\(\int \frac {(d+e x^2)^{3/2} (A+B x^2+C x^4)}{(a+b x^2+c x^4)^2} \, dx\) [161]

Optimal result

Integrand size = 38, antiderivative size = 867 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=-\frac {(A b c-2 a B c+a b C) e x \sqrt {d+e x^2}}{2 a c \left (b^2-4 a c\right )}+\frac {x \left (A \left (b^2-2 a c\right )-a (b B-2 a C)+(A b c-2 a B c+a b C) x^2\right ) \left (d+e x^2\right )^{3/2}}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (A c^2 \left (b c d^2-6 a c d e+2 a b e^2\right )+a \left (C (c d-b e) \left (b c d+2 b^2 e-10 a c e\right )-B c^2 \left (2 c d^2-e (3 b d-4 a e)\right )\right )-\frac {A c^2 \left (12 a c^2 d^2-8 a b c d e-b^2 \left (c d^2-2 a e^2\right )\right )+a \left (b^3 c C d e-2 b^4 C e^2-4 b c^2 \left (B c d^2+3 a C d e+a B e^2\right )-4 a c^2 \left (4 a C e^2-c d (C d+B e)\right )+b^2 c \left (14 a C e^2+c d (C d+3 B e)\right )\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 a c^2 \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (A c^2 \left (b c d^2-6 a c d e+2 a b e^2\right )+a \left (C (c d-b e) \left (b c d+2 b^2 e-10 a c e\right )-B c^2 \left (2 c d^2-e (3 b d-4 a e)\right )\right )+\frac {A c^2 \left (12 a c^2 d^2-8 a b c d e-b^2 \left (c d^2-2 a e^2\right )\right )+a \left (b^3 c C d e-2 b^4 C e^2-4 b c^2 \left (B c d^2+3 a C d e+a B e^2\right )-4 a c^2 \left (4 a C e^2-c d (C d+B e)\right )+b^2 c \left (14 a C e^2+c d (C d+3 B e)\right )\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 a c^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {C e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2} \] Output:

-1/2*(A*b*c-2*B*a*c+C*a*b)*e*x*(e*x^2+d)^(1/2)/a/c/(-4*a*c+b^2)+1/2*x*(A*( 
-2*a*c+b^2)-a*(B*b-2*C*a)+(A*b*c-2*B*a*c+C*a*b)*x^2)*(e*x^2+d)^(3/2)/a/(-4 
*a*c+b^2)/(c*x^4+b*x^2+a)+1/2*(A*c^2*(2*a*b*e^2-6*a*c*d*e+b*c*d^2)+a*(C*(- 
b*e+c*d)*(-10*a*c*e+2*b^2*e+b*c*d)-B*c^2*(2*c*d^2-e*(-4*a*e+3*b*d)))-(A*c^ 
2*(12*a*c^2*d^2-8*a*b*c*d*e-b^2*(-2*a*e^2+c*d^2))+a*(b^3*c*C*d*e-2*b^4*C*e 
^2-4*b*c^2*(B*a*e^2+B*c*d^2+3*C*a*d*e)-4*a*c^2*(4*C*a*e^2-c*d*(B*e+C*d))+b 
^2*c*(14*C*a*e^2+c*d*(3*B*e+C*d))))/(-4*a*c+b^2)^(1/2))*arctan((2*c*d-(b-( 
-4*a*c+b^2)^(1/2))*e)^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(e*x^2+d)^(1/2) 
)/a/c^2/(-4*a*c+b^2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^( 
1/2))*e)^(1/2)+1/2*(A*c^2*(2*a*b*e^2-6*a*c*d*e+b*c*d^2)+a*(C*(-b*e+c*d)*(- 
10*a*c*e+2*b^2*e+b*c*d)-B*c^2*(2*c*d^2-e*(-4*a*e+3*b*d)))+(A*c^2*(12*a*c^2 
*d^2-8*a*b*c*d*e-b^2*(-2*a*e^2+c*d^2))+a*(b^3*c*C*d*e-2*b^4*C*e^2-4*b*c^2* 
(B*a*e^2+B*c*d^2+3*C*a*d*e)-4*a*c^2*(4*C*a*e^2-c*d*(B*e+C*d))+b^2*c*(14*C* 
a*e^2+c*d*(3*B*e+C*d))))/(-4*a*c+b^2)^(1/2))*arctan((2*c*d-(b+(-4*a*c+b^2) 
^(1/2))*e)^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(e*x^2+d)^(1/2))/a/c^2/(-4 
*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1 
/2)+C*e^(3/2)*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/c^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(39198\) vs. \(2(867)=1734\).

