\(\int \frac {(d+e x^2)^{3/2} (A+B x^2+C x^4)}{a+b x^2+c x^4} \, dx\) [155]

Optimal result

Integrand size = 38, antiderivative size = 825 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx=\frac {(3 c C d+4 B c e-4 b C e) x \sqrt {d+e x^2}}{8 c^2}+\frac {C x \left (d+e x^2\right )^{3/2}}{4 c}+\frac {\left (2 b^2 c C d e+2 c^2 (A c-a C) d e-b^3 C e^2-b c \left (c C d^2+A c e^2-2 a C e^2\right )+B c \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )+\frac {b^4 C e^2-b^3 c e (2 C d+B e)-b c^2 \left (B c d^2+2 A c d e-6 a C d e-3 a B e^2\right )-b^2 c \left (4 a C e^2-c \left (C d^2+2 B d e+A e^2\right )\right )+2 c^2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d+2 B e)\right )\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (2 b^2 c C d e+2 c^2 (A c-a C) d e-b^3 C e^2-b c \left (c C d^2+A c e^2-2 a C e^2\right )+B c \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-\frac {b^4 C e^2-b^3 c e (2 C d+B e)-b c^2 \left (B c d^2+2 A c d e-6 a C d e-3 a B e^2\right )-b^2 c \left (4 a C e^2-c \left (C d^2+2 B d e+A e^2\right )\right )+2 c^2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d+2 B e)\right )\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (8 b^2 C e^2-4 c e (3 b C d+2 b B e+2 a C e)+c^2 \left (3 C d^2+4 e (3 B d+2 A e)\right )\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 c^3 \sqrt {e}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(40458\) vs. \(2(825)=1650\).

Time = 17.47 (sec) , antiderivative size = 40458, normalized size of antiderivative = 49.04 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx=\text {Result too large to show} \] Input:

Integrate[((d + e*x^2)^(3/2)*(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 = 2.78 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(3*C*d*x*Sqrt[d + e*x^2])/(8*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*Sqrt[d + e*x^2])/(4*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*Sqrt[d + e*x^2])/(4 
*c^2) + (C*x*(d + e*x^2)^(3/2))/(4*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^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^ 
2 - 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2*c*d - (b - Sqr 
t[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(2*c 
^3*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^2* 
d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + 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^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c* 
d - (b + Sqrt[b^2 - 4*a*c])*e]) + (3*C*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e* 
x^2]])/(8*c*Sqrt[e]) + ((B*c - b*C - (b*B*c - b^2*C - 2*c*(A*c - a*C))/Sqr 
t[b^2 - 4*a*c])*Sqrt[e]*(3*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[ 
e]*x)/Sqrt[d + e*x^2]])/(4*c^3) + ((B*c - b*C + (b*B*c - b^2*C - 2*c*(A*c 
- a*C))/Sqrt[b^2 - 4*a*c])*Sqrt[e]*(3*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*Arc 
Tanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(4*c^3)
 

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 = 1.51 (sec) , antiderivative size = 702, normalized size of antiderivative = 0.85

