\(\int \frac {(d+e x^2)^{5/2}}{(a d+(b d+a e) x^2+b e x^4)^{7/2}} \, dx\) [26]

Optimal result

Integrand size = 37, antiderivative size = 316 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {b \left (8 b^2 d^2-26 a b d e+33 a^2 e^2\right ) x \sqrt {d+e x^2}}{15 a^3 (b d-a e)^3 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b x \sqrt {d+e x^2}}{5 a (b d-a e) \left (a+b x^2\right )^2 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b (4 b d-9 a e) x \sqrt {d+e x^2}}{15 a^2 (b d-a e)^2 \left (a+b x^2\right ) \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e^3 \sqrt {a+b x^2} \sqrt {d+e x^2} \text {arctanh}\left (\frac {\sqrt {b d-a e} x}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {d} (b d-a e)^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \] Output:

1/15*b*(33*a^2*e^2-26*a*b*d*e+8*b^2*d^2)*x*(e*x^2+d)^(1/2)/a^3/(-a*e+b*d)^ 
3/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/5*b*x*(e*x^2+d)^(1/2)/a/(-a*e+b*d)/( 
b*x^2+a)^2/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/15*b*(-9*a*e+4*b*d)*x*(e*x^ 
2+d)^(1/2)/a^2/(-a*e+b*d)^2/(b*x^2+a)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)-e^ 
3*(b*x^2+a)^(1/2)*(e*x^2+d)^(1/2)*arctanh((-a*e+b*d)^(1/2)*x/d^(1/2)/(b*x^ 
2+a)^(1/2))/d^(1/2)/(-a*e+b*d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.71 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {\left (d+e x^2\right )^{7/2} \left (-\frac {b x \left (a+b x^2\right ) \left (45 a^4 e^2+8 b^4 d^2 x^4+2 a b^3 d x^2 \left (10 d-13 e x^2\right )+15 a^3 b e \left (-3 d+5 e x^2\right )+a^2 b^2 \left (15 d^2-65 d e x^2+33 e^2 x^4\right )\right )}{a^3 (-b d+a e)^3}-\frac {15 e^3 \left (a+b x^2\right )^{7/2} \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )}{\sqrt {d} (-b d+a e)^{7/2}}\right )}{15 \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{7/2}} \] Input:

Integrate[(d + e*x^2)^(5/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(7/2),x]
 

Output:

((d + e*x^2)^(7/2)*(-((b*x*(a + b*x^2)*(45*a^4*e^2 + 8*b^4*d^2*x^4 + 2*a*b 
^3*d*x^2*(10*d - 13*e*x^2) + 15*a^3*b*e*(-3*d + 5*e*x^2) + a^2*b^2*(15*d^2 
 - 65*d*e*x^2 + 33*e^2*x^4)))/(a^3*(-(b*d) + a*e)^3)) - (15*e^3*(a + b*x^2 
)^(7/2)*ArcTan[(-(e*x*Sqrt[a + b*x^2]) + Sqrt[b]*(d + e*x^2))/(Sqrt[d]*Sqr 
t[-(b*d) + a*e])])/(Sqrt[d]*(-(b*d) + a*e)^(7/2))))/(15*((a + b*x^2)*(d + 
e*x^2))^(7/2))
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.82, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {1395, 316, 25, 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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (x^2 (a e+b d)+a d+b e x^4\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1395

