\(\int \frac {1}{\sqrt {d+e x^2} (a d+(b d+a e) x^2+b e x^4)^{5/2}} \, dx\) [21]

Optimal result

Integrand size = 37, antiderivative size = 402 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=-\frac {e x}{4 d (b d-a e) \sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}}+\frac {b (4 b d+3 a e) x \sqrt {d+e x^2}}{12 a d (b d-a e)^2 \left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}}+\frac {e \left (8 b^2 d^2+36 a b d e-9 a^2 e^2\right ) x}{24 a d^2 (b d-a e)^3 \sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (16 b^3 d^3-88 a b^2 d^2 e-42 a^2 b d e^2+9 a^3 e^3\right ) x \sqrt {d+e x^2}}{24 a^2 d^2 (b d-a e)^4 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {e^2 \left (48 b^2 d^2-16 a b d e+3 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{8 d^{5/2} (b d-a e)^{9/2}} \] Output:

-1/4*e*x/d/(-a*e+b*d)/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2)+1/ 
12*b*(3*a*e+4*b*d)*x*(e*x^2+d)^(1/2)/a/d/(-a*e+b*d)^2/(a*d+(a*e+b*d)*x^2+b 
*e*x^4)^(3/2)+1/24*e*(-9*a^2*e^2+36*a*b*d*e+8*b^2*d^2)*x/a/d^2/(-a*e+b*d)^ 
3/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/24*b*(9*a^3*e^3-42*a 
^2*b*d*e^2-88*a*b^2*d^2*e+16*b^3*d^3)*x*(e*x^2+d)^(1/2)/a^2/d^2/(-a*e+b*d) 
^4/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/8*e^2*(3*a^2*e^2-16*a*b*d*e+48*b^2* 
d^2)*arctanh((-a*e+b*d)^(1/2)*x*(e*x^2+d)^(1/2)/d^(1/2)/(a*d+(a*e+b*d)*x^2 
+b*e*x^4)^(1/2))/d^(5/2)/(-a*e+b*d)^(9/2)
 

Mathematica [A] (verified)

Time = 3.50 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\frac {\sqrt {d+e x^2} \left (\frac {\sqrt {d} x \left (a+b x^2\right ) \left (16 b^5 d^3 x^2 \left (d+e x^2\right )^2+8 a b^4 d^2 \left (3 d-11 e x^2\right ) \left (d+e x^2\right )^2+3 a^5 e^4 \left (5 d+3 e x^2\right )+3 a^3 b^2 e^3 x^2 \left (-32 d^2-23 d e x^2+3 e^2 x^4\right )+6 a^4 b e^3 \left (-8 d^2-2 d e x^2+3 e^2 x^4\right )-6 a^2 b^3 d e \left (16 d^3+32 d^2 e x^2+24 d e^2 x^4+7 e^3 x^6\right )\right )}{a^2 (b d-a e)^4}-\frac {9 e^2 (-4 b d+a e)^2 \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}}+\frac {24 a b d e^3 \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 \text {arctanh}\left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}}\right )}{24 d^{5/2} \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{5/2}} \] Input:

Integrate[1/(Sqrt[d + e*x^2]*(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(5/2)),x]
 

Output:

(Sqrt[d + e*x^2]*((Sqrt[d]*x*(a + b*x^2)*(16*b^5*d^3*x^2*(d + e*x^2)^2 + 8 
*a*b^4*d^2*(3*d - 11*e*x^2)*(d + e*x^2)^2 + 3*a^5*e^4*(5*d + 3*e*x^2) + 3* 
a^3*b^2*e^3*x^2*(-32*d^2 - 23*d*e*x^2 + 3*e^2*x^4) + 6*a^4*b*e^3*(-8*d^2 - 
 2*d*e*x^2 + 3*e^2*x^4) - 6*a^2*b^3*d*e*(16*d^3 + 32*d^2*e*x^2 + 24*d*e^2* 
x^4 + 7*e^3*x^6)))/(a^2*(b*d - a*e)^4) - (9*e^2*(-4*b*d + a*e)^2*(a + b*x^ 
2)^(5/2)*(d + e*x^2)^2*ArcTan[(-(e*x*Sqrt[a + b*x^2]) + Sqrt[b]*(d + e*x^2 
))/(Sqrt[d]*Sqrt[-(b*d) + a*e])])/(-(b*d) + a*e)^(9/2) + (24*a*b*d*e^3*(a 
+ b*x^2)^(5/2)*(d + e*x^2)^2*ArcTanh[(-(e*x*Sqrt[a + b*x^2]) + Sqrt[b]*(d 
+ e*x^2))/(Sqrt[d]*Sqrt[b*d - a*e])])/(b*d - a*e)^(9/2)))/(24*d^(5/2)*((a 
+ b*x^2)*(d + e*x^2))^(5/2))
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {1395, 316, 402, 25, 402, 25, 27, 402, 27, 291, 221}

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.

