\(\int \frac {1}{(d+e x^2)^{3/2} (a+c x^4)^2} \, dx\) [395]

Optimal result

Integrand size = 21, antiderivative size = 504 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )^2} \, dx=-\frac {e^2 \left (c d^2-2 a e^2\right ) x}{2 a d \left (c d^2+a e^2\right )^2 \sqrt {d+e x^2}}+\frac {c x \left (d-e x^2\right )}{4 a \left (c d^2+a e^2\right ) \sqrt {d+e x^2} \left (a+c x^4\right )}-\frac {3 \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 a e^3+\left (c d^2+3 a e^2\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} \left (c d^2+a e^2\right )^{5/2}}-\frac {3 \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 a e^3+\left (c d^2+3 a e^2\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} \left (c d^2+a e^2\right )^{5/2}} \] Output:

-1/2*e^2*(-2*a*e^2+c*d^2)*x/a/d/(a*e^2+c*d^2)^2/(e*x^2+d)^(1/2)+1/4*c*x*(- 
e*x^2+d)/a/(a*e^2+c*d^2)/(e*x^2+d)^(1/2)/(c*x^4+a)-3/16*c^(1/2)*(a^(1/2)*e 
+(a*e^2+c*d^2)^(1/2))^(1/2)*(2*a*e^3+(3*a*e^2+c*d^2)*(e-(a*e^2+c*d^2)^(1/2 
)/a^(1/2)))*arctan(2^(1/2)*a^(1/4)*c^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2)) 
^(1/2)*x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))-c*d*x^2) 
)*2^(1/2)/a^(5/4)/(a*e^2+c*d^2)^(5/2)-3/16*c^(1/2)*(-a^(1/2)*e+(a*e^2+c*d^ 
2)^(1/2))^(1/2)*(2*a*e^3+(3*a*e^2+c*d^2)*(e+(a*e^2+c*d^2)^(1/2)/a^(1/2)))* 
arctanh(2^(1/2)*a^(1/4)*c^(1/2)*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*( 
e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e-(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/ 
a^(5/4)/(a*e^2+c*d^2)^(5/2)
 

Mathematica [C] (verified)

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

Time = 0.89 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )^2} \, dx=-\frac {2 e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {17 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+16 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-6 c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]+\frac {\frac {x \left (-4 a^2 e^4+a c e^2 \left (d^2+d e x^2-4 e^2 x^4\right )+c^2 d^2 \left (-d^2+d e x^2+2 e^2 x^4\right )\right )}{d \sqrt {d+e x^2} \left (a+c x^4\right )}+e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {31 a c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+32 a^2 e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+6 c^2 d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+12 a c d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-a c e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-c d^3+3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}-3 c d \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ]}{a}}{4 \left (c d^2+a e^2\right )^2} \] Input:

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

Output:

-1/4*(2*e^(7/2)*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 & , (17*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + 
 e*x^2] - #1] + 16*a*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - # 
1] - 6*c*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*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) & ] + ((x*(-4*a^2*e^4 + a*c*e^2*(d^2 + 
 d*e*x^2 - 4*e^2*x^4) + c^2*d^2*(-d^2 + d*e*x^2 + 2*e^2*x^4)))/(d*Sqrt[d + 
 e*x^2]*(a + c*x^4)) + 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 & , (31*a*c*d^2*e^2*Log[d + 2*e*x^2 - 
 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 32*a^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e 
]*x*Sqrt[d + e*x^2] - #1] + 6*c^2*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d 
 + e*x^2] - #1]*#1 + 12*a*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e 
*x^2] - #1]*#1 - 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) & ])/a 
)/(c*d^2 + a*e^2)^2
 

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.

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

\(\Big \downarrow \) 1571

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

Input:

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

Leaf count of result is larger than twice the leaf count of optimal. \(1115\) vs. \(2(416)=832\).

