\(\int \frac {\sqrt {d+e x^2}}{a-c x^4} \, dx\) [344]

Optimal result

Integrand size = 22, antiderivative size = 145 \[ \int \frac {\sqrt {d+e x^2}}{a-c x^4} \, dx=\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \sqrt {c}}+\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \sqrt {c}} \] Output:

1/2*(c^(1/2)*d-a^(1/2)*e)^(1/2)*arctan((c^(1/2)*d-a^(1/2)*e)^(1/2)*x/a^(1/ 
4)/(e*x^2+d)^(1/2))/a^(3/4)/c^(1/2)+1/2*(c^(1/2)*d+a^(1/2)*e)^(1/2)*arctan 
h((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(3/4)/c^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.13 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {d+e x^2}}{a-c x^4} \, dx=-\frac {1}{2} e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+2 d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ] \] Input:

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

Output:

-1/2*(e^(3/2)*RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 - 16*a*e^2*#1^2 - 
4*c*d*#1^3 + c*#1^4 & , (d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] 
 - #1] + 2*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + Log[ 
d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2)/(c*d^3 - 3*c*d^2*#1 
+ 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1489, 27, 301, 224, 219, 291, 218, 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.

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

\(\Big \downarrow \) 1489

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {\left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {a} e+\sqrt {c} d}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {a}}\)

Input:

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

Output:

(((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*x)/(a^(1/4 
)*Sqrt[d + e*x^2])])/(a^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + (Sqrt[e]*ArcT 
anh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c])/(2*Sqrt[a]) + (-((Sqrt[e]*ArcTa 
nh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c]) + ((d + (Sqrt[a]*e)/Sqrt[c])*Arc 
Tanh[(Sqrt[Sqrt[c]*d + Sqrt[a]*e]*x)/(a^(1/4)*Sqrt[d + e*x^2])])/(a^(1/4)* 
Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/(2*Sqrt[a])
 

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 218

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

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 1489

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r 
= Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S 
imp[c/(2*r)   Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, 
q}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(-\frac {d \left (\frac {\left (-a e +\sqrt {d^{2} a c}\right ) \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{\sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}-\frac {\left (a e +\sqrt {d^{2} a c}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{\sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2 \sqrt {d^{2} a c}}\)	\(137\)
default	\(\text {Expression too large to display}\)	\(1626\)

Input:

int((e*x^2+d)^(1/2)/(-c*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (101) = 202\).

Time = 0.19 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.36 \[ \int \frac {\sqrt {d+e x^2}}{a-c x^4} \, dx=\frac {1}{8} \, \sqrt {\frac {a c \sqrt {\frac {d^{2}}{a^{3} c}} + e}{a c}} \log \left (\frac {a c d x^{2} \sqrt {\frac {d^{2}}{a^{3} c}} + 2 \, \sqrt {e x^{2} + d} a^{2} c x \sqrt {\frac {a c \sqrt {\frac {d^{2}}{a^{3} c}} + e}{a c}} \sqrt {\frac {d^{2}}{a^{3} c}} + 2 \, d e x^{2} + d^{2}}{x^{2}}\right ) - \frac {1}{8} \, \sqrt {\frac {a c \sqrt {\frac {d^{2}}{a^{3} c}} + e}{a c}} \log \left (\frac {a c d x^{2} \sqrt {\frac {d^{2}}{a^{3} c}} - 2 \, \sqrt {e x^{2} + d} a^{2} c x \sqrt {\frac {a c \sqrt {\frac {d^{2}}{a^{3} c}} + e}{a c}} \sqrt {\frac {d^{2}}{a^{3} c}} + 2 \, d e x^{2} + d^{2}}{x^{2}}\right ) + \frac {1}{8} \, \sqrt {-\frac {a c \sqrt {\frac {d^{2}}{a^{3} c}} - e}{a c}} \log \left (-\frac {a c d x^{2} \sqrt {\frac {d^{2}}{a^{3} c}} + 2 \, \sqrt {e x^{2} + d} a^{2} c x \sqrt {-\frac {a c \sqrt {\frac {d^{2}}{a^{3} c}} - e}{a c}} \sqrt {\frac {d^{2}}{a^{3} c}} - 2 \, d e x^{2} - d^{2}}{x^{2}}\right ) - \frac {1}{8} \, \sqrt {-\frac {a c \sqrt {\frac {d^{2}}{a^{3} c}} - e}{a c}} \log \left (-\frac {a c d x^{2} \sqrt {\frac {d^{2}}{a^{3} c}} - 2 \, \sqrt {e x^{2} + d} a^{2} c x \sqrt {-\frac {a c \sqrt {\frac {d^{2}}{a^{3} c}} - e}{a c}} \sqrt {\frac {d^{2}}{a^{3} c}} - 2 \, d e x^{2} - d^{2}}{x^{2}}\right ) \] Input:

integrate((e*x^2+d)^(1/2)/(-c*x^4+a),x, algorithm="fricas")
 

Output:

1/8*sqrt((a*c*sqrt(d^2/(a^3*c)) + e)/(a*c))*log((a*c*d*x^2*sqrt(d^2/(a^3*c 
)) + 2*sqrt(e*x^2 + d)*a^2*c*x*sqrt((a*c*sqrt(d^2/(a^3*c)) + e)/(a*c))*sqr 
t(d^2/(a^3*c)) + 2*d*e*x^2 + d^2)/x^2) - 1/8*sqrt((a*c*sqrt(d^2/(a^3*c)) + 
 e)/(a*c))*log((a*c*d*x^2*sqrt(d^2/(a^3*c)) - 2*sqrt(e*x^2 + d)*a^2*c*x*sq 
rt((a*c*sqrt(d^2/(a^3*c)) + e)/(a*c))*sqrt(d^2/(a^3*c)) + 2*d*e*x^2 + d^2) 
/x^2) + 1/8*sqrt(-(a*c*sqrt(d^2/(a^3*c)) - e)/(a*c))*log(-(a*c*d*x^2*sqrt( 
d^2/(a^3*c)) + 2*sqrt(e*x^2 + d)*a^2*c*x*sqrt(-(a*c*sqrt(d^2/(a^3*c)) - e) 
/(a*c))*sqrt(d^2/(a^3*c)) - 2*d*e*x^2 - d^2)/x^2) - 1/8*sqrt(-(a*c*sqrt(d^ 
2/(a^3*c)) - e)/(a*c))*log(-(a*c*d*x^2*sqrt(d^2/(a^3*c)) - 2*sqrt(e*x^2 + 
d)*a^2*c*x*sqrt(-(a*c*sqrt(d^2/(a^3*c)) - e)/(a*c))*sqrt(d^2/(a^3*c)) - 2* 
d*e*x^2 - d^2)/x^2)
 

Sympy [F]

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

integrate((e*x**2+d)**(1/2)/(-c*x**4+a),x)
 

Output:

-Integral(sqrt(d + e*x**2)/(-a + c*x**4), x)
 

Maxima [F]

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

integrate((e*x^2+d)^(1/2)/(-c*x^4+a),x, algorithm="maxima")
 

Output:

-integrate(sqrt(e*x^2 + d)/(c*x^4 - a), x)
 

Giac [F(-1)]

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

integrate((e*x^2+d)^(1/2)/(-c*x^4+a),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int((d + e*x^2)^(1/2)/(a - c*x^4),x)
 

Output:

int((d + e*x^2)^(1/2)/(a - c*x^4), x)
 

Reduce [F]

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

int((e*x^2+d)^(1/2)/(-c*x^4+a),x)
 

Output:

int(sqrt(d + e*x**2)/(a - c*x**4),x)