\(\displaystyle \int \frac {1}{\sqrt {d+e x^2} \left (x^2 (a e+b d)+a d+b e x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1395

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \frac {1}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^3}dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {-6 b e x^2+4 b d-3 a e}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^2}dx}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}-\frac {\int -\frac {8 b^2 d^2-24 a b e d+9 a^2 e^2+4 b e (4 b d+3 a e) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^2}dx}{3 a (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {8 b^2 d^2-24 a b e d+9 a^2 e^2+4 b e (4 b d+3 a e) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^2}dx}{3 a (b d-a e)}+\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-3 a^2 e^2-40 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right ) (b d-a e)}-\frac {\int -\frac {e \left (2 b \left (8 b^2 d^2-40 a b e d-3 a^2 e^2\right ) x^2+a \left (8 b^2 d^2+36 a b e d-9 a^2 e^2\right )\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^2}dx}{a (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\int \frac {e \left (2 b \left (8 b^2 d^2-40 a b e d-3 a^2 e^2\right ) x^2+a \left (8 b^2 d^2+36 a b e d-9 a^2 e^2\right )\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^2}dx}{a (b d-a e)}+\frac {b x \left (-3 a^2 e^2-40 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right ) (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \int \frac {2 b \left (8 b^2 d^2-40 a b e d-3 a^2 e^2\right ) x^2+a \left (8 b^2 d^2+36 a b e d-9 a^2 e^2\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^2}dx}{a (b d-a e)}+\frac {b x \left (-3 a^2 e^2-40 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right ) (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \left (\frac {\int \frac {3 a^2 e \left (48 b^2 d^2-16 a b e d+3 a^2 e^2\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (9 a^3 e^3-42 a^2 b d e^2-88 a b^2 d^2 e+16 b^3 d^3\right )}{2 d \left (d+e x^2\right ) (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (-3 a^2 e^2-40 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right ) (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \left (\frac {3 a^2 e \left (3 a^2 e^2-16 a b d e+48 b^2 d^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (9 a^3 e^3-42 a^2 b d e^2-88 a b^2 d^2 e+16 b^3 d^3\right )}{2 d \left (d+e x^2\right ) (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (-3 a^2 e^2-40 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right ) (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \left (\frac {3 a^2 e \left (3 a^2 e^2-16 a b d e+48 b^2 d^2\right ) \int \frac {1}{d-\frac {(b d-a e) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (9 a^3 e^3-42 a^2 b d e^2-88 a b^2 d^2 e+16 b^3 d^3\right )}{2 d \left (d+e x^2\right ) (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (-3 a^2 e^2-40 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right ) (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-3 a^2 e^2-40 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right ) (b d-a e)}+\frac {e \left (\frac {3 a^2 e \left (3 a^2 e^2-16 a b d e+48 b^2 d^2\right ) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{2 d^{3/2} (b d-a e)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (9 a^3 e^3-42 a^2 b d e^2-88 a b^2 d^2 e+16 b^3 d^3\right )}{2 d \left (d+e x^2\right ) (b d-a e)}\right )}{a (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+4 b d)}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}-\frac {e x}{4 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[1/(Sqrt[d + e*x^2]*(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(5/2)),x]
 

Output:

(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*(-1/4*(e*x)/(d*(b*d - a*e)*(a + b*x^2)^(3 
/2)*(d + e*x^2)^2) + ((b*(4*b*d + 3*a*e)*x)/(3*a*(b*d - a*e)*(a + b*x^2)^( 
3/2)*(d + e*x^2)) + ((b*(8*b^2*d^2 - 40*a*b*d*e - 3*a^2*e^2)*x)/(a*(b*d - 
a*e)*Sqrt[a + b*x^2]*(d + e*x^2)) + (e*(((16*b^3*d^3 - 88*a*b^2*d^2*e - 42 
*a^2*b*d*e^2 + 9*a^3*e^3)*x*Sqrt[a + b*x^2])/(2*d*(b*d - a*e)*(d + e*x^2)) 
 + (3*a^2*e*(48*b^2*d^2 - 16*a*b*d*e + 3*a^2*e^2)*ArcTanh[(Sqrt[b*d - a*e] 
*x)/(Sqrt[d]*Sqrt[a + b*x^2])])/(2*d^(3/2)*(b*d - a*e)^(3/2))))/(a*(b*d - 
a*e)))/(3*a*(b*d - a*e)))/(4*d*(b*d - a*e))))/Sqrt[a*d + (b*d + a*e)*x^2 + 
 b*e*x^4]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

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])
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(8016\) vs. \(2(364)=728\).

