\(\int \frac {(d+e x^2)^{3/2}}{(a d+(b d+a e) x^2+b e x^4)^{5/2}} \, dx\) [19]

Optimal result

Integrand size = 37, antiderivative size = 197 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\frac {b x \left (d+e x^2\right )^{3/2}}{3 a (b d-a e) \left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}}+\frac {b (2 b d-5 a e) x \sqrt {d+e x^2}}{3 a^2 (b d-a e)^2 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {e^2 \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{\sqrt {d} (b d-a e)^{5/2}} \] Output:

1/3*b*x*(e*x^2+d)^(3/2)/a/(-a*e+b*d)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2)+1/3 
*b*(-5*a*e+2*b*d)*x*(e*x^2+d)^(1/2)/a^2/(-a*e+b*d)^2/(a*d+(a*e+b*d)*x^2+b* 
e*x^4)^(1/2)+e^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^(1/2)/(-a*e+b*d)^(5/2)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\frac {\left (d+e x^2\right )^{3/2} \left (b \sqrt {d} \sqrt {-b d+a e} x \left (-6 a^2 e+2 b^2 d x^2+a b \left (3 d-5 e x^2\right )\right )-3 a^2 e^2 \left (a+b x^2\right )^{3/2} \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )\right )}{3 a^2 \sqrt {d} (-b d+a e)^{5/2} \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{3/2}} \] Input:

Integrate[(d + e*x^2)^(3/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(5/2),x]
 

Output:

((d + e*x^2)^(3/2)*(b*Sqrt[d]*Sqrt[-(b*d) + a*e]*x*(-6*a^2*e + 2*b^2*d*x^2 
 + a*b*(3*d - 5*e*x^2)) - 3*a^2*e^2*(a + b*x^2)^(3/2)*ArcTan[(-(e*x*Sqrt[a 
 + b*x^2]) + Sqrt[b]*(d + e*x^2))/(Sqrt[d]*Sqrt[-(b*d) + a*e])]))/(3*a^2*S 
qrt[d]*(-(b*d) + a*e)^(5/2)*((a + b*x^2)*(d + e*x^2))^(3/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1395, 316, 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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

Int[(d + e*x^2)^(3/2)/(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]*((b*x)/(3*a*(b*d - a*e)*(a + b*x^2)^(3/2) 
) + ((b*(2*b*d - 5*a*e)*x)/(a*(b*d - a*e)*Sqrt[a + b*x^2]) + (3*a*e^2*ArcT 
anh[(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))))/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. \(839\) vs. \(2(173)=346\).

Time = 0.29 (sec) , antiderivative size = 840, normalized size of antiderivative = 4.26

method	result	size
default	\(-\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, e \,b^{2} \left (4 \sqrt {-d e}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, a \,b^{2} e \,x^{3}-4 \sqrt {-d e}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, b^{3} d \,x^{3}+6 \sqrt {-d e}\, \sqrt {\frac {a e -b d}{e}}\, \sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, a \,b^{2} e \,x^{3}+3 \sqrt {b \,x^{2}+a}\, \ln \left (\frac {2 \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, e +2 \sqrt {-d e}\, b x +2 a e}{e x -\sqrt {-d e}}\right ) \sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, a^{2} b \,e^{2} x^{2}-3 \sqrt {b \,x^{2}+a}\, \ln \left (\frac {2 \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, e -2 \sqrt {-d e}\, b x +2 a e}{e x +\sqrt {-d e}}\right ) \sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, a^{2} b \,e^{2} x^{2}+6 \sqrt {-d e}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, a^{2} b e x -6 \sqrt {-d e}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, a \,b^{2} d x +6 \sqrt {-d e}\, \sqrt {\frac {a e -b d}{e}}\, \sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, a^{2} b e x +3 \sqrt {b \,x^{2}+a}\, \ln \left (\frac {2 \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, e +2 \sqrt {-d e}\, b x +2 a e}{e x -\sqrt {-d e}}\right ) \sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, a^{3} e^{2}-3 \sqrt {b \,x^{2}+a}\, \ln \left (\frac {2 \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, e -2 \sqrt {-d e}\, b x +2 a e}{e x +\sqrt {-d e}}\right ) \sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, a^{3} e^{2}\right )}{6 \sqrt {e \,x^{2}+d}\, \left (b \,x^{2}+a \right ) \sqrt {-d e}\, \left (\sqrt {-d e}\, b +e \sqrt {-a b}\right ) \left (\sqrt {-d e}\, b -e \sqrt {-a b}\right ) \left (a e -b d \right ) a^{2} \sqrt {\frac {a e -b d}{e}}\, \left (b x -\sqrt {-a b}\right ) \sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, \left (b x +\sqrt {-a b}\right )}\)	\(840\)

