\(\int \frac {\sqrt {d+e x^2}}{(a d+(b d+a e) x^2+b e x^4)^{5/2}} \, dx\) [20]

Optimal result

Integrand size = 37, antiderivative size = 293 \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\frac {b x \sqrt {d+e x^2}}{3 a (b d-a e) \left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}}+\frac {e (2 b d+3 a e) x}{6 a d (b d-a e)^2 \sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (4 b^2 d^2-16 a b d e-3 a^2 e^2\right ) x \sqrt {d+e x^2}}{6 a^2 d (b d-a e)^3 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {e^2 (6 b d-a e) \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{2 d^{3/2} (b d-a e)^{7/2}} \] Output:

1/3*b*x*(e*x^2+d)^(1/2)/a/(-a*e+b*d)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2)+1/6 
*e*(3*a*e+2*b*d)*x/a/d/(-a*e+b*d)^2/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e 
*x^4)^(1/2)+1/6*b*(-3*a^2*e^2-16*a*b*d*e+4*b^2*d^2)*x*(e*x^2+d)^(1/2)/a^2/ 
d/(-a*e+b*d)^3/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/2*e^2*(-a*e+6*b*d)*arct 
anh((-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^(3/2)/(-a*e+b*d)^(7/2)
 

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\frac {\left (d+e x^2\right )^{3/2} \left (\frac {\sqrt {d} x \left (a+b x^2\right ) \left (3 a^4 e^3+6 a^3 b e^3 x^2-4 b^4 d^2 x^2 \left (d+e x^2\right )+3 a^2 b^2 e \left (6 d^2+6 d e x^2+e^2 x^4\right )+2 a b^3 d \left (-3 d^2+5 d e x^2+8 e^2 x^4\right )\right )}{a^2 (-b d+a e)^3}+\frac {3 e^2 (6 b d-a e) \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}}\right )}{6 d^{3/2} \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{5/2}} \] Input:

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

Output:

((d + e*x^2)^(3/2)*((Sqrt[d]*x*(a + b*x^2)*(3*a^4*e^3 + 6*a^3*b*e^3*x^2 - 
4*b^4*d^2*x^2*(d + e*x^2) + 3*a^2*b^2*e*(6*d^2 + 6*d*e*x^2 + e^2*x^4) + 2* 
a*b^3*d*(-3*d^2 + 5*d*e*x^2 + 8*e^2*x^4)))/(a^2*(-(b*d) + a*e)^3) + (3*e^2 
*(6*b*d - a*e)*(a + b*x^2)^(5/2)*(d + e*x^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)^(7/ 
2)))/(6*d^(3/2)*((a + b*x^2)*(d + e*x^2))^(5/2))
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1395, 316, 402, 25, 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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {b x (3 a e+2 b d)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}-\frac {\int -\frac {4 b^2 d^2-12 a b e d+3 a^2 e^2+2 b e (2 b d+3 a e) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )}dx}{3 a (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {4 b^2 d^2-12 a b e d+3 a^2 e^2+2 b e (2 b d+3 a e) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )}dx}{3 a (b d-a e)}+\frac {b x (3 a e+2 b d)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-3 a^2 e^2-16 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {\int -\frac {3 a^2 e^2 (6 b d-a e)}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{a (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+2 b d)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 a e^2 (6 b d-a e) \int \frac {1}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{b d-a e}+\frac {b x \left (-3 a^2 e^2-16 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+2 b d)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {3 a e^2 (6 b d-a e) \int \frac {1}{d-\frac {(b d-a e) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b d-a e}+\frac {b x \left (-3 a^2 e^2-16 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} (b d-a e)}}{3 a (b d-a e)}+\frac {b x (3 a e+2 b d)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-3 a^2 e^2-16 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} (b d-a e)}+\frac {3 a e^2 (6 b d-a e) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {d} (b d-a e)^{3/2}}}{3 a (b d-a e)}+\frac {b x (3 a e+2 b d)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[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]*(-1/2*(e*x)/(d*(b*d - a*e)*(a + b*x^2)^(3 
/2)*(d + e*x^2)) + ((b*(2*b*d + 3*a*e)*x)/(3*a*(b*d - a*e)*(a + b*x^2)^(3/ 
2)) + ((b*(4*b^2*d^2 - 16*a*b*d*e - 3*a^2*e^2)*x)/(a*(b*d - a*e)*Sqrt[a + 
b*x^2]) + (3*a*e^2*(6*b*d - a*e)*ArcTanh[(Sqrt[b*d - a*e]*x)/(Sqrt[d]*Sqrt 
[a + b*x^2])])/(Sqrt[d]*(b*d - a*e)^(3/2)))/(3*a*(b*d - a*e)))/(2*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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2066\) vs. \(2(261)=522\).