method	result	size
risch	\(\frac {x \left (2 C e c \,x^{2}+4 B c e -4 C b e +5 C c d \right ) \sqrt {e \,x^{2}+d}}{8 c^{2}}+\frac {\frac {\left (8 A \,c^{2} e^{2}-8 B b c \,e^{2}+12 B \,c^{2} d e -8 C a c \,e^{2}+8 b^{2} C \,e^{2}-12 C b c d e +3 C \,c^{2} d^{2}\right ) \ln \left (\sqrt {e \,x^{2}+d}+x \sqrt {e}\right )}{c \sqrt {e}}+\frac {4 \left (\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 (-C c \,e^{2} a^{2}+\left (b^{2} C \,e^{2}-b e \left (B e +2 C d \right ) c +\left (A \,e^{2}+2 B d e +C \,d^{2}\right ) c^{2}\right ) a -A \,c^{3} d^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{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 (-3 C b c \,e^{2}+2 c^{2} e \left (B e +2 C d \right )\right ) a^{2}+\left (b^{3} C \,e^{2}-c e \left (B e +2 C d \right ) b^{2}+c^{2} \left (A \,e^{2}+2 B d e +C \,d^{2}\right ) b +\left (-4 d e A -2 B \,d^{2}\right ) c^{3}\right ) a +A b \,c^{3} d^{2}\right ) d \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}}}{8 c^{2}}\)	\(702\)
pseudoelliptic	\(\frac {-\sqrt {e}\, \left (\left (-C c \,e^{2} a^{2}+\left (\left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}+2 c d \left (B c -b C \right ) e +C \,c^{2} d^{2}\right ) a -A \,c^{3} d^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+\left (\left (\left (2 B \,c^{2}-3 C b c \right ) e^{2}+4 C \,c^{2} d e \right ) a^{2}+\left (b \left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}-4 c \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right ) d e +\left (-2 B \,c^{3}+C b \,c^{2}\right ) d^{2}\right ) a +A b \,c^{3} d^{2}\right ) d \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 )+\sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (\sqrt {e}\, \left (\left (-C c \,e^{2} a^{2}+\left (\left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}+2 c d \left (B c -b C \right ) e +C \,c^{2} d^{2}\right ) a -A \,c^{3} d^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-\left (\left (\left (2 B \,c^{2}-3 C b c \right ) e^{2}+4 C \,c^{2} d e \right ) a^{2}+\left (b \left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}-4 c \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right ) d e +\left (-2 B \,c^{3}+C b \,c^{2}\right ) d^{2}\right ) a +A b \,c^{3} d^{2}\right ) d \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 )+\left (\left (-2 C a c \,e^{2}+\left (2 A \,c^{2}-2 b B c +2 b^{2} C \right ) e^{2}+3 c d \left (B c -b C \right ) e +\frac {3 C \,c^{2} d^{2}}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\sqrt {e}\, \sqrt {e \,x^{2}+d}\, \left (\left (-b C +c \left (\frac {C \,x^{2}}{2}+B \right )\right ) e +\frac {5 C c d}{4}\right ) c x \right ) \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}\right )}{2 \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {e}\, \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}}\, c^{3}}\)	\(778\)
default	\(\frac {C \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (\sqrt {e \,x^{2}+d}+x \sqrt {e}\right )}{2 \sqrt {e}}\right )}{4}\right )}{c}-\frac {\sqrt {e}\, \left (\left (-C c \,e^{2} a^{2}+\left (\left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}+2 c d \left (B c -b C \right ) e +C \,c^{2} d^{2}\right ) a -A \,c^{3} d^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+\left (\left (\left (2 B \,c^{2}-3 C b c \right ) e^{2}+4 C \,c^{2} d e \right ) a^{2}+\left (b \left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}-4 c \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right ) d e +\left (-2 B \,c^{3}+C b \,c^{2}\right ) d^{2}\right ) a +A b \,c^{3} d^{2}\right ) d \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 (\sqrt {e}\, \left (\left (-C c \,e^{2} a^{2}+\left (\left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}+2 c d \left (B c -b C \right ) e +C \,c^{2} d^{2}\right ) a -A \,c^{3} d^{2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-\left (\left (\left (2 B \,c^{2}-3 C b c \right ) e^{2}+4 C \,c^{2} d e \right ) a^{2}+\left (b \left (A \,c^{2}-b B c +b^{2} C \right ) e^{2}-4 c \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right ) d e +\left (-2 B \,c^{3}+C b \,c^{2}\right ) d^{2}\right ) a +A b \,c^{3} d^{2}\right ) d \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 )+\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}}\, \left (2 \left (-C a e c +\left (A \,c^{2}-b B c +b^{2} C \right ) e +\frac {3 c d \left (B c -b C \right )}{2}\right ) e \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+e^{\frac {3}{2}} \sqrt {e \,x^{2}+d}\, c \left (B c -b C \right ) x \right )\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}{2 c^{3} \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {e}\, \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}}}\)	\(806\)

Input:

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

Output:

1/8*x*(2*C*c*e*x^2+4*B*c*e-4*C*b*e+5*C*c*d)*(e*x^2+d)^(1/2)/c^2+1/8/c^2*(( 
8*A*c^2*e^2-8*B*b*c*e^2+12*B*c^2*d*e-8*C*a*c*e^2+8*C*b^2*e^2-12*C*b*c*d*e+ 
3*C*c^2*d^2)/c*ln((e*x^2+d)^(1/2)+x*e^(1/2))/e^(1/2)+4/c/(-4*(a*c-1/4*b^2) 
*d^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*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))*(- 
C*c*e^2*a^2+(b^2*C*e^2-b*e*(B*e+2*C*d)*c+(A*e^2+2*B*d*e+C*d^2)*c^2)*a-A*c^ 
3*d^2)*(-4*(a*c-1/4*b^2)*d^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*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))*((-3*C*b*c*e^2+2*c^2*e*(B*e+2*C*d))*a^2+(b^3*C*e^2-c* 
e*(B*e+2*C*d)*b^2+c^2*(A*e^2+2*B*d*e+C*d^2)*b+(-4*A*d*e-2*B*d^2)*c^3)*a+A* 
b*c^3*d^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))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.07 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx=\frac {5 \sqrt {e \,x^{2}+d}\, d e x +2 \sqrt {e \,x^{2}+d}\, e^{2} x^{3}+3 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}}{8 e} \] Input:

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

Output:

(5*sqrt(d + e*x**2)*d*e*x + 2*sqrt(d + e*x**2)*e**2*x**3 + 3*sqrt(e)*log(( 
sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d**2)/(8*e)