\(\int \frac {(d+e x^2)^{5/2} (A+B x^2+C x^4)}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx\) [146]

Optimal result

Integrand size = 49, antiderivative size = 445 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=-\frac {\left (35 a^3 C e^2-64 b^3 d (B d+2 A e)-40 a^2 b e (2 C d+B e)+48 a b^2 \left (C d^2+e (2 B d+A e)\right )\right ) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{128 b^4 \sqrt {d+e x^2}}+\frac {\left (35 a^2 C e^2-40 a b e (2 C d+B e)+48 b^2 \left (C d^2+e (2 B d+A e)\right )\right ) x^3 \sqrt {a d+(b d+a e) x^2+b e x^4}}{192 b^3 \sqrt {d+e x^2}}-\frac {e (7 a C e-8 b (2 C d+B e)) x^5 \sqrt {a d+(b d+a e) x^2+b e x^4}}{48 b^2 \sqrt {d+e x^2}}+\frac {C e^2 x^7 \sqrt {a d+(b d+a e) x^2+b e x^4}}{8 b \sqrt {d+e x^2}}+\frac {\left (16 A b^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )-a \left (64 b^3 B d^2-35 a^3 C e^2+40 a^2 b e (2 C d+B e)-48 a b^2 d (C d+2 B e)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x \sqrt {d+e x^2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{128 b^{9/2}} \] Output:

-1/128*(35*a^3*C*e^2-64*b^3*d*(2*A*e+B*d)-40*a^2*b*e*(B*e+2*C*d)+48*a*b^2* 
(C*d^2+e*(A*e+2*B*d)))*x*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/b^4/(e*x^2+d)^( 
1/2)+1/192*(35*a^2*C*e^2-40*a*b*e*(B*e+2*C*d)+48*b^2*(C*d^2+e*(A*e+2*B*d)) 
)*x^3*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/b^3/(e*x^2+d)^(1/2)-1/48*e*(7*C*a* 
e-8*b*(B*e+2*C*d))*x^5*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/b^2/(e*x^2+d)^(1/ 
2)+1/8*C*e^2*x^7*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/b/(e*x^2+d)^(1/2)+1/128 
*(16*A*b^2*(3*a^2*e^2-8*a*b*d*e+8*b^2*d^2)-a*(64*b^3*B*d^2-35*a^3*C*e^2+40 
*a^2*b*e*(B*e+2*C*d)-48*a*b^2*d*(2*B*e+C*d)))*arctanh(b^(1/2)*x*(e*x^2+d)^ 
(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2))/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {\sqrt {d+e x^2} \left (-\sqrt {b} x \left (a+b x^2\right ) \left (105 a^3 C e^2-10 a^2 b e \left (24 C d+12 B e+7 C e x^2\right )-16 b^3 \left (6 A e \left (4 d+e x^2\right )+4 B \left (3 d^2+3 d e x^2+e^2 x^4\right )+C x^2 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )\right )+8 a b^2 \left (2 e \left (18 B d+9 A e+5 B e x^2\right )+C \left (18 d^2+20 d e x^2+7 e^2 x^4\right )\right )\right )-3 \left (16 A b^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )+a \left (-64 b^3 B d^2+35 a^3 C e^2-40 a^2 b e (2 C d+B e)+48 a b^2 d (C d+2 B e)\right )\right ) \sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{384 b^{9/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

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

Output:

(Sqrt[d + e*x^2]*(-(Sqrt[b]*x*(a + b*x^2)*(105*a^3*C*e^2 - 10*a^2*b*e*(24* 
C*d + 12*B*e + 7*C*e*x^2) - 16*b^3*(6*A*e*(4*d + e*x^2) + 4*B*(3*d^2 + 3*d 
*e*x^2 + e^2*x^4) + C*x^2*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)) + 8*a*b^2*(2*e* 
(18*B*d + 9*A*e + 5*B*e*x^2) + C*(18*d^2 + 20*d*e*x^2 + 7*e^2*x^4)))) - 3* 
(16*A*b^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2) + a*(-64*b^3*B*d^2 + 35*a^3* 
C*e^2 - 40*a^2*b*e*(2*C*d + B*e) + 48*a*b^2*d*(C*d + 2*B*e)))*Sqrt[a + b*x 
^2]*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]))/(384*b^(9/2)*Sqrt[(a + b*x^2)*(d 
 + e*x^2)])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1395, 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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 2256