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \frac {1}{\left (b x^2+a\right )^{7/2} \left (e x^2+d\right )}dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {b x (4 b d-9 a e)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}-\frac {\int -\frac {8 b^2 d^2-18 a b e d+15 a^2 e^2+2 b e (4 b d-9 a e) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )}dx}{3 a (b d-a e)}}{5 a (b d-a e)}+\frac {b x}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {8 b^2 d^2-18 a b e d+15 a^2 e^2+2 b e (4 b d-9 a e) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )}dx}{3 a (b d-a e)}+\frac {b x (4 b d-9 a e)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (33 a^2 e^2-26 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {\int \frac {15 a^3 e^3}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{a (b d-a e)}}{3 a (b d-a e)}+\frac {b x (4 b d-9 a e)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (33 a^2 e^2-26 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {15 a^2 e^3 \int \frac {1}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{b d-a e}}{3 a (b d-a e)}+\frac {b x (4 b d-9 a e)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (33 a^2 e^2-26 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {15 a^2 e^3 \int \frac {1}{d-\frac {(b d-a e) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b d-a e}}{3 a (b d-a e)}+\frac {b x (4 b d-9 a e)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (33 a^2 e^2-26 a b d e+8 b^2 d^2\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {15 a^2 e^3 \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {d} (b d-a e)^{3/2}}}{3 a (b d-a e)}+\frac {b x (4 b d-9 a e)}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[(d + e*x^2)^(5/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(7/2),x]
 

Output:

(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*((b*x)/(5*a*(b*d - a*e)*(a + b*x^2)^(5/2) 
) + ((b*(4*b*d - 9*a*e)*x)/(3*a*(b*d - a*e)*(a + b*x^2)^(3/2)) + ((b*(8*b^ 
2*d^2 - 26*a*b*d*e + 33*a^2*e^2)*x)/(a*(b*d - a*e)*Sqrt[a + b*x^2]) - (15* 
a^2*e^3*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)))/(5*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. \(1515\) vs. \(2(282)=564\).

Time = 0.29 (sec) , antiderivative size = 1516, normalized size of antiderivative = 4.80

method	result	size
default	\(\text {Expression too large to display}\)	\(1516\)

Input:

int((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x,method=_RETURNVERB 
OSE)
 

Output:

-1/30*b^3*(15*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^3*x^4*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x+( 
-a*b)^(1/2)))^(1/2)*(b*x^2+a)^(1/2)-15*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^3*x^4*(-1/b*( 
b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)*(b*x^2+a)^(1/2)+50*(-d*e)^(1/ 
2)*b^3*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)*((a*e-b*d)/e)^( 
1/2)*a^2*e^2*x^5+16*a^2*b^3*e^2*x^5*(b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*(- 
d*e)^(1/2)-20*(-d*e)^(1/2)*b^4*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2) 
))^(1/2)*((a*e-b*d)/e)^(1/2)*a*d*e*x^5-32*a*b^4*d*e*x^5*(b*x^2+a)^(1/2)*(( 
a*e-b*d)/e)^(1/2)*(-d*e)^(1/2)+16*b^5*d^2*x^5*(b*x^2+a)^(1/2)*((a*e-b*d)/e 
)^(1/2)*(-d*e)^(1/2)+30*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^3*x^2*(-1/b*(b*x+(-a*b)^(1/2)) 
*(-b*x+(-a*b)^(1/2)))^(1/2)*(b*x^2+a)^(1/2)-30*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^3*x^2*( 
-1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)*(b*x^2+a)^(1/2)+110*(-d 
*e)^(1/2)*a^3*b^2*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)*((a* 
e-b*d)/e)^(1/2)*e^2*x^3+40*a^3*b^2*e^2*x^3*(b*x^2+a)^(1/2)*((a*e-b*d)/e)^( 
1/2)*(-d*e)^(1/2)-50*(-d*e)^(1/2)*a^2*b^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*x^3-80*a^2*b^3*d*e*x^3*(b*x^2+ 
a)^(1/2)*((a*e-b*d)/e)^(1/2)*(-d*e)^(1/2)+40*a*b^4*d^2*x^3*(b*x^2+a)^(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (282) = 564\).

Time = 0.15 (sec) , antiderivative size = 1512, normalized size of antiderivative = 4.78 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" 
fricas")
 

Output:

[-1/30*(15*(a^3*b^3*e^4*x^8 + a^6*d*e^3 + (a^3*b^3*d*e^3 + 3*a^4*b^2*e^4)* 
x^6 + 3*(a^4*b^2*d*e^3 + a^5*b*e^4)*x^4 + (3*a^5*b*d*e^3 + a^6*e^4)*x^2)*s 
qrt(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*sq 
rt(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)*(( 
8*b^6*d^4 - 34*a*b^5*d^3*e + 59*a^2*b^4*d^2*e^2 - 33*a^3*b^3*d*e^3)*x^5 + 
5*(4*a*b^5*d^4 - 17*a^2*b^4*d^3*e + 28*a^3*b^3*d^2*e^2 - 15*a^4*b^2*d*e^3) 
*x^3 + 15*(a^2*b^4*d^4 - 4*a^3*b^3*d^3*e + 6*a^4*b^2*d^2*e^2 - 3*a^5*b*d*e 
^3)*x)*sqrt(e*x^2 + d))/(a^6*b^4*d^6 - 4*a^7*b^3*d^5*e + 6*a^8*b^2*d^4*e^2 
 - 4*a^9*b*d^3*e^3 + a^10*d^2*e^4 + (a^3*b^7*d^5*e - 4*a^4*b^6*d^4*e^2 + 6 
*a^5*b^5*d^3*e^3 - 4*a^6*b^4*d^2*e^4 + a^7*b^3*d*e^5)*x^8 + (a^3*b^7*d^6 - 
 a^4*b^6*d^5*e - 6*a^5*b^5*d^4*e^2 + 14*a^6*b^4*d^3*e^3 - 11*a^7*b^3*d^2*e 
^4 + 3*a^8*b^2*d*e^5)*x^6 + 3*(a^4*b^6*d^6 - 3*a^5*b^5*d^5*e + 2*a^6*b^4*d 
^4*e^2 + 2*a^7*b^3*d^3*e^3 - 3*a^8*b^2*d^2*e^4 + a^9*b*d*e^5)*x^4 + (3*a^5 
*b^5*d^6 - 11*a^6*b^4*d^5*e + 14*a^7*b^3*d^4*e^2 - 6*a^8*b^2*d^3*e^3 - a^9 
*b*d^2*e^4 + a^10*d*e^5)*x^2), 1/15*(15*(a^3*b^3*e^4*x^8 + a^6*d*e^3 + (a^ 
3*b^3*d*e^3 + 3*a^4*b^2*e^4)*x^6 + 3*(a^4*b^2*d*e^3 + a^5*b*e^4)*x^4 + (3* 
a^5*b*d*e^3 + a^6*e^4)*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)*((8...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**(5/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" 
maxima")
 

Output:

integrate((e*x^2 + d)^(5/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" 
giac")
 

Output:

integrate((e*x^2 + d)^(5/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{7/2}} \,d x \] Input:

int((d + e*x^2)^(5/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int((d + e*x^2)^(5/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1279, normalized size of antiderivative = 4.05 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

int((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x)
 

Output:

( - 15*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**3 - 45*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**3*x**2 - 45*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)/(sqrt( 
d)*sqrt(b)))*a**4*b**2*e**3*x**4 - 15*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*e**3*x**6 - 15*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* 
*3 - 45*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**3*x**2 - 45*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**3*x**4 - 15*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*e**3*x**6 - 45*sqrt(a + b*x**2)*a**5*b*d 
*e**3*x + 90*sqrt(a + b*x**2)*a**4*b**2*d**2*e**2*x - 75*sqrt(a + b*x**2)* 
a**4*b**2*d*e**3*x**3 - 60*sqrt(a + b*x**2)*a**3*b**3*d**3*e*x + 140*sqrt( 
a + b*x**2)*a**3*b**3*d**2*e**2*x**3 - 33*sqrt(a + b*x**2)*a**3*b**3*d*e** 
3*x**5 + 15*sqrt(a + b*x**2)*a**2*b**4*d**4*x - 85*sqrt(a + b*x**2)*a**2*b 
**4*d**3*e*x**3 + 59*sqrt(a + b*x**2)*a**2*b**4*d**2*e**2*x**5 + 20*sqr...