\(\int \frac {(d+e x^2)^{5/2} (A+B x^2+C x^4)}{a+b x^2+c x^4} \, dx\) [154]

Optimal result

Integrand size = 38, antiderivative size = 1233 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx =\text {Too large to display} \] Output:

1/16*(8*b^2*C*e^2-2*c*e*(4*B*b*e+4*C*a*e+7*C*b*d)+c^2*(5*C*d^2+2*e*(4*A*e+ 
7*B*d)))*x*(e*x^2+d)^(1/2)/c^3+1/24*(6*B*c*e-6*C*b*e+5*C*c*d)*x*(e*x^2+d)^ 
(3/2)/c^2+1/6*C*x*(e*x^2+d)^(5/2)/c-(3*b^3*c*C*d*e^2-b^4*C*e^3-c^2*(A*c-C* 
a)*e*(-a*e^2+3*c*d^2)+b*c^2*d*(3*A*c*e^2-6*C*a*e^2+C*c*d^2)-b^2*c*e*(A*c*e 
^2-3*C*a*e^2+3*C*c*d^2)-B*c*(c^3*d^3-b^3*e^3-3*c^2*d*e*(a*e+b*d)+b*c*e^2*( 
2*a*e+3*b*d))+(b^5*C*e^3-b^4*c*e^2*(B*e+3*C*d)+b^2*c^2*(4*a*e^2*(B*e+3*C*d 
)-c*d*(C*d^2+3*e*(A*e+B*d)))-b^3*c*e*(5*C*a*e^2-c*(3*C*d^2+e*(A*e+3*B*d))) 
-2*c^3*(A*c*d*(-3*a*e^2+c*d^2)+a*(a*e^2*(B*e+3*C*d)-c*d^2*(3*B*e+C*d)))+b* 
c^2*(B*c*d*(-9*a*e^2+c*d^2)-e*(a*C*(-5*a*e^2+9*c*d^2)-3*A*c*(-a*e^2+c*d^2) 
)))/(-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))/c^4/(b-(-4*a*c+b^2)^(1/2))^(1/ 
2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)-(3*b^3*c*C*d*e^2-b^4*C*e^3-c^2*( 
A*c-C*a)*e*(-a*e^2+3*c*d^2)+b*c^2*d*(3*A*c*e^2-6*C*a*e^2+C*c*d^2)-b^2*c*e* 
(A*c*e^2-3*C*a*e^2+3*C*c*d^2)-B*c*(c^3*d^3-b^3*e^3-3*c^2*d*e*(a*e+b*d)+b*c 
*e^2*(2*a*e+3*b*d))-(b^5*C*e^3-b^4*c*e^2*(B*e+3*C*d)+b*c^2*(B*c*d*(-9*a*e^ 
2+c*d^2)-a*C*e*(-5*a*e^2+9*c*d^2)+3*A*c*e*(-a*e^2+c*d^2))+b^2*c^2*(4*a*e^2 
*(B*e+3*C*d)-c*d*(C*d^2+3*e*(A*e+B*d)))-b^3*c*e*(5*C*a*e^2-c*(3*C*d^2+e*(A 
*e+3*B*d)))-2*c^3*(A*c*d*(-3*a*e^2+c*d^2)+a*(a*e^2*(B*e+3*C*d)-c*d^2*(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))/c^4/(b+(-4*a*c+b^2)^(...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(63315\) vs. \(2(1233)=2466\).