Time = 17.90 (sec) , antiderivative size = 39198, normalized size of antiderivative = 45.21 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Result too large to show} \] Input:

Integrate[((d + e*x^2)^(3/2)*(A + B*x^2 + C*x^4))/(a + b*x^2 + c*x^4)^2,x]
 

Output:

Result too large to show
 

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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

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

Input:

Int[((d + e*x^2)^(3/2)*(A + B*x^2 + C*x^4))/(a + b*x^2 + c*x^4)^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

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

rule 2256

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

Maple [A] (verified)

Time = 3.68 (sec) , antiderivative size = 948, normalized size of antiderivative = 1.09

method	result	size
pseudoelliptic	\(-\frac {\left (\left (8 C \,a^{3} c \,e^{2}+\left (\left (-2 B d e -2 C \,d^{2}\right ) c^{2}+C b c d e -2 b^{2} C \,e^{2}\right ) a^{2}+\left (-6 A c d +b \left (A e +B d \right )\right ) c^{2} d a +A \,b^{2} c^{2} d^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+4 \left (\left (\left (-2 B \,e^{2}-5 C d e \right ) c^{2}+3 C b c \,e^{2}\right ) a^{3}+\left (\left (-3 d e A -B \,d^{2}\right ) c^{3}+c^{2} \left (A \,e^{2}+2 B d e +C \,d^{2}\right ) b +\frac {b^{2} c C d e}{4}-\frac {b^{3} C \,e^{2}}{2}\right ) a^{2}-\frac {b \,c^{2} d \left (-8 A c d +b \left (A e +B d \right )\right ) a}{4}-\frac {A \,b^{3} c^{2} d^{2}}{4}\right ) d \right ) \left (c \,x^{4}+b \,x^{2}+a \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )-\left (\left (\left (8 C \,a^{3} c \,e^{2}+\left (\left (-2 B d e -2 C \,d^{2}\right ) c^{2}+C b c d e -2 b^{2} C \,e^{2}\right ) a^{2}+\left (-6 A c d +b \left (A e +B d \right )\right ) c^{2} d a +A \,b^{2} c^{2} d^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-4 \left (\left (\left (-2 B \,e^{2}-5 C d e \right ) c^{2}+3 C b c \,e^{2}\right ) a^{3}+\left (\left (-3 d e A -B \,d^{2}\right ) c^{3}+c^{2} \left (A \,e^{2}+2 B d e +C \,d^{2}\right ) b +\frac {b^{2} c C d e}{4}-\frac {b^{3} C \,e^{2}}{2}\right ) a^{2}-\frac {b \,c^{2} d \left (-8 A c d +b \left (A e +B d \right )\right ) a}{4}-\frac {A \,b^{3} c^{2} d^{2}}{4}\right ) d \right ) \left (c \,x^{4}+b \,x^{2}+a \right ) \sqrt {2}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )+2 \left (8 \left (a c -\frac {b^{2}}{4}\right ) e^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}+a \right ) a C \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+c \left (\left (\left (-2 C d -2 \left (C \,x^{2}+B \right ) e \right ) c +C b e \right ) a^{2}+\left (\left (\left (2 B \,x^{2}+2 A \right ) d +2 A e \,x^{2}\right ) c^{2}+c \left (\left (-C \,x^{2}+B \right ) d +e \left (-B \,x^{2}+A \right )\right ) b +C \,b^{2} e \,x^{2}\right ) a -A b c d \left (c \,x^{2}+b \right )\right ) \sqrt {e \,x^{2}+d}\, x \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}{4 \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (4 a c -b^{2}\right ) a \left (c \,x^{4}+b \,x^{2}+a \right ) c^{2}}\)	\(948\)
default	\(\text {Expression too large to display}\)	\(1216\)

