\(\int \frac {1}{(d+e x^2)^{5/2} (a+c x^4)} \, dx\) [388]

Optimal result

Integrand size = 21, antiderivative size = 494 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\frac {e^2 x}{3 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^{3/2}}+\frac {2 e^2 \left (4 c d^2+a e^2\right ) x}{3 d^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x^2}}-\frac {\sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 c d^2 e+\left (c d^2-a e^2\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} d \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 c d^2 e+\left (c d^2-a e^2\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} d \left (c d^2+a e^2\right )^{5/2}} \] Output:

1/3*e^2*x/d/(a*e^2+c*d^2)/(e*x^2+d)^(3/2)+2/3*e^2*(a*e^2+4*c*d^2)*x/d^2/(a 
*e^2+c*d^2)^2/(e*x^2+d)^(1/2)-1/4*c^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^ 
(1/2)*(2*c*d^2*e+(-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^(1/4)/d/( 
a*e^2+c*d^2)^(5/2)-1/4*c^(1/2)*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(2*c 
*d^2*e+(-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^(1/4)/d/(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 = 0.39 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\frac {a e^4 x \left (3 d+2 e x^2\right )+c d^2 e^2 x \left (9 d+8 e x^2\right )-3 c d^2 e^{3/2} \left (d+e x^2\right )^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-4 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{3 \left (c d^3+a d e^2\right )^2 \left (d+e x^2\right )^{3/2}} \] Input:

Integrate[1/((d + e*x^2)^(5/2)*(a + c*x^4)),x]
 

Output:

(a*e^4*x*(3*d + 2*e*x^2) + c*d^2*e^2*x*(9*d + 8*e*x^2) - 3*c*d^2*e^(3/2)*( 
d + e*x^2)^(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 & , (c*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e 
*x^2] - #1] - 4*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]* 
#1 + 2*a*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*d* 
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) & ])/(3*(c*d^3 + a*d*e^2)^2*(d + e 
*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1487, 209, 208, 2257, 2009}

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.

\(\displaystyle \int \frac {1}{\left (a+c x^4\right ) \left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1487

\(\displaystyle \frac {e^2 \int \frac {1}{\left (e x^2+d\right )^{5/2}}dx}{a e^2+c d^2}+\frac {c \int \frac {d-e x^2}{\left (e x^2+d\right )^{3/2} \left (c x^4+a\right )}dx}{a e^2+c d^2}\)

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 2257

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \left (-\frac {\left (\sqrt {-a} e+\sqrt {c} d\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2}}-\frac {\left (\sqrt {c} d-\sqrt {-a} e\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2}}-\frac {e x \left (\sqrt {c} d-\sqrt {-a} e\right )}{2 \sqrt {c} d \sqrt {d+e x^2} \left (\sqrt {-a} \sqrt {c} d-a e\right )}+\frac {e x \left (\sqrt {-a} e+\sqrt {c} d\right )}{2 \sqrt {c} d \sqrt {d+e x^2} \left (\sqrt {-a} \sqrt {c} d+a e\right )}\right )}{a e^2+c d^2}+\frac {e^2 \left (\frac {2 x}{3 d^2 \sqrt {d+e x^2}}+\frac {x}{3 d \left (d+e x^2\right )^{3/2}}\right )}{a e^2+c d^2}\)

Input:

Int[1/((d + e*x^2)^(5/2)*(a + c*x^4)),x]
 

Output:

(e^2*(x/(3*d*(d + e*x^2)^(3/2)) + (2*x)/(3*d^2*Sqrt[d + e*x^2])))/(c*d^2 + 
 a*e^2) + (c*(-1/2*(e*(Sqrt[c]*d - Sqrt[-a]*e)*x)/(Sqrt[c]*d*(Sqrt[-a]*Sqr 
t[c]*d - a*e)*Sqrt[d + e*x^2]) + (e*(Sqrt[c]*d + Sqrt[-a]*e)*x)/(2*Sqrt[c] 
*d*(Sqrt[-a]*Sqrt[c]*d + a*e)*Sqrt[d + e*x^2]) - ((Sqrt[c]*d + Sqrt[-a]*e) 
*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x^2])])/(2 
*(-a)^(3/4)*(Sqrt[c]*d - Sqrt[-a]*e)^(3/2)) - ((Sqrt[c]*d - Sqrt[-a]*e)*Ar 
cTanh[(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x^2])])/(2*( 
-a)^(3/4)*(Sqrt[c]*d + Sqrt[-a]*e)^(3/2))))/(c*d^2 + a*e^2)
 

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

rule 1487

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Simp[e^2 
/(c*d^2 + a*e^2)   Int[(d + e*x^2)^q, x], x] + Simp[c/(c*d^2 + a*e^2)   Int 
[(d + e*x^2)^(q + 1)*((d - e*x^2)/(a + c*x^4)), x], x] /; FreeQ[{a, c, d, e 
}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q] && LtQ[q, -1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2257

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(923\) vs. \(2(406)=812\).