Time = 0.55 (sec) , antiderivative size = 8017, normalized size of antiderivative = 19.94

method	result	size
default	\(\text {Expression too large to display}\)	\(8017\)

Input:

int(1/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x,method=_RETURNVE 
RBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1333 vs. \(2 (364) = 728\).

Time = 0.36 (sec) , antiderivative size = 2691, normalized size of antiderivative = 6.69 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(3*(48*a^4*b^2*d^5*e^2 - 16*a^5*b*d^4*e^3 + 3*a^6*d^3*e^4 + (48*a^2* 
b^4*d^2*e^5 - 16*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*x^10 + (144*a^2*b^4*d^3*e^ 
4 + 48*a^3*b^3*d^2*e^5 - 23*a^4*b^2*d*e^6 + 6*a^5*b*e^7)*x^8 + (144*a^2*b^ 
4*d^4*e^3 + 240*a^3*b^3*d^3*e^4 - 39*a^4*b^2*d^2*e^5 + 2*a^5*b*d*e^6 + 3*a 
^6*e^7)*x^6 + (48*a^2*b^4*d^5*e^2 + 272*a^3*b^3*d^4*e^3 + 51*a^4*b^2*d^3*e 
^4 - 30*a^5*b*d^2*e^5 + 9*a^6*d*e^6)*x^4 + (96*a^3*b^3*d^5*e^2 + 112*a^4*b 
^2*d^4*e^3 - 42*a^5*b*d^3*e^4 + 9*a^6*d^2*e^5)*x^2)*sqrt(b*d^2 - a*d*e)*lo 
g((2*b*d^2*x^2 + (2*b*d*e - a*e^2)*x^4 + a*d^2 + 2*sqrt(b*e*x^4 + (b*d + a 
*e)*x^2 + a*d)*sqrt(b*d^2 - a*d*e)*sqrt(e*x^2 + d)*x)/(e^2*x^4 + 2*d*e*x^2 
 + d^2)) + 2*((16*b^6*d^5*e^2 - 104*a*b^5*d^4*e^3 + 46*a^2*b^4*d^3*e^4 + 5 
1*a^3*b^3*d^2*e^5 - 9*a^4*b^2*d*e^6)*x^7 + (32*b^6*d^6*e - 184*a*b^5*d^5*e 
^2 + 8*a^2*b^4*d^4*e^3 + 75*a^3*b^3*d^3*e^4 + 87*a^4*b^2*d^2*e^5 - 18*a^5* 
b*d*e^6)*x^5 + (16*b^6*d^7 - 56*a*b^5*d^6*e - 152*a^2*b^4*d^5*e^2 + 96*a^3 
*b^3*d^4*e^3 + 84*a^4*b^2*d^3*e^4 + 21*a^5*b*d^2*e^5 - 9*a^6*d*e^6)*x^3 + 
3*(8*a*b^5*d^7 - 40*a^2*b^4*d^6*e + 32*a^3*b^3*d^5*e^2 - 16*a^4*b^2*d^4*e^ 
3 + 21*a^5*b*d^3*e^4 - 5*a^6*d^2*e^5)*x)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + 
a*d)*sqrt(e*x^2 + d))/(a^4*b^5*d^11 - 5*a^5*b^4*d^10*e + 10*a^6*b^3*d^9*e^ 
2 - 10*a^7*b^2*d^8*e^3 + 5*a^8*b*d^7*e^4 - a^9*d^6*e^5 + (a^2*b^7*d^8*e^3 
- 5*a^3*b^6*d^7*e^4 + 10*a^4*b^5*d^6*e^5 - 10*a^5*b^4*d^5*e^6 + 5*a^6*b^3* 
d^4*e^7 - a^7*b^2*d^3*e^8)*x^10 + (3*a^2*b^7*d^9*e^2 - 13*a^3*b^6*d^8*e...
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (\left (a + b x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}} \sqrt {d + e x^{2}}}\, dx \] Input:

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

Output:

Integral(1/(((a + b*x**2)*(d + e*x**2))**(5/2)*sqrt(d + e*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {5}{2}} \sqrt {e x^{2} + d}} \,d x } \] Input:

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

Output:

integrate(1/((b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(5/2)*sqrt(e*x^2 + d)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {5}{2}} \sqrt {e x^{2} + d}} \,d x } \] Input:

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

Output:

integrate(1/((b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(5/2)*sqrt(e*x^2 + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {e\,x^2+d}\,{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, \left (a d +\left (a e +b d \right ) x^{2}+b e \,x^{4}\right )^{\frac {5}{2}}}d x \] Input:

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

Output:

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