\(\int \frac {(d+e x^2)^{3/2}}{(a d+(b d+a e) x^2+b e x^4)^{7/2}} \, dx\) [27]

Optimal result

Integrand size = 37, antiderivative size = 437 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=-\frac {e x}{2 d (b d-a e) \left (a+b x^2\right )^2 \sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (16 b^3 d^3-72 a b^2 d^2 e+146 a^2 b d e^2+15 a^3 e^3\right ) x \sqrt {d+e x^2}}{30 a^3 d (b d-a e)^4 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b (2 b d+5 a e) x \sqrt {d+e x^2}}{10 a d (b d-a e)^2 \left (a+b x^2\right )^2 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (8 b^2 d^2-28 a b d e-15 a^2 e^2\right ) x \sqrt {d+e x^2}}{30 a^2 d (b d-a e)^3 \left (a+b x^2\right ) \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e^3 (8 b d-a e) \sqrt {a+b x^2} \sqrt {d+e x^2} \text {arctanh}\left (\frac {\sqrt {b d-a e} x}{\sqrt {d} \sqrt {a+b x^2}}\right )}{2 d^{3/2} (b d-a e)^{9/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \] Output:

-1/2*e*x/d/(-a*e+b*d)/(b*x^2+a)^2/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x 
^4)^(1/2)+1/30*b*(15*a^3*e^3+146*a^2*b*d*e^2-72*a*b^2*d^2*e+16*b^3*d^3)*x* 
(e*x^2+d)^(1/2)/a^3/d/(-a*e+b*d)^4/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/10* 
b*(5*a*e+2*b*d)*x*(e*x^2+d)^(1/2)/a/d/(-a*e+b*d)^2/(b*x^2+a)^2/(a*d+(a*e+b 
*d)*x^2+b*e*x^4)^(1/2)+1/30*b*(-15*a^2*e^2-28*a*b*d*e+8*b^2*d^2)*x*(e*x^2+ 
d)^(1/2)/a^2/d/(-a*e+b*d)^3/(b*x^2+a)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)-1/ 
2*e^3*(-a*e+8*b*d)*(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^(3/2)/(-a*e+b*d)^(9/2)/(a*d+(a*e+b*d)*x^2+b 
*e*x^4)^(1/2)
 

Mathematica [A] (verified)

Time = 1.78 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.77 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {\left (d+e x^2\right )^{5/2} \left (\frac {\sqrt {d} x \left (a+b x^2\right ) \left (15 a^6 e^4+45 a^5 b e^4 x^2+16 b^6 d^3 x^4 \left (d+e x^2\right )+45 a^4 b^2 e^2 \left (2 d+e x^2\right )^2+8 a b^5 d^2 x^2 \left (5 d^2-4 d e x^2-9 e^2 x^4\right )+5 a^3 b^3 e \left (-24 d^3+40 d^2 e x^2+64 d e^2 x^4+3 e^3 x^6\right )+2 a^2 b^4 d \left (15 d^3-75 d^2 e x^2-17 d e^2 x^4+73 e^3 x^6\right )\right )}{a^3 (b d-a e)^4}+\frac {15 e^3 (8 b d-a e) \left (a+b x^2\right )^{7/2} \left (d+e x^2\right ) \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}}\right )}{30 d^{3/2} \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{7/2}} \] Input:

Integrate[(d + e*x^2)^(3/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(7/2),x]
 

Output:

((d + e*x^2)^(5/2)*((Sqrt[d]*x*(a + b*x^2)*(15*a^6*e^4 + 45*a^5*b*e^4*x^2 
+ 16*b^6*d^3*x^4*(d + e*x^2) + 45*a^4*b^2*e^2*(2*d + e*x^2)^2 + 8*a*b^5*d^ 
2*x^2*(5*d^2 - 4*d*e*x^2 - 9*e^2*x^4) + 5*a^3*b^3*e*(-24*d^3 + 40*d^2*e*x^ 
2 + 64*d*e^2*x^4 + 3*e^3*x^6) + 2*a^2*b^4*d*(15*d^3 - 75*d^2*e*x^2 - 17*d* 
e^2*x^4 + 73*e^3*x^6)))/(a^3*(b*d - a*e)^4) + (15*e^3*(8*b*d - a*e)*(a + b 
*x^2)^(7/2)*(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)^(9/2)))/(30*d^(3/2)*((a 
+ b*x^2)*(d + e*x^2))^(7/2))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.83, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {1395, 316, 402, 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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (x^2 (a e+b d)+a d+b e x^4\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {-6 b e x^2+2 b d-a e}{\left (b x^2+a\right )^{7/2} \left (e x^2+d\right )}dx}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {b x (5 a e+2 b d)}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}-\frac {\int -\frac {8 b^2 d^2-20 a b e d+5 a^2 e^2+4 b e (2 b d+5 a e) x^2}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )}dx}{5 a (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {8 b^2 d^2-20 a b e d+5 a^2 e^2+4 b e (2 b d+5 a e) x^2}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )}dx}{5 a (b d-a e)}+\frac {b x (5 a e+2 b d)}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-15 a^2 e^2-28 a b d e+8 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}-\frac {\int -\frac {16 b^3 d^3-56 a b^2 e d^2+90 a^2 b e^2 d-15 a^3 e^3+2 b e \left (8 b^2 d^2-28 a b e d-15 a^2 e^2\right ) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )}dx}{3 a (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+2 b d)}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\int \frac {16 b^3 d^3-56 a b^2 e d^2+90 a^2 b e^2 d-15 a^3 e^3+2 b e \left (8 b^2 d^2-28 a b e d-15 a^2 e^2\right ) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )}dx}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-28 a b d e+8 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+2 b d)}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {b x \left (15 a^3 e^3+146 a^2 b d e^2-72 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {\int \frac {15 a^3 e^3 (8 b d-a e)}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{a (b d-a e)}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-28 a b d e+8 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+2 b d)}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {b x \left (15 a^3 e^3+146 a^2 b d e^2-72 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {15 a^2 e^3 (8 b d-a e) \int \frac {1}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{b d-a e}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-28 a b d e+8 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+2 b d)}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {b x \left (15 a^3 e^3+146 a^2 b d e^2-72 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {15 a^2 e^3 (8 b d-a e) \int \frac {1}{d-\frac {(b d-a e) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b d-a e}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-28 a b d e+8 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+2 b d)}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-15 a^2 e^2-28 a b d e+8 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} (b d-a e)}+\frac {\frac {b x \left (15 a^3 e^3+146 a^2 b d e^2-72 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} (b d-a e)}-\frac {15 a^2 e^3 (8 b d-a e) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {d} (b d-a e)^{3/2}}}{3 a (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+2 b d)}{5 a \left (a+b x^2\right )^{5/2} (b d-a e)}}{2 d (b d-a e)}-\frac {e x}{2 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right ) (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[(d + e*x^2)^(3/2)/(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]*(-1/2*(e*x)/(d*(b*d - a*e)*(a + b*x^2)^(5 
/2)*(d + e*x^2)) + ((b*(2*b*d + 5*a*e)*x)/(5*a*(b*d - a*e)*(a + b*x^2)^(5/ 
2)) + ((b*(8*b^2*d^2 - 28*a*b*d*e - 15*a^2*e^2)*x)/(3*a*(b*d - a*e)*(a + b 
*x^2)^(3/2)) + ((b*(16*b^3*d^3 - 72*a*b^2*d^2*e + 146*a^2*b*d*e^2 + 15*a^3 
*e^3)*x)/(a*(b*d - a*e)*Sqrt[a + b*x^2]) - (15*a^2*e^3*(8*b*d - a*e)*ArcTa 
nh[(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)))/(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. \(3525\) vs. \(2(395)=790\).

Time = 0.33 (sec) , antiderivative size = 3526, normalized size of antiderivative = 8.07

method	result	size
default	\(\text {Expression too large to display}\)	\(3526\)

Input:

int((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x,method=_RETURNVERB 
OSE)
 

Output:

1/60*(225*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^2*d*e^4*x^4*(b*x^2+a)^(1/2)*(-1/b*(b*x+(-a*b)^ 
(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)+120*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^3*d^2*e^3*x^4*(b*x 
^2+a)^(1/2)*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b)^(1/2)))^(1/2)-90*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*b*d*e^4*x^2*(b*x^2+a)^(1/2)*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x+(-a*b) 
^(1/2)))^(1/2)-144*a*b^5*d^2*e^2*x^7*(-d*e)^(1/2)*(b*x^2+a)^(1/2)*((a*e-b* 
d)/e)^(1/2)+280*a^3*b^3*d*e^3*x^5*(-d*e)^(1/2)*(b*x^2+a)^(1/2)*((a*e-b*d)/ 
e)^(1/2)+360*a^3*b^3*d*e^3*x^5*(-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)-248*a^2*b^4*d^2*e^2*x^5*(-d*e)^( 
1/2)*(b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)+180*a^2*b^4*d^2*e^2*x^5*(-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 
)-64*a*b^5*d^3*e*x^5*(-d*e)^(1/2)*(b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)+180* 
a^4*b^2*d*e^3*x^3*(-d*e)^(1/2)*(b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)+180*a^4 
*b^2*d*e^3*x^3*(-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)+40*a^3*b^3*d^2*e^2*x^3*(-d*e)^(1/2)*(b*x^2+a)^(1 
/2)*((a*e-b*d)/e)^(1/2)+360*a^3*b^3*d^2*e^2*x^3*(-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)+112*a^2*b^4*d*e 
^3*x^7*(-d*e)^(1/2)*(b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)+30*a^3*b^3*e^4*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1237 vs. \(2 (395) = 790\).