Input:

int((e*x^2+d)^(3/2)*(C*x^4+B*x^2+A)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERB 
OSE)
 

Output:

-1/4*(((8*C*a^3*c*e^2+((-2*B*d*e-2*C*d^2)*c^2+C*b*c*d*e-2*b^2*C*e^2)*a^2+( 
-6*A*c*d+b*(A*e+B*d))*c^2*d*a+A*b^2*c^2*d^2)*(-4*(a*c-1/4*b^2)*d^2)^(1/2)+ 
4*(((-2*B*e^2-5*C*d*e)*c^2+3*C*b*c*e^2)*a^3+((-3*A*d*e-B*d^2)*c^3+c^2*(A*e 
^2+2*B*d*e+C*d^2)*b+1/4*b^2*c*C*d*e-1/2*b^3*C*e^2)*a^2-1/4*b*c^2*d*(-8*A*c 
*d+b*(A*e+B*d))*a-1/4*A*b^3*c^2*d^2)*d)*(c*x^4+b*x^2+a)*((-2*a*e+b*d+(-4*( 
a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)*2^(1/2)*arctanh(a*(e*x^2+d)^(1/2)/x*2^(1 
/2)/((2*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))-(((8*C*a^3*c*e^2+( 
(-2*B*d*e-2*C*d^2)*c^2+C*b*c*d*e-2*b^2*C*e^2)*a^2+(-6*A*c*d+b*(A*e+B*d))*c 
^2*d*a+A*b^2*c^2*d^2)*(-4*(a*c-1/4*b^2)*d^2)^(1/2)-4*(((-2*B*e^2-5*C*d*e)* 
c^2+3*C*b*c*e^2)*a^3+((-3*A*d*e-B*d^2)*c^3+c^2*(A*e^2+2*B*d*e+C*d^2)*b+1/4 
*b^2*c*C*d*e-1/2*b^3*C*e^2)*a^2-1/4*b*c^2*d*(-8*A*c*d+b*(A*e+B*d))*a-1/4*A 
*b^3*c^2*d^2)*d)*(c*x^4+b*x^2+a)*2^(1/2)*arctan(a*(e*x^2+d)^(1/2)/x*2^(1/2 
)/((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))+2*(8*(a*c-1/4*b^2)* 
e^(3/2)*(c*x^4+b*x^2+a)*a*C*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))+c*(((-2*C*d 
-2*(C*x^2+B)*e)*c+C*b*e)*a^2+(((2*B*x^2+2*A)*d+2*A*e*x^2)*c^2+c*((-C*x^2+B 
)*d+e*(-B*x^2+A))*b+C*b^2*e*x^2)*a-A*b*c*d*(c*x^2+b))*(e*x^2+d)^(1/2)*x)*( 
(-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)*(-4*(a*c-1/4*b^2)*d^2)^ 
(1/2))*((2*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))/((2*a*e-b*d+(-4 
*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)/(-4*(a*c-1/4*b^2)*d^2)^(1/2)/((-2*a*e+ 
b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)/(4*a*c-b^2)/a/(c*x^4+b*x^2+a...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((e*x**2+d)**(3/2)*(C*x**4+B*x**2+A)/(c*x**4+b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^(3/2)/(c*x^4 + b*x^2 + a)^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4241 vs. \(2 (811) = 1622\).