Time = 18.18 (sec) , antiderivative size = 63315, normalized size of antiderivative = 51.35 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 6.10 (sec) , antiderivative size = 1198, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2256, 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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 C \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^2+d}}\right ) d^3}{16 c \sqrt {e}}+\frac {5 C x \sqrt {e x^2+d} d^2}{16 c}+\frac {5 C x \left (e x^2+d\right )^{3/2} d}{24 c}+\frac {C x \left (e x^2+d\right )^{5/2}}{6 c}+\frac {\left (B c-b C-\frac {-C b^2+B c b-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) e x \left (e x^2+d\right )^{3/2}}{8 c^2}+\frac {\left (B c-b C+\frac {-C b^2+B c b-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) e x \left (e x^2+d\right )^{3/2}}{8 c^2}+\frac {\left (B c-b C-\frac {-C b^2+B c b-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) \left (2 c^3 d^3-3 c^2 e \left (b d-\sqrt {b^2-4 a c} d+2 a e\right ) d-b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3+c e^2 \left (3 d b^2-3 \sqrt {b^2-4 a c} d b+3 a e b-a \sqrt {b^2-4 a c} e\right )\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 {e x^2+d}}\right )}{2 c^4 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (B c-b C+\frac {-C b^2+B c b-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) \left (2 c^3 d^3-3 c^2 e \left (b d+\sqrt {b^2-4 a c} d+2 a e\right ) d-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3+c e^2 \left (3 d b^2+3 \left (\sqrt {b^2-4 a c} d+a e\right ) b+a \sqrt {b^2-4 a c} e\right )\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 {e x^2+d}}\right )}{2 c^4 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (B c-b C-\frac {-C b^2+B c b-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) \sqrt {e} \left (15 c^2 d^2+4 b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (5 b d-5 \sqrt {b^2-4 a c} d+4 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^2+d}}\right )}{16 c^4}+\frac {\left (B c-b C+\frac {-C b^2+B c b-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) \sqrt {e} \left (15 c^2 d^2+4 b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (5 b d+5 \sqrt {b^2-4 a c} d+4 a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^2+d}}\right )}{16 c^4}+\frac {\left (B c-b C-\frac {-C b^2+B c b-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) e \left (7 c d-2 \left (b-\sqrt {b^2-4 a c}\right ) e\right ) x \sqrt {e x^2+d}}{16 c^3}+\frac {\left (B c-b C+\frac {-C b^2+B c b-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) e \left (7 c d-2 \left (b+\sqrt {b^2-4 a c}\right ) e\right ) x \sqrt {e x^2+d}}{16 c^3}\)

Input:

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

Output:

(5*C*d^2*x*Sqrt[d + e*x^2])/(16*c) + ((B*c - b*C - (b*B*c - b^2*C - 2*c*(A 
*c - a*C))/Sqrt[b^2 - 4*a*c])*e*(7*c*d - 2*(b - Sqrt[b^2 - 4*a*c])*e)*x*Sq 
rt[d + e*x^2])/(16*c^3) + ((B*c - b*C + (b*B*c - b^2*C - 2*c*(A*c - a*C))/ 
Sqrt[b^2 - 4*a*c])*e*(7*c*d - 2*(b + Sqrt[b^2 - 4*a*c])*e)*x*Sqrt[d + e*x^ 
2])/(16*c^3) + (5*C*d*x*(d + e*x^2)^(3/2))/(24*c) + ((B*c - b*C - (b*B*c - 
 b^2*C - 2*c*(A*c - a*C))/Sqrt[b^2 - 4*a*c])*e*x*(d + e*x^2)^(3/2))/(8*c^2 
) + ((B*c - b*C + (b*B*c - b^2*C - 2*c*(A*c - a*C))/Sqrt[b^2 - 4*a*c])*e*x 
*(d + e*x^2)^(3/2))/(8*c^2) + (C*x*(d + e*x^2)^(5/2))/(6*c) + ((B*c - b*C 
- (b*B*c - b^2*C - 2*c*(A*c - a*C))/Sqrt[b^2 - 4*a*c])*(2*c^3*d^3 - b^2*(b 
 - Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d - Sqrt[b^2 - 4*a*c]*d + 2*a*e) 
+ c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*a*b*e - a*Sqrt[b^2 - 4*a*c] 
*e))*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 
 - 4*a*c]]*Sqrt[d + e*x^2])])/(2*c^4*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c* 
d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((B*c - b*C + (b*B*c - b^2*C - 2*c*(A*c 
- a*C))/Sqrt[b^2 - 4*a*c])*(2*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 
3*c^2*d*e*(b*d + Sqrt[b^2 - 4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d + a*Sqrt[b^ 
2 - 4*a*c]*e + 3*b*(Sqrt[b^2 - 4*a*c]*d + a*e)))*ArcTan[(Sqrt[2*c*d - (b + 
 Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/ 
(2*c^4*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e] 
) + (5*C*d^3*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(16*c*Sqrt[e]) + ((B...
 

