\(\int \frac {1}{\sqrt {d+e x^2} (a-c x^4)^3} \, dx\) [363]

Optimal result

Integrand size = 22, antiderivative size = 319 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^3} \, dx=\frac {c x \left (d-e x^2\right ) \sqrt {d+e x^2}}{8 a \left (c d^2-a e^2\right ) \left (a-c x^4\right )^2}+\frac {c x \sqrt {d+e x^2} \left (d \left (7 c d^2-13 a e^2\right )-6 e \left (c d^2-2 a e^2\right ) x^2\right )}{32 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^4\right )}+\frac {\left (21 c d^2-50 \sqrt {a} \sqrt {c} d e+32 a e^2\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (21 c d^2+50 \sqrt {a} \sqrt {c} d e+32 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \] Output:

1/8*c*x*(-e*x^2+d)*(e*x^2+d)^(1/2)/a/(-a*e^2+c*d^2)/(-c*x^4+a)^2+1/32*c*x* 
(e*x^2+d)^(1/2)*(d*(-13*a*e^2+7*c*d^2)-6*e*(-2*a*e^2+c*d^2)*x^2)/a^2/(-a*e 
^2+c*d^2)^2/(-c*x^4+a)+1/64*(21*c*d^2-50*a^(1/2)*c^(1/2)*d*e+32*a*e^2)*arc 
tan((c^(1/2)*d-a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(11/4)/(c^(1/ 
2)*d-a^(1/2)*e)^(5/2)+1/64*(21*c*d^2+50*a^(1/2)*c^(1/2)*d*e+32*a*e^2)*arct 
anh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(11/4)/(c^(1/ 
2)*d+a^(1/2)*e)^(5/2)
 

Mathematica [C] (verified)

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

Time = 1.53 (sec) , antiderivative size = 1214, normalized size of antiderivative = 3.81 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[1/(Sqrt[d + e*x^2]*(a - c*x^4)^3),x]
 

Output:

-1/32*((c^3*x*Sqrt[d + e*x^2]*(a^2*e^2*(17*d - 16*e*x^2) + c^2*d^2*x^4*(7* 
d - 6*e*x^2) + a*c*(-11*d^3 + 10*d^2*e*x^2 - 13*d*e^2*x^4 + 12*e^3*x^6)))/ 
(a - c*x^4)^2 + (64*a*e^(7/2)*(-(c*d^2) + a*e^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 & , (141*c^2*d^4*Lo 
g[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 656*a*c*d^2*e^2*Log[d 
+ 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 512*a^2*e^4*Log[d + 2*e*x^ 
2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 110*c^2*d^3*Log[d + 2*e*x^2 - 2*Sq 
rt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 128*a*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[ 
e]*x*Sqrt[d + e*x^2] - #1]*#1 + 29*c^2*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*S 
qrt[d + e*x^2] - #1]*#1^2 - 32*a*c*e^2*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) & ])/d^3 + (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 & , (4*c^4*d^8*Log[d + 2*e*x^2 - 2*Sqrt[e]* 
x*Sqrt[d + e*x^2] - #1] + 9017*a*c^3*d^6*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x 
*Sqrt[d + e*x^2] - #1] - 51008*a^2*c^2*d^4*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e] 
*x*Sqrt[d + e*x^2] - #1] + 74752*a^3*c*d^2*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e] 
*x*Sqrt[d + e*x^2] - #1] - 32768*a^4*e^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqr 
t[d + e*x^2] - #1] + 34*c^4*d^7*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x 
^2] - #1]*#1 - 7120*a*c^3*d^5*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e 
*x^2] - #1]*#1 + 15296*a^2*c^2*d^3*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sq...
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^3 \sqrt {d+e x^2}} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^3 \sqrt {d+e x^2}}dx\)

Input:

Int[1/(Sqrt[d + e*x^2]*(a - c*x^4)^3),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1571

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U 
nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
 

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.29

method	result	size
pseudoelliptic	\(-\frac {17 \left (\frac {7 \left (\left (\frac {16}{7} a^{2} e^{4}-\frac {47}{14} a c \,d^{2} e^{2}+\frac {3}{2} c^{2} d^{4}\right ) \sqrt {d^{2} a c}+a c \,d^{2} e \left (a \,e^{2}-\frac {4 c \,d^{2}}{7}\right )\right ) \left (-c \,x^{4}+a \right )^{2} \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{17}+\sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (\frac {7 \left (\left (-\frac {16}{7} a^{2} e^{4}+\frac {47}{14} a c \,d^{2} e^{2}-\frac {3}{2} c^{2} d^{4}\right ) \sqrt {d^{2} a c}+a c \,d^{2} e \left (a \,e^{2}-\frac {4 c \,d^{2}}{7}\right )\right ) \left (-c \,x^{4}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{17}+x \sqrt {e \,x^{2}+d}\, \left (e^{2} \left (-\frac {16 e \,x^{2}}{17}+d \right ) a^{2}-\frac {11 \left (-\frac {12}{11} e^{3} x^{6}+\frac {13}{11} d \,e^{2} x^{4}-\frac {10}{11} d^{2} e \,x^{2}+d^{3}\right ) c a}{17}+\frac {7 x^{4} \left (-\frac {6 e \,x^{2}}{7}+d \right ) d^{2} c^{2}}{17}\right ) c \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\right )\right )}{32 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, a^{2} \left (-c \,x^{4}+a \right )^{2}}\)	\(411\)
default	\(\text {Expression too large to display}\)	\(4764\)

Input:

int(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-17/32/(a*e^2-c*d^2)^2*(7/17*((16/7*a^2*e^4-47/14*a*c*d^2*e^2+3/2*c^2*d^4) 
*(d^2*a*c)^(1/2)+a*c*d^2*e*(a*e^2-4/7*c*d^2))*(-c*x^4+a)^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)^(1/2)*(7/17*((-16/7*a^2*e^4+47/14*a*c*d^2*e 
^2-3/2*c^2*d^4)*(d^2*a*c)^(1/2)+a*c*d^2*e*(a*e^2-4/7*c*d^2))*(-c*x^4+a)^2* 
arctanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a)^(1/2))+x*(e*x^2+d)^( 
1/2)*(e^2*(-16/17*e*x^2+d)*a^2-11/17*(-12/11*e^3*x^6+13/11*d*e^2*x^4-10/11 
*d^2*e*x^2+d^3)*c*a+7/17*x^4*(-6/7*e*x^2+d)*d^2*c^2)*c*(d^2*a*c)^(1/2)*((a 
*e+(d^2*a*c)^(1/2))*a)^(1/2)))/(d^2*a*c)^(1/2)/((-a*e+(d^2*a*c)^(1/2))*a)^ 
(1/2)/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)/a^2/(-c*x^4+a)^2
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x**2+d)**(1/2)/(-c*x**4+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^3} \, dx=\int { -\frac {1}{{\left (c x^{4} - a\right )}^{3} \sqrt {e x^{2} + d}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^3,x, algorithm="maxima")
 

Output:

-integrate(1/((c*x^4 - a)^3*sqrt(e*x^2 + d)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x^2+d)^(1/2)/(-c*x^4+a)^3,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^3} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^3\,\sqrt {e\,x^2+d}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a-c x^4\right )^3} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, \left (-c \,x^{4}+a \right )^{3}}d x \] Input:

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

Output:

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