Time = 0.91 (sec) , antiderivative size = 4241, normalized size of antiderivative = 4.89 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*C*e^(3/2)*log((sqrt(e)*x - sqrt(e*x^2 + d))^2)/c^2 - ((sqrt(e)*x - sq 
rt(e*x^2 + d))^6*C*a*b*c^2*d^2*sqrt(e) - 2*(sqrt(e)*x - sqrt(e*x^2 + d))^6 
*B*a*c^3*d^2*sqrt(e) + (sqrt(e)*x - sqrt(e*x^2 + d))^6*A*b*c^3*d^2*sqrt(e) 
 - 3*(sqrt(e)*x - sqrt(e*x^2 + d))^6*C*a*b^2*c*d*e^(3/2) + 6*(sqrt(e)*x - 
sqrt(e*x^2 + d))^6*C*a^2*c^2*d*e^(3/2) + 3*(sqrt(e)*x - sqrt(e*x^2 + d))^6 
*B*a*b*c^2*d*e^(3/2) - 6*(sqrt(e)*x - sqrt(e*x^2 + d))^6*A*a*c^3*d*e^(3/2) 
 + 2*(sqrt(e)*x - sqrt(e*x^2 + d))^6*C*a*b^3*e^(5/2) - 6*(sqrt(e)*x - sqrt 
(e*x^2 + d))^6*C*a^2*b*c*e^(5/2) - 2*(sqrt(e)*x - sqrt(e*x^2 + d))^6*B*a*b 
^2*c*e^(5/2) + 4*(sqrt(e)*x - sqrt(e*x^2 + d))^6*B*a^2*c^2*e^(5/2) + 2*(sq 
rt(e)*x - sqrt(e*x^2 + d))^6*A*a*b*c^2*e^(5/2) - 3*(sqrt(e)*x - sqrt(e*x^2 
 + d))^4*C*a*b*c^2*d^3*sqrt(e) + 6*(sqrt(e)*x - sqrt(e*x^2 + d))^4*B*a*c^3 
*d^3*sqrt(e) - 3*(sqrt(e)*x - sqrt(e*x^2 + d))^4*A*b*c^3*d^3*sqrt(e) + 7*( 
sqrt(e)*x - sqrt(e*x^2 + d))^4*C*a*b^2*c*d^2*e^(3/2) - 6*(sqrt(e)*x - sqrt 
(e*x^2 + d))^4*C*a^2*c^2*d^2*e^(3/2) - 11*(sqrt(e)*x - sqrt(e*x^2 + d))^4* 
B*a*b*c^2*d^2*e^(3/2) + 4*(sqrt(e)*x - sqrt(e*x^2 + d))^4*A*b^2*c^2*d^2*e^ 
(3/2) + 6*(sqrt(e)*x - sqrt(e*x^2 + d))^4*A*a*c^3*d^2*e^(3/2) - 4*(sqrt(e) 
*x - sqrt(e*x^2 + d))^4*C*a*b^3*d*e^(5/2) + 4*(sqrt(e)*x - sqrt(e*x^2 + d) 
)^4*B*a*b^2*c*d*e^(5/2) + 16*(sqrt(e)*x - sqrt(e*x^2 + d))^4*B*a^2*c^2*d*e 
^(5/2) - 16*(sqrt(e)*x - sqrt(e*x^2 + d))^4*A*a*b*c^2*d*e^(5/2) + 8*(sqrt( 
e)*x - sqrt(e*x^2 + d))^4*C*a^2*b^2*e^(7/2) - 16*(sqrt(e)*x - sqrt(e*x^...
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int(((d + e*x^2)^(3/2)*(A + B*x^2 + C*x^4))/(a + b*x^2 + c*x^4)^2,x)
 

Output:

int(((d + e*x^2)^(3/2)*(A + B*x^2 + C*x^4))/(a + b*x^2 + c*x^4)^2, x)
 

Reduce [F]

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

int((e*x^2+d)^(3/2)*(C*x^4+B*x^2+A)/(c*x^4+b*x^2+a)^2,x)
 

Output:

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