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 = 3.32 (sec) , antiderivative size = 954, normalized size of antiderivative = 0.77

method	result	size
risch	\(\frac {x \left (8 C \,e^{2} c^{2} x^{4}+12 B \,c^{2} e^{2} x^{2}-12 C b c \,e^{2} x^{2}+26 C \,c^{2} d e \,x^{2}+24 A \,c^{2} e^{2}-24 B b c \,e^{2}+54 B \,c^{2} d e -24 C a c \,e^{2}+24 b^{2} C \,e^{2}-54 C b c d e +33 C \,c^{2} d^{2}\right ) \sqrt {e \,x^{2}+d}}{48 c^{3}}-\frac {\frac {\left (16 A b \,c^{2} e^{3}-40 A \,c^{3} e^{2} d +16 B a \,c^{2} e^{3}-16 B \,b^{2} c \,e^{3}+40 B b \,c^{2} d \,e^{2}-30 B \,c^{3} d^{2} e -32 C a b c \,e^{3}+40 C a \,e^{2} c^{2} d +16 b^{3} C \,e^{3}-40 C \,b^{2} c d \,e^{2}+30 C b \,d^{2} e \,c^{2}-5 C \,c^{3} d^{3}\right ) \ln \left (\sqrt {e \,x^{2}+d}+x \sqrt {e}\right )}{c \sqrt {e}}-\frac {8 \left (\left (\left (\left (B c -2 b C \right ) e +3 C c d \right ) c \,e^{2} a^{2}+\left (b \left (A \,c^{2}-b B c +b^{2} C \right ) e^{3}-3 c d \left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}-3 c^{2} d^{2} \left (B c -b C \right ) e -C \,c^{3} d^{3}\right ) a +A \,c^{4} d^{3}\right ) \left (\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 ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}-\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 ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-2 \left (\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 ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}+\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 ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\right ) d \left (-C \,a^{3} c^{2} e^{3}+c e \left (\left (A \,c^{2}-\frac {3}{2} b B c +2 b^{2} C \right ) e^{2}+3 c \left (B c -\frac {3 b C}{2}\right ) d e +3 C \,c^{2} d^{2}\right ) a^{2}+\left (-\frac {b^{2} \left (A \,c^{2}-b B c +b^{2} C \right ) e^{3}}{2}+\frac {3 b c d \left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}}{2}-3 c^{2} \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right ) d^{2} e -c^{3} d^{3} \left (B c -\frac {b C}{2}\right )\right ) a +\frac {A b \,c^{4} d^{3}}{2}\right )\right ) \sqrt {2}}{c \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}\, \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}}{16 c^{3}}\)	\(954\)
pseudoelliptic	\(-\frac {\left (2 \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (\left (\left (B \,c^{2}-2 C b c \right ) a +b \left (A \,c^{2}-b B c +b^{2} C \right )\right ) e^{3}-\frac {5 c d \left (A \,c^{2}-b B c -C c a +b^{2} C \right ) e^{2}}{2}-\frac {15 c^{2} d^{2} \left (B c -b C \right ) e}{8}-\frac {5 C \,c^{3} d^{3}}{16}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\sqrt {e}\, \left (-\sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, c \left (\left (-C c a +b^{2} C -\left (\frac {C \,x^{2}}{2}+B \right ) c b +c^{2} \left (\frac {1}{3} C \,x^{4}+\frac {1}{2} B \,x^{2}+A \right )\right ) e^{2}+\frac {9 \left (-b C +c \left (\frac {13 C \,x^{2}}{27}+B \right )\right ) c d e}{4}+\frac {11 C \,c^{2} d^{2}}{8}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, x \sqrt {e \,x^{2}+d}+\left (\left (\left (B \,c^{2}-2 C b c \right ) a +b \left (A \,c^{2}-b B c +b^{2} C \right )\right ) a \,e^{3}-3 a c d \left (A \,c^{2}-b B c -C c a +b^{2} C \right ) e^{2}-3 a \,c^{2} d^{2} \left (B c -b C \right ) e +c^{3} d^{3} \left (A c -C a \right )\right ) \sqrt {2}\, \left (\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 ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}-\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 ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\right )\right )\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+2 \left (\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 ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}+\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 ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\right ) \left (\left (-C \,a^{2} c^{2}+c \left (A \,c^{2}-\frac {3}{2} b B c +2 b^{2} C \right ) a -\frac {b^{2} \left (A \,c^{2}-b B c +b^{2} C \right )}{2}\right ) a \,e^{3}+\frac {3 c d a \left (\left (2 B \,c^{2}-3 C b c \right ) a +b \left (A \,c^{2}-b B c +b^{2} C \right )\right ) e^{2}}{2}-3 c^{2} d^{2} \left (A \,c^{2}-\frac {1}{2} b B c -C c a +\frac {1}{2} b^{2} C \right ) a e +\frac {c^{3} d^{3} \left (\left (-2 B c +b C \right ) a +A b c \right )}{2}\right ) \sqrt {e}\, d \sqrt {2}}{2 \sqrt {e}\, \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}\, \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, c^{4}}\)	\(991\)
default	\(\text {Expression too large to display}\)	\(1029\)