Time = 1.95 (sec) , antiderivative size = 1116, normalized size of antiderivative = 2.21

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1116\)
default	\(\text {Expression too large to display}\)	\(5390\)

Input:

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

Output:

-9/8/(e*x^2+d)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1 
/2)-2*a*e)^(1/2)/a^(5/2)/(a*e^2+c*d^2)^(5/2)*(-1/4*(4*(a*e^2+c*d^2)^(1/2)* 
a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)*(e*x^2+d)^(1/2)*(2*(a*(a*e^ 
2+c*d^2))^(1/2)+2*a*e)^(1/2)*((-(a*e^2+1/3*c*d^2)*(c*x^4+a)*(a*e^2+c*d^2)^ 
(1/2)-1/3*(c*(5*e^2*x^4+d^2)*a^(3/2)+c^2*d^2*x^4*a^(1/2)+5*e^2*a^(5/2))*e) 
*(a*(a*e^2+c*d^2))^(1/2)+((a*e^2+1/3*c*d^2)*a*(c*x^4+a)*(a*e^2+c*d^2)^(1/2 
)+5/3*e*(1/5*c*(5*e^2*x^4+d^2)*a^(5/2)+1/5*c^2*d^2*x^4*a^(3/2)+a^(7/2)*e^2 
))*e)*ln((a^(1/2)*(e*x^2+d)-(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a 
*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)+1/4*(4*(a*e^2+c*d^2)^(1/2)*a^(1/ 
2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)*(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d 
^2))^(1/2)+2*a*e)^(1/2)*((-(a*e^2+1/3*c*d^2)*(c*x^4+a)*(a*e^2+c*d^2)^(1/2) 
-1/3*(c*(5*e^2*x^4+d^2)*a^(3/2)+c^2*d^2*x^4*a^(1/2)+5*e^2*a^(5/2))*e)*(a*( 
a*e^2+c*d^2))^(1/2)+((a*e^2+1/3*c*d^2)*a*(c*x^4+a)*(a*e^2+c*d^2)^(1/2)+5/3 
*e*(1/5*c*(5*e^2*x^4+d^2)*a^(5/2)+1/5*c^2*d^2*x^4*a^(3/2)+a^(7/2)*e^2))*e) 
*ln((a^(1/2)*(e*x^2+d)+(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^( 
1/2)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)-8/9*x*(a*e^2+c*d^2)^(1/2)*(1/4*c^2*d^ 
2*(e*x^2+d)*(-2*e*x^2+d)*a^(3/2)+(-1/4*c*(-4*e^2*x^4+d*e*x^2+d^2)*a^(5/2)+ 
a^(7/2)*e^2)*e^2)*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2) 
-2*a*e)^(1/2)+(arctan((2*a^(1/2)*(e*x^2+d)^(1/2)+(2*(a*(a*e^2+c*d^2))^(1/2 
)+2*a*e)^(1/2)*x)/x/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7841 vs. \(2 (419) = 838\).

Time = 170.30 (sec) , antiderivative size = 7841, normalized size of antiderivative = 15.56 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )^2} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, a^{2} d +\sqrt {e \,x^{2}+d}\, a^{2} e \,x^{2}+2 \sqrt {e \,x^{2}+d}\, a c d \,x^{4}+2 \sqrt {e \,x^{2}+d}\, a c e \,x^{6}+\sqrt {e \,x^{2}+d}\, c^{2} d \,x^{8}+\sqrt {e \,x^{2}+d}\, c^{2} e \,x^{10}}d x \] Input:

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

Output:

int(1/(sqrt(d + e*x**2)*a**2*d + sqrt(d + e*x**2)*a**2*e*x**2 + 2*sqrt(d + 
 e*x**2)*a*c*d*x**4 + 2*sqrt(d + e*x**2)*a*c*e*x**6 + sqrt(d + e*x**2)*c** 
2*d*x**8 + sqrt(d + e*x**2)*c**2*e*x**10),x)