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \left (\frac {C e^2 x^8}{\sqrt {b x^2+a}}+\frac {e (2 C d+B e) x^6}{\sqrt {b x^2+a}}+\frac {\left (C d^2+e (2 B d+A e)\right ) x^4}{\sqrt {b x^2+a}}+\frac {d (B d+2 A e) x^2}{\sqrt {b x^2+a}}+\frac {A d^2}{\sqrt {b x^2+a}}\right )dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {35 a^4 C e^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{9/2}}-\frac {5 a^3 e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (B e+2 C d)}{16 b^{7/2}}-\frac {35 a^3 C e^2 x \sqrt {a+b x^2}}{128 b^4}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (e (A e+2 B d)+C d^2\right )}{8 b^{5/2}}+\frac {5 a^2 e x \sqrt {a+b x^2} (B e+2 C d)}{16 b^3}+\frac {35 a^2 C e^2 x^3 \sqrt {a+b x^2}}{192 b^3}-\frac {a d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 A e+B d)}{2 b^{3/2}}+\frac {A d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-\frac {3 a x \sqrt {a+b x^2} \left (e (A e+2 B d)+C d^2\right )}{8 b^2}+\frac {x^3 \sqrt {a+b x^2} \left (e (A e+2 B d)+C d^2\right )}{4 b}+\frac {d x \sqrt {a+b x^2} (2 A e+B d)}{2 b}-\frac {5 a e x^3 \sqrt {a+b x^2} (B e+2 C d)}{24 b^2}-\frac {7 a C e^2 x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {e x^5 \sqrt {a+b x^2} (B e+2 C d)}{6 b}+\frac {C e^2 x^7 \sqrt {a+b x^2}}{8 b}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

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

Output:

(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*((-35*a^3*C*e^2*x*Sqrt[a + b*x^2])/(128*b 
^4) + (d*(B*d + 2*A*e)*x*Sqrt[a + b*x^2])/(2*b) + (5*a^2*e*(2*C*d + B*e)*x 
*Sqrt[a + b*x^2])/(16*b^3) - (3*a*(C*d^2 + e*(2*B*d + A*e))*x*Sqrt[a + b*x 
^2])/(8*b^2) + (35*a^2*C*e^2*x^3*Sqrt[a + b*x^2])/(192*b^3) - (5*a*e*(2*C* 
d + B*e)*x^3*Sqrt[a + b*x^2])/(24*b^2) + ((C*d^2 + e*(2*B*d + A*e))*x^3*Sq 
rt[a + b*x^2])/(4*b) - (7*a*C*e^2*x^5*Sqrt[a + b*x^2])/(48*b^2) + (e*(2*C* 
d + B*e)*x^5*Sqrt[a + b*x^2])/(6*b) + (C*e^2*x^7*Sqrt[a + b*x^2])/(8*b) + 
(A*d^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] + (35*a^4*C*e^2*ArcTa 
nh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(9/2)) - (a*d*(B*d + 2*A*e)*ArcTan 
h[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2)) - (5*a^3*e*(2*C*d + B*e)*ArcTa 
nh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2)) + (3*a^2*(C*d^2 + e*(2*B*d + 
 A*e))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))))/Sqrt[a*d + (b*d 
 + a*e)*x^2 + b*e*x^4]
 

