\(\int \frac {1}{(d+e x^2)^{3/2} (a+c x^4)} \, dx\) [387]

Optimal result

Integrand size = 21, antiderivative size = 399 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )} \, dx=\frac {e^2 x}{d \left (c d^2+a e^2\right ) \sqrt {d+e x^2}}-\frac {\sqrt {c} \left (2 \sqrt {a} e-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^{3/2}}-\frac {\sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 \sqrt {a} e+\sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^{3/2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.19 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )} \, dx=\frac {\frac {2 e^2 x}{\sqrt {d+e x^2}}-c d e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-6 d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{2 c d^3+2 a d e^2} \] Input:

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

Output:

((2*e^2*x)/Sqrt[d + e*x^2] - c*d*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] - 6*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) & ])/(2*c*d 
^3 + 2*a*d*e^2)
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1487, 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.

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

\(\Big \downarrow \) 1487

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 2257

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^2*x)/(d*(c*d^2 + a*e^2)*Sqrt[d + e*x^2]) + (c*(-1/2*((Sqrt[c]*d + Sqrt[ 
-a]*e)*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x^2] 
)])/((-a)^(3/4)*Sqrt[c]*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]) - ((Sqrt[c]*d - Sqrt 
[-a]*e)*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x^ 
2])])/(2*(-a)^(3/4)*Sqrt[c]*Sqrt[Sqrt[c]*d + Sqrt[-a]*e])))/(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 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. \(772\) vs. \(2(317)=634\).

Time = 1.01 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.94

method	result	size
pseudoelliptic	\(-\frac {-\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \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 (\left (-2 e \sqrt {a}-\sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+a e \sqrt {a \,e^{2}+c \,d^{2}}+2 a^{\frac {3}{2}} e^{2}\right ) \sqrt {e \,x^{2}+d}\, \ln \left (\frac {\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}-\sqrt {a}\, \left (e \,x^{2}+d \right )}{x^{2}}\right )}{4}+\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \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 (\left (-2 e \sqrt {a}-\sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+a e \sqrt {a \,e^{2}+c \,d^{2}}+2 a^{\frac {3}{2}} e^{2}\right ) \sqrt {e \,x^{2}+d}\, \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}-2 a^{\frac {3}{2}} e^{2} 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}\, \sqrt {a \,e^{2}+c \,d^{2}}+d^{2} \left (a \sqrt {a \,e^{2}+c \,d^{2}}-2 a^{\frac {3}{2}} e \right ) \left (\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 )-\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 )\right ) c \sqrt {e \,x^{2}+d}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}} a^{\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 {e \,x^{2}+d}\, d}\)	\(773\)
default	\(\text {Expression too large to display}\)	\(2100\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3803 vs. \(2 (319) = 638\).

Time = 10.93 (sec) , antiderivative size = 3803, normalized size of antiderivative = 9.53 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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