\(\int \frac {A+B x^2+C x^4}{(d+e x^2)^{5/2} (a+c x^4)} \, dx\) [18]

Optimal result

Integrand size = 33, antiderivative size = 637 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\frac {\left (C d^2-B d e+A e^2\right ) x}{3 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^{3/2}}+\frac {\left (c d^2 \left (2 C d^2-e (5 B d-8 A e)\right )-a e^2 \left (4 C d^2-e (B d+2 A e)\right )\right ) x}{3 d^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x^2}}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (c d \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c} d \left (c d^2+a e^2\right )^{5/2}}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (c d \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c} d \left (c d^2+a e^2\right )^{5/2}} \] Output:

1/3*(A*e^2-B*d*e+C*d^2)*x/d/(a*e^2+c*d^2)/(e*x^2+d)^(3/2)+1/3*(c*d^2*(2*C* 
d^2-e*(-8*A*e+5*B*d))-a*e^2*(4*C*d^2-e*(2*A*e+B*d)))*x/d^2/(a*e^2+c*d^2)^2 
/(e*x^2+d)^(1/2)+1/4*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(c*d*(-2*A*c*d* 
e-B*a*e^2+B*c*d^2+2*C*a*d*e)-(e-(a*e^2+c*d^2)^(1/2)/a^(1/2))*(A*c*(-a*e^2+ 
c*d^2)+a*(C*a*e^2-c*d*(-2*B*e+C*d))))*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)/c^(1/2)/d/(a*e^2+c*d^2)^(5/2)+ 
1/4*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(c*d*(-2*A*c*d*e-B*a*e^2+B*c*d^ 
2+2*C*a*d*e)-(e+(a*e^2+c*d^2)^(1/2)/a^(1/2))*(A*c*(-a*e^2+c*d^2)+a*(C*a*e^ 
2-c*d*(-2*B*e+C*d))))*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)/c^(1/2)/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 = 1.51 (sec) , antiderivative size = 839, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )} \, dx=\frac {2 x \left (c C d^4 \left (3 d+2 e x^2\right )-a C d^2 e^2 \left (3 d+4 e x^2\right )+a e^4 \left (3 A d+B d x^2+2 A e x^2\right )+c d^2 e \left (-B d \left (6 d+5 e x^2\right )+A e \left (9 d+8 e x^2\right )\right )\right )+3 d^2 \sqrt {e} \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 {B c^2 d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-2 A c^2 d^3 e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+2 a c C d^3 e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-a B c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-2 B 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}+8 A c^2 d^2 e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-8 a c C d^2 e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+10 a B 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}-4 a A c e^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a^2 C e^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+B c^2 d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 A c^2 d e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a c C d e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a B 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 ]}{6 \left (c d^3+a d e^2\right )^2 \left (d+e x^2\right )^{3/2}} \] Input:

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

Output:

(2*x*(c*C*d^4*(3*d + 2*e*x^2) - a*C*d^2*e^2*(3*d + 4*e*x^2) + a*e^4*(3*A*d 
 + B*d*x^2 + 2*A*e*x^2) + c*d^2*e*(-(B*d*(6*d + 5*e*x^2)) + A*e*(9*d + 8*e 
*x^2))) + 3*d^2*Sqrt[e]*(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 & , (B*c^2*d^4*Log[d + 2*e 
*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2*A*c^2*d^3*e*Log[d + 2*e*x^2 - 
 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 2*a*c*C*d^3*e*Log[d + 2*e*x^2 - 2*Sqr 
t[e]*x*Sqrt[d + e*x^2] - #1] - a*B*c*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x 
*Sqrt[d + e*x^2] - #1] - 2*B*c^2*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d 
+ e*x^2] - #1]*#1 + 8*A*c^2*d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e 
*x^2] - #1]*#1 - 8*a*c*C*d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 
2] - #1]*#1 + 10*a*B*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] 
 - #1]*#1 - 4*a*A*c*e^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1 
]*#1 + 4*a^2*C*e^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 
+ B*c^2*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 2*A 
*c^2*d*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 2*a*c* 
C*d*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - a*B*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) & ])/(6*(c*d^3 + a*d*e^2)^2*(d + 
e*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 553, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {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.