Defintions of rubi rules used

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

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 = 0.08 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {x \left (-48 C \,e^{2} x^{6} b^{3}-64 B \,b^{3} e^{2} x^{4}+56 C a \,b^{2} e^{2} x^{4}-128 C \,b^{3} d e \,x^{4}-96 A \,b^{3} e^{2} x^{2}+80 B a \,b^{2} e^{2} x^{2}-192 B \,b^{3} d e \,x^{2}-70 C \,a^{2} b \,e^{2} x^{2}+160 C a \,b^{2} d e \,x^{2}-96 C \,b^{3} d^{2} x^{2}+144 A a \,b^{2} e^{2}-384 d e A \,b^{3}-120 B \,a^{2} b \,e^{2}+288 B a \,b^{2} d e -192 b^{3} B \,d^{2}+105 a^{3} C \,e^{2}-240 C \,a^{2} b d e +144 C a \,b^{2} d^{2}\right ) \left (b \,x^{2}+a \right ) \sqrt {e \,x^{2}+d}}{384 b^{4} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}+\frac {\left (48 A \,a^{2} b^{2} e^{2}-128 A a \,b^{3} d e +128 A \,b^{4} d^{2}-40 B \,a^{3} b \,e^{2}+96 B \,a^{2} b^{2} d e -64 B a \,b^{3} d^{2}+35 C \,a^{4} e^{2}-80 C \,a^{3} b d e +48 C \,a^{2} b^{2} d^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {e \,x^{2}+d}}{128 b^{\frac {9}{2}} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}\)	\(385\)
default	\(\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \left (70 C \,a^{2} b^{\frac {3}{2}} e^{2} x^{3} \sqrt {b \,x^{2}+a}-144 A a \,b^{\frac {5}{2}} e^{2} x \sqrt {b \,x^{2}+a}+384 A \,b^{\frac {7}{2}} d e x \sqrt {b \,x^{2}+a}+120 B \,a^{2} b^{\frac {3}{2}} e^{2} x \sqrt {b \,x^{2}+a}-144 C a \,b^{\frac {5}{2}} d^{2} x \sqrt {b \,x^{2}+a}-56 C a \,b^{\frac {5}{2}} e^{2} x^{5} \sqrt {b \,x^{2}+a}+128 C \,b^{\frac {7}{2}} d e \,x^{5} \sqrt {b \,x^{2}+a}-80 B a \,b^{\frac {5}{2}} e^{2} x^{3} \sqrt {b \,x^{2}+a}+192 B \,b^{\frac {7}{2}} d e \,x^{3} \sqrt {b \,x^{2}+a}-105 C \,a^{3} e^{2} x \sqrt {b \,x^{2}+a}\, \sqrt {b}-384 A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a \,b^{3} d e +288 B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{2} b^{2} d e +384 A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) b^{4} d^{2}+105 C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{4} e^{2}+144 A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{2} b^{2} e^{2}-120 B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{3} b \,e^{2}-192 B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a \,b^{3} d^{2}+144 C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{2} b^{2} d^{2}+48 C \,b^{\frac {7}{2}} e^{2} x^{7} \sqrt {b \,x^{2}+a}+64 B \,b^{\frac {7}{2}} e^{2} x^{5} \sqrt {b \,x^{2}+a}+96 A \,b^{\frac {7}{2}} e^{2} x^{3} \sqrt {b \,x^{2}+a}+96 C \,b^{\frac {7}{2}} d^{2} x^{3} \sqrt {b \,x^{2}+a}+192 B \,b^{\frac {7}{2}} d^{2} x \sqrt {b \,x^{2}+a}-288 B a \,b^{\frac {5}{2}} d e x \sqrt {b \,x^{2}+a}+240 C \,a^{2} b^{\frac {3}{2}} d e x \sqrt {b \,x^{2}+a}-160 C a \,b^{\frac {5}{2}} d e \,x^{3} \sqrt {b \,x^{2}+a}-240 C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{3} b d e \right )}{384 b^{\frac {9}{2}} \sqrt {e \,x^{2}+d}\, \sqrt {b \,x^{2}+a}}\)	\(651\)