Input:

int((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x,method=_RETURNVERB 
OSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (173) = 346\).

Time = 0.10 (sec) , antiderivative size = 1023, normalized size of antiderivative = 5.19 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x, algorithm=" 
fricas")
 

Output:

[1/6*(3*(a^2*b^2*e^3*x^6 + a^4*d*e^2 + (a^2*b^2*d*e^2 + 2*a^3*b*e^3)*x^4 + 
 (2*a^3*b*d*e^2 + a^4*e^3)*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)*((2*b^4*d^3 - 7*a*b^3*d^2*e + 5*a^2*b^2*d*e 
^2)*x^3 + 3*(a*b^3*d^3 - 3*a^2*b^2*d^2*e + 2*a^3*b*d*e^2)*x)*sqrt(e*x^2 + 
d))/(a^4*b^3*d^5 - 3*a^5*b^2*d^4*e + 3*a^6*b*d^3*e^2 - a^7*d^2*e^3 + (a^2* 
b^5*d^4*e - 3*a^3*b^4*d^3*e^2 + 3*a^4*b^3*d^2*e^3 - a^5*b^2*d*e^4)*x^6 + ( 
a^2*b^5*d^5 - a^3*b^4*d^4*e - 3*a^4*b^3*d^3*e^2 + 5*a^5*b^2*d^2*e^3 - 2*a^ 
6*b*d*e^4)*x^4 + (2*a^3*b^4*d^5 - 5*a^4*b^3*d^4*e + 3*a^5*b^2*d^3*e^2 + a^ 
6*b*d^2*e^3 - a^7*d*e^4)*x^2), -1/3*(3*(a^2*b^2*e^3*x^6 + a^4*d*e^2 + (a^2 
*b^2*d*e^2 + 2*a^3*b*e^3)*x^4 + (2*a^3*b*d*e^2 + a^4*e^3)*x^2)*sqrt(-b*d^2 
 + a*d*e)*arctan(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/(b*d*e*x^4 + a*d^2 + (b*d^2 + a*d*e)*x^2)) - sqrt(b*e* 
x^4 + (b*d + a*e)*x^2 + a*d)*((2*b^4*d^3 - 7*a*b^3*d^2*e + 5*a^2*b^2*d*e^2 
)*x^3 + 3*(a*b^3*d^3 - 3*a^2*b^2*d^2*e + 2*a^3*b*d*e^2)*x)*sqrt(e*x^2 + d) 
)/(a^4*b^3*d^5 - 3*a^5*b^2*d^4*e + 3*a^6*b*d^3*e^2 - a^7*d^2*e^3 + (a^2*b^ 
5*d^4*e - 3*a^3*b^4*d^3*e^2 + 3*a^4*b^3*d^2*e^3 - a^5*b^2*d*e^4)*x^6 + (a^ 
2*b^5*d^5 - a^3*b^4*d^4*e - 3*a^4*b^3*d^3*e^2 + 5*a^5*b^2*d^2*e^3 - 2*a^6* 
b*d*e^4)*x^4 + (2*a^3*b^4*d^5 - 5*a^4*b^3*d^4*e + 3*a^5*b^2*d^3*e^2 + a...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**(3/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x, algorithm=" 
maxima")
 

Output:

integrate((e*x^2 + d)^(3/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(5/2), x)
 

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x, algorithm=" 
giac")
 