Input:

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

Output:

1/48*x*(8*C*c^2*e^2*x^4+12*B*c^2*e^2*x^2-12*C*b*c*e^2*x^2+26*C*c^2*d*e*x^2 
+24*A*c^2*e^2-24*B*b*c*e^2+54*B*c^2*d*e-24*C*a*c*e^2+24*C*b^2*e^2-54*C*b*c 
*d*e+33*C*c^2*d^2)*(e*x^2+d)^(1/2)/c^3-1/16/c^3*((16*A*b*c^2*e^3-40*A*c^3* 
d*e^2+16*B*a*c^2*e^3-16*B*b^2*c*e^3+40*B*b*c^2*d*e^2-30*B*c^3*d^2*e-32*C*a 
*b*c*e^3+40*C*a*c^2*d*e^2+16*C*b^3*e^3-40*C*b^2*c*d*e^2+30*C*b*c^2*d^2*e-5 
*C*c^3*d^3)/c*ln((e*x^2+d)^(1/2)+x*e^(1/2))/e^(1/2)-8/c/(-4*(a*c-1/4*b^2)* 
d^2)^(1/2)*((((B*c-2*C*b)*e+3*C*c*d)*c*e^2*a^2+(b*(A*c^2-B*b*c+C*b^2)*e^3- 
3*c*d*(A*c^2-B*b*c+C*b^2)*e^2-3*c^2*d^2*(B*c-C*b)*e-C*c^3*d^3)*a+A*c^4*d^3 
)*(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))*((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)-arcta 
n(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*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*(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*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2)) 
*a)^(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))*((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2 
))*d*(-C*a^3*c^2*e^3+c*e*((A*c^2-3/2*b*B*c+2*b^2*C)*e^2+3*c*(B*c-3/2*b*C)* 
d*e+3*C*c^2*d^2)*a^2+(-1/2*b^2*(A*c^2-B*b*c+C*b^2)*e^3+3/2*b*c*d*(A*c^2-B* 
b*c+C*b^2)*e^2-3*c^2*(A*c^2-1/2*b*B*c+1/2*b^2*C)*d^2*e-c^3*d^3*(B*c-1/2*b* 
C))*a+1/2*A*b*c^4*d^3))*2^(1/2)/((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.06 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx=\frac {33 \sqrt {e \,x^{2}+d}\, d^{2} e x +26 \sqrt {e \,x^{2}+d}\, d \,e^{2} x^{3}+8 \sqrt {e \,x^{2}+d}\, e^{3} x^{5}+15 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{3}}{48 e} \] Input:

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

Output:

(33*sqrt(d + e*x**2)*d**2*e*x + 26*sqrt(d + e*x**2)*d*e**2*x**3 + 8*sqrt(d 
 + e*x**2)*e**3*x**5 + 15*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt( 
d))*d**3)/(48*e)