Input:

int((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

-1/384*x*(-48*C*b^3*e^2*x^6-64*B*b^3*e^2*x^4+56*C*a*b^2*e^2*x^4-128*C*b^3* 
d*e*x^4-96*A*b^3*e^2*x^2+80*B*a*b^2*e^2*x^2-192*B*b^3*d*e*x^2-70*C*a^2*b*e 
^2*x^2+160*C*a*b^2*d*e*x^2-96*C*b^3*d^2*x^2+144*A*a*b^2*e^2-384*A*b^3*d*e- 
120*B*a^2*b*e^2+288*B*a*b^2*d*e-192*B*b^3*d^2+105*C*a^3*e^2-240*C*a^2*b*d* 
e+144*C*a*b^2*d^2)*(b*x^2+a)/b^4/((e*x^2+d)*(b*x^2+a))^(1/2)*(e*x^2+d)^(1/ 
2)+1/128*(48*A*a^2*b^2*e^2-128*A*a*b^3*d*e+128*A*b^4*d^2-40*B*a^3*b*e^2+96 
*B*a^2*b^2*d*e-64*B*a*b^3*d^2+35*C*a^4*e^2-80*C*a^3*b*d*e+48*C*a^2*b^2*d^2 
)/b^(9/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))*(b*x^2+a)^(1/2)/((e*x^2+d)*(b*x^2+ 
a))^(1/2)*(e*x^2+d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.15 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx =\text {Too large to display} \] Input:

integrate((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="fricas")
 

Output:

[1/768*(3*(16*(3*C*a^2*b^2 - 4*B*a*b^3 + 8*A*b^4)*d^3 - 16*(5*C*a^3*b - 6* 
B*a^2*b^2 + 8*A*a*b^3)*d^2*e + (35*C*a^4 - 40*B*a^3*b + 48*A*a^2*b^2)*d*e^ 
2 + (16*(3*C*a^2*b^2 - 4*B*a*b^3 + 8*A*b^4)*d^2*e - 16*(5*C*a^3*b - 6*B*a^ 
2*b^2 + 8*A*a*b^3)*d*e^2 + (35*C*a^4 - 40*B*a^3*b + 48*A*a^2*b^2)*e^3)*x^2 
)*sqrt(b)*log((2*b*e*x^4 + (2*b*d + a*e)*x^2 + 2*sqrt(b*e*x^4 + (b*d + a*e 
)*x^2 + a*d)*sqrt(e*x^2 + d)*sqrt(b)*x + a*d)/(e*x^2 + d)) + 2*(48*C*b^4*e 
^2*x^7 + 8*(16*C*b^4*d*e - (7*C*a*b^3 - 8*B*b^4)*e^2)*x^5 + 2*(48*C*b^4*d^ 
2 - 16*(5*C*a*b^3 - 6*B*b^4)*d*e + (35*C*a^2*b^2 - 40*B*a*b^3 + 48*A*b^4)* 
e^2)*x^3 - 3*(16*(3*C*a*b^3 - 4*B*b^4)*d^2 - 16*(5*C*a^2*b^2 - 6*B*a*b^3 + 
 8*A*b^4)*d*e + (35*C*a^3*b - 40*B*a^2*b^2 + 48*A*a*b^3)*e^2)*x)*sqrt(b*e* 
x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d))/(b^5*e*x^2 + b^5*d), -1/384* 
(3*(16*(3*C*a^2*b^2 - 4*B*a*b^3 + 8*A*b^4)*d^3 - 16*(5*C*a^3*b - 6*B*a^2*b 
^2 + 8*A*a*b^3)*d^2*e + (35*C*a^4 - 40*B*a^3*b + 48*A*a^2*b^2)*d*e^2 + (16 
*(3*C*a^2*b^2 - 4*B*a*b^3 + 8*A*b^4)*d^2*e - 16*(5*C*a^3*b - 6*B*a^2*b^2 + 
 8*A*a*b^3)*d*e^2 + (35*C*a^4 - 40*B*a^3*b + 48*A*a^2*b^2)*e^3)*x^2)*sqrt( 
-b)*arctan(sqrt(e*x^2 + d)*sqrt(-b)*x/sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d 
)) - (48*C*b^4*e^2*x^7 + 8*(16*C*b^4*d*e - (7*C*a*b^3 - 8*B*b^4)*e^2)*x^5 
+ 2*(48*C*b^4*d^2 - 16*(5*C*a*b^3 - 6*B*b^4)*d*e + (35*C*a^2*b^2 - 40*B*a* 
b^3 + 48*A*b^4)*e^2)*x^3 - 3*(16*(3*C*a*b^3 - 4*B*b^4)*d^2 - 16*(5*C*a^2*b 
^2 - 6*B*a*b^3 + 8*A*b^4)*d*e + (35*C*a^3*b - 40*B*a^2*b^2 + 48*A*a*b^3...
 