Output:

integrate((e*x^2 + d)^(3/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}}{{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{5/2}} \,d x \] Input:

int((d + e*x^2)^(3/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int((d + e*x^2)^(3/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 768, normalized size of antiderivative = 3.90 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\frac {-3 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}-\sqrt {e}\, \sqrt {b \,x^{2}+a}-\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a^{4} e^{2}-6 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}-\sqrt {e}\, \sqrt {b \,x^{2}+a}-\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a^{3} b \,e^{2} x^{2}-3 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}-\sqrt {e}\, \sqrt {b \,x^{2}+a}-\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a^{2} b^{2} e^{2} x^{4}-3 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}+\sqrt {e}\, \sqrt {b \,x^{2}+a}+\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a^{4} e^{2}-6 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}+\sqrt {e}\, \sqrt {b \,x^{2}+a}+\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a^{3} b \,e^{2} x^{2}-3 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}+\sqrt {e}\, \sqrt {b \,x^{2}+a}+\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a^{2} b^{2} e^{2} x^{4}-6 \sqrt {b \,x^{2}+a}\, a^{3} b d \,e^{2} x +9 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{2} e x -5 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,e^{2} x^{3}-3 \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{3} x +7 \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{2} e \,x^{3}-2 \sqrt {b \,x^{2}+a}\, b^{4} d^{3} x^{3}+3 \sqrt {b}\, a^{4} d \,e^{2}-5 \sqrt {b}\, a^{3} b \,d^{2} e +6 \sqrt {b}\, a^{3} b d \,e^{2} x^{2}+2 \sqrt {b}\, a^{2} b^{2} d^{3}-10 \sqrt {b}\, a^{2} b^{2} d^{2} e \,x^{2}+3 \sqrt {b}\, a^{2} b^{2} d \,e^{2} x^{4}+4 \sqrt {b}\, a \,b^{3} d^{3} x^{2}-5 \sqrt {b}\, a \,b^{3} d^{2} e \,x^{4}+2 \sqrt {b}\, b^{4} d^{3} x^{4}}{3 a^{2} d \left (a^{3} b^{2} e^{3} x^{4}-3 a^{2} b^{3} d \,e^{2} x^{4}+3 a \,b^{4} d^{2} e \,x^{4}-b^{5} d^{3} x^{4}+2 a^{4} b \,e^{3} x^{2}-6 a^{3} b^{2} d \,e^{2} x^{2}+6 a^{2} b^{3} d^{2} e \,x^{2}-2 a \,b^{4} d^{3} x^{2}+a^{5} e^{3}-3 a^{4} b d \,e^{2}+3 a^{3} b^{2} d^{2} e -a^{2} b^{3} d^{3}\right )} \] Input:

int((e*x^2+d)^(3/2)/(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**4*e**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**3*b*e**2*x**2 - 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**2*b**2*e**2*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*e**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**3*b*e**2*x**2 - 3*s 
qrt(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**2*b**2*e**2*x**4 - 6*sqrt(a + b*x 
**2)*a**3*b*d*e**2*x + 9*sqrt(a + b*x**2)*a**2*b**2*d**2*e*x - 5*sqrt(a + 
b*x**2)*a**2*b**2*d*e**2*x**3 - 3*sqrt(a + b*x**2)*a*b**3*d**3*x + 7*sqrt( 
a + b*x**2)*a*b**3*d**2*e*x**3 - 2*sqrt(a + b*x**2)*b**4*d**3*x**3 + 3*sqr 
t(b)*a**4*d*e**2 - 5*sqrt(b)*a**3*b*d**2*e + 6*sqrt(b)*a**3*b*d*e**2*x**2 
+ 2*sqrt(b)*a**2*b**2*d**3 - 10*sqrt(b)*a**2*b**2*d**2*e*x**2 + 3*sqrt(b)* 
a**2*b**2*d*e**2*x**4 + 4*sqrt(b)*a*b**3*d**3*x**2 - 5*sqrt(b)*a*b**3*d**2 
*e*x**4 + 2*sqrt(b)*b**4*d**3*x**4)/(3*a**2*d*(a**5*e**3 - 3*a**4*b*d*e**2 
 + 2*a**4*b*e**3*x**2 + 3*a**3*b**2*d**2*e - 6*a**3*b**2*d*e**2*x**2 + a** 
3*b**2*e**3*x**4 - a**2*b**3*d**3 + 6*a**2*b**3*d**2*e*x**2 - 3*a**2*b*...