Time = 1.00 (sec) , antiderivative size = 924, normalized size of antiderivative = 1.87

method	result	size
pseudoelliptic	\(-\frac {\frac {\left (\left (\left (-a \,e^{2}+c \,d^{2}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+3 \sqrt {a}\, c \,d^{2} e -a^{\frac {3}{2}} e^{3}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (\left (a^{2} e^{2}-d^{2} a c \right ) \sqrt {a \,e^{2}+c \,d^{2}}-3 c \,d^{2} e \,a^{\frac {3}{2}}+e^{3} a^{\frac {5}{2}}\right ) e \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )-\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{4}-\frac {\left (\left (\left (-a \,e^{2}+c \,d^{2}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+3 \sqrt {a}\, c \,d^{2} e -a^{\frac {3}{2}} e^{3}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (\left (a^{2} e^{2}-d^{2} a c \right ) \sqrt {a \,e^{2}+c \,d^{2}}-3 c \,d^{2} e \,a^{\frac {3}{2}}+e^{3} a^{\frac {5}{2}}\right ) e \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{4}-6 x \sqrt {a \,e^{2}+c \,d^{2}}\, \left (d^{2} \left (\frac {8 e \,x^{2}}{9}+d \right ) c \,a^{\frac {3}{2}}+\frac {\left (\frac {2 e \,x^{2}}{3}+d \right ) e^{2} a^{\frac {5}{2}}}{3}\right ) e^{2} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}+\left (\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right ) d^{2} c \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (\left (a^{2} e^{2}-d^{2} a c \right ) \sqrt {a \,e^{2}+c \,d^{2}}+3 c \,d^{2} e \,a^{\frac {3}{2}}-e^{3} a^{\frac {5}{2}}\right )}{2 a^{\frac {3}{2}} \left (a \,e^{2}+c \,d^{2}\right )^{\frac {5}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}} d^{2}}\)	\(924\)
default	\(\text {Expression too large to display}\)	\(3806\)

Input:

int(1/(e*x^2+d)^(5/2)/(c*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

-1/2/a^(3/2)/(a*e^2+c*d^2)^(5/2)/(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)*(1/4*(((-a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2)+3*a 
^(1/2)*c*d^2*e-a^(3/2)*e^3)*(a*(a*e^2+c*d^2))^(1/2)+((a^2*e^2-a*c*d^2)*(a* 
e^2+c*d^2)^(1/2)-3*c*d^2*e*a^(3/2)+e^3*a^(5/2))*e)*(e*x^2+d)^(3/2)*(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)*(2*(a*(a*e^ 
2+c*d^2))^(1/2)+2*a*e)^(1/2)*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*(((-a 
*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2)+3*a^(1/2)*c*d^2*e-a^(3/2)*e^3)*(a*(a*e^2+c 
*d^2))^(1/2)+((a^2*e^2-a*c*d^2)*(a*e^2+c*d^2)^(1/2)-3*c*d^2*e*a^(3/2)+e^3* 
a^(5/2))*e)*(e*x^2+d)^(3/2)*(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)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*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)-6*x*(a*e^2+c*d^2)^(1/2)*(d^2*(8/9*e*x^2+d)*c*a^( 
3/2)+1/3*(2/3*e*x^2+d)*e^2*a^(5/2))*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*(a*e^2+c*d^2))^(1/2)+2* 
a*e)^(1/2)*x-2*a^(1/2)*(e*x^2+d)^(1/2))/x/(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))^(1/2)-2*a*e)^(1/2)))*d^2*c*(e*x^2+d)^(3/2)*((a^2*e^2- 
a*c*d^2)*(a*e^2+c*d^2)^(1/2)+3*c*d^2*e*a^(3/2)-e^3*a^(5/2)))/(e*x^2+d)^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6557 vs. \(2 (408) = 816\).

Time = 46.72 (sec) , antiderivative size = 6557, normalized size of antiderivative = 13.27 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x^2+d)^(5/2)/(c*x^4+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\int \frac {1}{\left (a + c x^{4}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(e*x**2+d)**(5/2)/(c*x**4+a),x)
 

Output:

Integral(1/((a + c*x**4)*(d + e*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(5/2)/(c*x^4+a),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^4 + a)*(e*x^2 + d)^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x^2+d)^(5/2)/(c*x^4+a),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\int \frac {1}{\left (c\,x^4+a\right )\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

int(1/((a + c*x^4)*(d + e*x^2)^(5/2)),x)
 

Output:

int(1/((a + c*x^4)*(d + e*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, a \,d^{2}+2 \sqrt {e \,x^{2}+d}\, a d e \,x^{2}+\sqrt {e \,x^{2}+d}\, a \,e^{2} x^{4}+\sqrt {e \,x^{2}+d}\, c \,d^{2} x^{4}+2 \sqrt {e \,x^{2}+d}\, c d e \,x^{6}+\sqrt {e \,x^{2}+d}\, c \,e^{2} x^{8}}d x \] Input:

int(1/(e*x^2+d)^(5/2)/(c*x^4+a),x)
 

Output:

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