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

\(\Big \downarrow \) 2257

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (\sqrt {-a} B \sqrt {c}+a C-A c\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} \left (\sqrt {c} d-\sqrt {-a} e\right )^{5/2}}-\frac {\left (\sqrt {-a} B \sqrt {c}-a C+A c\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} \left (\sqrt {-a} e+\sqrt {c} d\right )^{5/2}}-\frac {e x \left (2 \sqrt {-a} e+5 \sqrt {c} d\right ) \left (\sqrt {-a} B \sqrt {c}-a C+A c\right )}{6 \sqrt {-a} c d^2 \sqrt {d+e x^2} \left (\sqrt {-a} e+\sqrt {c} d\right )^2}-\frac {e x \left (5 \sqrt {c} d-2 \sqrt {-a} e\right ) \left (\sqrt {-a} B \sqrt {c}+a C-A c\right )}{6 \sqrt {-a} c d^2 \sqrt {d+e x^2} \left (\sqrt {c} d-\sqrt {-a} e\right )^2}-\frac {e x \left (\sqrt {-a} B \sqrt {c}-a C+A c\right )}{6 c d \left (d+e x^2\right )^{3/2} \left (\sqrt {-a} \sqrt {c} d-a e\right )}-\frac {e x \left (\sqrt {-a} B \sqrt {c}+a C-A c\right )}{6 c d \left (d+e x^2\right )^{3/2} \left (\sqrt {-a} \sqrt {c} d+a e\right )}+\frac {2 C x}{3 c d^2 \sqrt {d+e x^2}}+\frac {C x}{3 c d \left (d+e x^2\right )^{3/2}}\)

Input:

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

Output:

(C*x)/(3*c*d*(d + e*x^2)^(3/2)) - ((Sqrt[-a]*B*Sqrt[c] + A*c - a*C)*e*x)/( 
6*c*d*(Sqrt[-a]*Sqrt[c]*d - a*e)*(d + e*x^2)^(3/2)) - ((Sqrt[-a]*B*Sqrt[c] 
 - A*c + a*C)*e*x)/(6*c*d*(Sqrt[-a]*Sqrt[c]*d + a*e)*(d + e*x^2)^(3/2)) + 
(2*C*x)/(3*c*d^2*Sqrt[d + e*x^2]) - ((Sqrt[-a]*B*Sqrt[c] - A*c + a*C)*e*(5 
*Sqrt[c]*d - 2*Sqrt[-a]*e)*x)/(6*Sqrt[-a]*c*d^2*(Sqrt[c]*d - Sqrt[-a]*e)^2 
*Sqrt[d + e*x^2]) - ((Sqrt[-a]*B*Sqrt[c] + A*c - a*C)*e*(5*Sqrt[c]*d + 2*S 
qrt[-a]*e)*x)/(6*Sqrt[-a]*c*d^2*(Sqrt[c]*d + Sqrt[-a]*e)^2*Sqrt[d + e*x^2] 
) + ((Sqrt[-a]*B*Sqrt[c] - A*c + a*C)*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)^ 
(5/2)) - ((Sqrt[-a]*B*Sqrt[c] + A*c - a*C)*ArcTanh[(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)^(5/2))
 

Defintions of rubi rules used

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. \(1286\) vs. \(2(549)=1098\).

Time = 2.22 (sec) , antiderivative size = 1287, normalized size of antiderivative = 2.02

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1287\)
default	\(\text {Expression too large to display}\)	\(3916\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((C*x**4+B*x**2+A)/(e*x**2+d)**(5/2)/(c*x**4+a),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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