Time = 0.46 (sec) , antiderivative size = 2067, normalized size of antiderivative = 7.05

method	result	size
default	\(\text {Expression too large to display}\)	\(2067\)

Input:

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

Output:

-1/12*(-3*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e 
)/(e*x+(-d*e)^(1/2)))*a^5*d*e^3+3*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2 
)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*a^5*e^4*x^2-3*ln(2*((b*x^2+a 
)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a^ 
5*e^4*x^2+3*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a 
*e)/(e*x-(-d*e)^(1/2)))*a^5*d*e^3-6*a^2*b^2*e^3*x^5*(b*x^2+a)^(1/2)*((a*e- 
b*d)/e)^(1/2)*(-d*e)^(1/2)-18*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+ 
(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*a^4*b*d^2*e^2+18*ln(2*((b*x^2+a) 
^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a^4 
*b*d^2*e^2+3*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+ 
a*e)/(e*x-(-d*e)^(1/2)))*a^3*b^2*e^4*x^6-3*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d 
)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a^3*b^2*e^4*x^6+6*l 
n(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d* 
e)^(1/2)))*a^4*b*e^4*x^4-6*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d 
*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a^4*b*e^4*x^4-32*a*b^3*d*e^2*x^5*(- 
1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)*((a*e-b*d)/e)^(1/2)*(-d* 
e)^(1/2)-36*a^2*b^2*d*e^2*x^3*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2)) 
)^(1/2)*((a*e-b*d)/e)^(1/2)*(-d*e)^(1/2)-20*a*b^3*d^2*e*x^3*(-1/b*(b*x+(-a 
*b)^(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)*((a*e-b*d)/e)^(1/2)*(-d*e)^(1/2)-36* 
a^2*b^2*d^2*e*x*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)*((a...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (261) = 522\).

Time = 0.16 (sec) , antiderivative size = 1727, normalized size of antiderivative = 5.89 \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(3*(6*a^4*b*d^3*e^2 - a^5*d^2*e^3 + (6*a^2*b^3*d*e^4 - a^3*b^2*e^5)* 
x^8 + 2*(6*a^2*b^3*d^2*e^3 + 5*a^3*b^2*d*e^4 - a^4*b*e^5)*x^6 + (6*a^2*b^3 
*d^3*e^2 + 23*a^3*b^2*d^2*e^3 + 2*a^4*b*d*e^4 - a^5*e^5)*x^4 + 2*(6*a^3*b^ 
2*d^3*e^2 + 5*a^4*b*d^2*e^3 - a^5*d*e^4)*x^2)*sqrt(b*d^2 - a*d*e)*log((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*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*((4*b^5*d^4*e - 20*a*b^4*d^3* 
e^2 + 13*a^2*b^3*d^2*e^3 + 3*a^3*b^2*d*e^4)*x^5 + 2*(2*b^5*d^5 - 7*a*b^4*d 
^4*e - 4*a^2*b^3*d^3*e^2 + 6*a^3*b^2*d^2*e^3 + 3*a^4*b*d*e^4)*x^3 + 3*(2*a 
*b^4*d^5 - 8*a^2*b^3*d^4*e + 6*a^3*b^2*d^3*e^2 - a^4*b*d^2*e^3 + a^5*d*e^4 
)*x)*sqrt(e*x^2 + d))/(a^4*b^4*d^8 - 4*a^5*b^3*d^7*e + 6*a^6*b^2*d^6*e^2 - 
 4*a^7*b*d^5*e^3 + a^8*d^4*e^4 + (a^2*b^6*d^6*e^2 - 4*a^3*b^5*d^5*e^3 + 6* 
a^4*b^4*d^4*e^4 - 4*a^5*b^3*d^3*e^5 + a^6*b^2*d^2*e^6)*x^8 + 2*(a^2*b^6*d^ 
7*e - 3*a^3*b^5*d^6*e^2 + 2*a^4*b^4*d^5*e^3 + 2*a^5*b^3*d^4*e^4 - 3*a^6*b^ 
2*d^3*e^5 + a^7*b*d^2*e^6)*x^6 + (a^2*b^6*d^8 - 9*a^4*b^4*d^6*e^2 + 16*a^5 
*b^3*d^5*e^3 - 9*a^6*b^2*d^4*e^4 + a^8*d^2*e^6)*x^4 + 2*(a^3*b^5*d^8 - 3*a 
^4*b^4*d^7*e + 2*a^5*b^3*d^6*e^2 + 2*a^6*b^2*d^5*e^3 - 3*a^7*b*d^4*e^4 + a 
^8*d^3*e^5)*x^2), -1/6*(3*(6*a^4*b*d^3*e^2 - a^5*d^2*e^3 + (6*a^2*b^3*d*e^ 
4 - a^3*b^2*e^5)*x^8 + 2*(6*a^2*b^3*d^2*e^3 + 5*a^3*b^2*d*e^4 - a^4*b*e^5) 
*x^6 + (6*a^2*b^3*d^3*e^2 + 23*a^3*b^2*d^2*e^3 + 2*a^4*b*d*e^4 - a^5*e^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 2.20 (sec) , antiderivative size = 2710, normalized size of antiderivative = 9.25 \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x 
**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**6*d*e**4 - 3*sqrt(d)*sqrt( 
a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt 
(b)*x)/(sqrt(d)*sqrt(b)))*a**6*e**5*x**2 + 6*sqrt(d)*sqrt(a*e - b*d)*atan( 
(sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)* 
sqrt(b)))*a**5*b*d**2*e**3 - 6*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b* 
d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**5 
*b*e**5*x**4 + 72*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)* 
sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*b**2*d**3*e* 
*2 + 84*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b 
*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*b**2*d**2*e**3*x**2 + 
9*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) 
 - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*b**2*d*e**4*x**4 - 3*sqrt(d) 
*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e 
)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*b**2*e**5*x**6 + 144*sqrt(d)*sqrt(a*e 
 - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b) 
*x)/(sqrt(d)*sqrt(b)))*a**3*b**3*d**3*e**2*x**2 + 150*sqrt(d)*sqrt(a*e - b 
*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/ 
(sqrt(d)*sqrt(b)))*a**3*b**3*d**2*e**3*x**4 + 6*sqrt(d)*sqrt(a*e - b*d)*at 
an((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sq...