Sympy [F(-1)]

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

integrate((e*x**2+d)**(5/2)*(C*x**4+B*x**2+A)/(a*d+(a*e+b*d)*x**2+b*e*x**4 
)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((d + e*x^2)^(5/2)*(A + B*x^2 + C*x^4))/(a*d + x^2*(a*e + b*d) + b*e*x 
^4)^(1/2),x)
 

Output:

int(((d + e*x^2)^(5/2)*(A + B*x^2 + C*x^4))/(a*d + x^2*(a*e + b*d) + b*e*x 
^4)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^{5/2} \left (A+B x^2+C x^4\right )}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {-105 \sqrt {b \,x^{2}+a}\, a^{3} b c \,e^{2} x -24 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} e^{2} x +240 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c d e x +70 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,e^{2} x^{3}+96 \sqrt {b \,x^{2}+a}\, a \,b^{4} d e x +16 \sqrt {b \,x^{2}+a}\, a \,b^{4} e^{2} x^{3}-144 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,d^{2} x -160 \sqrt {b \,x^{2}+a}\, a \,b^{3} c d e \,x^{3}-56 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,e^{2} x^{5}+192 \sqrt {b \,x^{2}+a}\, b^{5} d^{2} x +192 \sqrt {b \,x^{2}+a}\, b^{5} d e \,x^{3}+64 \sqrt {b \,x^{2}+a}\, b^{5} e^{2} x^{5}+96 \sqrt {b \,x^{2}+a}\, b^{4} c \,d^{2} x^{3}+128 \sqrt {b \,x^{2}+a}\, b^{4} c d e \,x^{5}+48 \sqrt {b \,x^{2}+a}\, b^{4} c \,e^{2} x^{7}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} c \,e^{2}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} e^{2}-240 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b c d e -96 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} d e +144 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c \,d^{2}+192 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} d^{2}}{384 b^{5}} \] Input:

int((e*x^2+d)^(5/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x)
 

Output:

( - 105*sqrt(a + b*x**2)*a**3*b*c*e**2*x - 24*sqrt(a + b*x**2)*a**2*b**3*e 
**2*x + 240*sqrt(a + b*x**2)*a**2*b**2*c*d*e*x + 70*sqrt(a + b*x**2)*a**2* 
b**2*c*e**2*x**3 + 96*sqrt(a + b*x**2)*a*b**4*d*e*x + 16*sqrt(a + b*x**2)* 
a*b**4*e**2*x**3 - 144*sqrt(a + b*x**2)*a*b**3*c*d**2*x - 160*sqrt(a + b*x 
**2)*a*b**3*c*d*e*x**3 - 56*sqrt(a + b*x**2)*a*b**3*c*e**2*x**5 + 192*sqrt 
(a + b*x**2)*b**5*d**2*x + 192*sqrt(a + b*x**2)*b**5*d*e*x**3 + 64*sqrt(a 
+ b*x**2)*b**5*e**2*x**5 + 96*sqrt(a + b*x**2)*b**4*c*d**2*x**3 + 128*sqrt 
(a + b*x**2)*b**4*c*d*e*x**5 + 48*sqrt(a + b*x**2)*b**4*c*e**2*x**7 + 105* 
sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*c*e**2 + 24*sqrt( 
b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**2*e**2 - 240*sqrt(b 
)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*c*d*e - 96*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**3*d*e + 144*sqrt(b)*log( 
(sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*c*d**2 + 192*sqrt(b)*log 
((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**4*d**2)/(384*b**5)