Time = 0.38 (sec) , antiderivative size = 2500, normalized size of antiderivative = 5.72 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" 
fricas")
 

Output:

[-1/60*(15*(8*a^6*b*d^3*e^3 - a^7*d^2*e^4 + (8*a^3*b^4*d*e^5 - a^4*b^3*e^6 
)*x^10 + (16*a^3*b^4*d^2*e^4 + 22*a^4*b^3*d*e^5 - 3*a^5*b^2*e^6)*x^8 + (8* 
a^3*b^4*d^3*e^3 + 47*a^4*b^3*d^2*e^4 + 18*a^5*b^2*d*e^5 - 3*a^6*b*e^6)*x^6 
 + (24*a^4*b^3*d^3*e^3 + 45*a^5*b^2*d^2*e^4 + 2*a^6*b*d*e^5 - a^7*e^6)*x^4 
 + (24*a^5*b^2*d^3*e^3 + 13*a^6*b*d^2*e^4 - 2*a^7*d*e^5)*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*((16*b^7*d^5*e - 88*a*b^6*d^4*e^2 + 218*a^2*b^5*d^3 
*e^3 - 131*a^3*b^4*d^2*e^4 - 15*a^4*b^3*d*e^5)*x^7 + (16*b^7*d^6 - 48*a*b^ 
6*d^5*e - 2*a^2*b^5*d^4*e^2 + 354*a^3*b^4*d^3*e^3 - 275*a^4*b^3*d^2*e^4 - 
45*a^5*b^2*d*e^5)*x^5 + 5*(8*a*b^6*d^6 - 38*a^2*b^5*d^5*e + 70*a^3*b^4*d^4 
*e^2 - 4*a^4*b^3*d^3*e^3 - 27*a^5*b^2*d^2*e^4 - 9*a^6*b*d*e^5)*x^3 + 15*(2 
*a^2*b^5*d^6 - 10*a^3*b^4*d^5*e + 20*a^4*b^3*d^4*e^2 - 12*a^5*b^2*d^3*e^3 
+ a^6*b*d^2*e^4 - a^7*d*e^5)*x)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt 
(e*x^2 + d))/(a^6*b^5*d^9 - 5*a^7*b^4*d^8*e + 10*a^8*b^3*d^7*e^2 - 10*a^9* 
b^2*d^6*e^3 + 5*a^10*b*d^5*e^4 - a^11*d^4*e^5 + (a^3*b^8*d^7*e^2 - 5*a^4*b 
^7*d^6*e^3 + 10*a^5*b^6*d^5*e^4 - 10*a^6*b^5*d^4*e^5 + 5*a^7*b^4*d^3*e^6 - 
 a^8*b^3*d^2*e^7)*x^10 + (2*a^3*b^8*d^8*e - 7*a^4*b^7*d^7*e^2 + 5*a^5*b^6* 
d^6*e^3 + 10*a^6*b^5*d^5*e^4 - 20*a^7*b^4*d^4*e^5 + 13*a^8*b^3*d^3*e^6 - 3 
*a^9*b^2*d^2*e^7)*x^8 + (a^3*b^8*d^9 + a^4*b^7*d^8*e - 17*a^5*b^6*d^7*e...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**(3/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" 
maxima")
 

Output:

integrate((e*x^2 + d)^(3/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" 
giac")
 

Output:

integrate((e*x^2 + d)^(3/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}}{{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{7/2}} \,d x \] Input:

int((d + e*x^2)^(3/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int((d + e*x^2)^(3/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 8.74 (sec) , antiderivative size = 4031, normalized size of antiderivative = 9.22 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

int((e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x)
 

Output:

( - 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**8*d*e**5 - 45*sqrt(d)*sqr 
t(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sq 
rt(b)*x)/(sqrt(d)*sqrt(b)))*a**8*e**6*x**2 + 300*sqrt(d)*sqrt(a*e - b*d)*a 
tan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt 
(d)*sqrt(b)))*a**7*b*d**2*e**4 + 165*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**7*b*d*e**5*x**2 - 135*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**7*b*e 
**6*x**4 + 480*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqr 
t(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**6*b**2*d**3*e**3 
+ 1380*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*b**2*d**2*e**4*x**2 + 7 
65*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*b**2*d*e**5*x**4 - 135*sqrt 
(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqr 
t(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**6*b**2*e**6*x**6 + 1440*sqrt(d)*sqrt 
(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqr 
t(b)*x)/(sqrt(d)*sqrt(b)))*a**5*b**3*d**3*e**3*x**2 + 2340*sqrt(d)*sqrt(a* 
e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqr...