e}\, \left (-\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right )+\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right )+\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right )-\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right )-2 \,\mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right )\right )}{2 e} \] Input:

int((e*x^2+d)^(3/2)/(-e^2*x^4+d^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)) + sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqr 
t(d) + sqrt(e)*x)/sqrt(d)) + sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt( 
2) - sqrt(d) + sqrt(e)*x)/sqrt(d)) - sqrt(2)*log((sqrt(d + e*x**2) + sqrt( 
d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d)) - 2*log((sqrt(d + e*x**2) + sqr 
t(e)*x)/sqrt(d))))/(2*e)