\(\int \frac {1}{x^2 (c+d x)^{5/2} (a-b x^2)^3} \, dx\) [740]

Optimal result

Integrand size = 23, antiderivative size = 483 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=-\frac {2 d}{3 c^2 \left (a-\frac {b c^2}{d^2}\right )^3 (c+d x)^{3/2}}+\frac {4 d^7 \left (4 b c^2-a d^2\right )}{c^3 \left (b c^2-a d^2\right )^4 \sqrt {c+d x}}-\frac {\sqrt {c+d x}}{a^3 c^3 x}-\frac {b^2 \sqrt {c+d x} \left (a^2 d^3-b^2 c^3 x+3 a b c d (c-d x)\right )}{4 a^2 \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )^2}+\frac {b^2 \sqrt {c+d x} \left (23 a^3 d^5+14 b^3 c^5 x+2 a^2 b c d^3 (45 c-43 d x)-a b^2 c^3 d (25 c+16 d x)\right )}{16 a^3 \left (b c^2-a d^2\right )^4 \left (a-b x^2\right )}+\frac {5 d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{7/2}}-\frac {5 b^{7/4} \left (12 b c^2-38 \sqrt {a} \sqrt {b} c d+33 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{9/2}}+\frac {5 b^{7/4} \left (12 b c^2+38 \sqrt {a} \sqrt {b} c d+33 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{7/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{9/2}} \] Output:

-2/3*d/c^2/(a-b*c^2/d^2)^3/(d*x+c)^(3/2)+4*d^7*(-a*d^2+4*b*c^2)/c^3/(-a*d^ 
2+b*c^2)^4/(d*x+c)^(1/2)-(d*x+c)^(1/2)/a^3/c^3/x-1/4*b^2*(d*x+c)^(1/2)*(a^ 
2*d^3-b^2*c^3*x+3*a*b*c*d*(-d*x+c))/a^2/(-a*d^2+b*c^2)^3/(-b*x^2+a)^2+1/16 
*b^2*(d*x+c)^(1/2)*(23*a^3*d^5+14*b^3*c^5*x+2*a^2*b*c*d^3*(-43*d*x+45*c)-a 
*b^2*c^3*d*(16*d*x+25*c))/a^3/(-a*d^2+b*c^2)^4/(-b*x^2+a)+5*d*arctanh((d*x 
+c)^(1/2)/c^(1/2))/a^3/c^(7/2)-5/32*b^(7/4)*(12*b*c^2-38*a^(1/2)*b^(1/2)*c 
*d+33*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^ 
(7/2)/(b^(1/2)*c-a^(1/2)*d)^(9/2)+5/32*b^(7/4)*(12*b*c^2+38*a^(1/2)*b^(1/2 
)*c*d+33*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2)) 
/a^(7/2)/(b^(1/2)*c+a^(1/2)*d)^(9/2)
 

Mathematica [A] (verified)

Time = 3.57 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \left (90 b^6 c^8 x^4 (c+d x)^2+16 a^6 d^8 \left (3 c^2+20 c d x+15 d^2 x^2\right )-15 a b^5 c^6 x^2 (c+d x)^2 \left (10 c^2+5 c d x+16 d^2 x^2\right )+3 a^2 b^4 c^4 (c+d x)^2 \left (16 c^4+37 c^3 d x+136 c^2 d^2 x^2+90 c d^3 x^3+10 d^4 x^4\right )-32 a^5 b d^6 \left (6 c^4+37 c^3 d x+33 c^2 d^2 x^2+20 c d^3 x^3+15 d^4 x^4\right )+a^4 b^2 d^4 \left (288 c^6+495 c^5 d x+510 c^4 d^2 x^2+2287 c^3 d^3 x^3+1968 c^2 d^4 x^4+320 c d^5 x^5+240 d^6 x^6\right )-a^3 b^3 c^2 d^2 \left (192 c^6+678 c^5 d x+1062 c^4 d^2 x^2+789 c^3 d^3 x^3+336 c^2 d^4 x^4+1115 c d^5 x^5+960 d^6 x^6\right )\right )}{c^3 \left (b c^2-a d^2\right )^4 x (c+d x)^{3/2} \left (a-b x^2\right )^2}+\frac {15 b^2 \left (12 b c^2+38 \sqrt {a} \sqrt {b} c d+33 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^4 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {15 b^2 \left (12 b c^2-38 \sqrt {a} \sqrt {b} c d+33 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^4 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}+\frac {480 \sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{7/2}}}{96 a^{7/2}} \] Input:

Integrate[1/(x^2*(c + d*x)^(5/2)*(a - b*x^2)^3),x]
 

Output:

((-2*Sqrt[a]*(90*b^6*c^8*x^4*(c + d*x)^2 + 16*a^6*d^8*(3*c^2 + 20*c*d*x + 
15*d^2*x^2) - 15*a*b^5*c^6*x^2*(c + d*x)^2*(10*c^2 + 5*c*d*x + 16*d^2*x^2) 
 + 3*a^2*b^4*c^4*(c + d*x)^2*(16*c^4 + 37*c^3*d*x + 136*c^2*d^2*x^2 + 90*c 
*d^3*x^3 + 10*d^4*x^4) - 32*a^5*b*d^6*(6*c^4 + 37*c^3*d*x + 33*c^2*d^2*x^2 
 + 20*c*d^3*x^3 + 15*d^4*x^4) + a^4*b^2*d^4*(288*c^6 + 495*c^5*d*x + 510*c 
^4*d^2*x^2 + 2287*c^3*d^3*x^3 + 1968*c^2*d^4*x^4 + 320*c*d^5*x^5 + 240*d^6 
*x^6) - a^3*b^3*c^2*d^2*(192*c^6 + 678*c^5*d*x + 1062*c^4*d^2*x^2 + 789*c^ 
3*d^3*x^3 + 336*c^2*d^4*x^4 + 1115*c*d^5*x^5 + 960*d^6*x^6)))/(c^3*(b*c^2 
- a*d^2)^4*x*(c + d*x)^(3/2)*(a - b*x^2)^2) + (15*b^2*(12*b*c^2 + 38*Sqrt[ 
a]*Sqrt[b]*c*d + 33*a*d^2)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c 
 + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/((Sqrt[b]*c + Sqrt[a]*d)^4*Sqrt[-(b*c) 
- Sqrt[a]*Sqrt[b]*d]) - (15*b^2*(12*b*c^2 - 38*Sqrt[a]*Sqrt[b]*c*d + 33*a* 
d^2)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - 
Sqrt[a]*d)])/((Sqrt[b]*c - Sqrt[a]*d)^4*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]) 
+ (480*Sqrt[a]*d*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(7/2))/(96*a^(7/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1476\) vs. \(2(483)=966\).

Time = 10.12 (sec) , antiderivative size = 1476, normalized size of antiderivative = 3.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {561, 27, 1674, 2009}

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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1674

\(\displaystyle 2 d \int \left (-\frac {d^6}{c^2 \left (b c^2-a d^2\right )^3 (c+d x)^2}+\frac {b^2 \left (3 b c^2-2 b (c+d x) c+a d^2\right ) d^4}{a \left (a d^2-b c^2\right )^2 \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )^3}+\frac {b^2 \left (3 b^2 c^4-9 a b d^2 c^2-2 b \left (b c^2-3 a d^2\right ) (c+d x) c-2 a^2 d^4\right ) d^2}{a^2 \left (b c^2-a d^2\right )^3 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )^2}-\frac {2 \left (4 b c^2 d^6-a d^8\right )}{c^3 \left (b c^2-a d^2\right )^4 (c+d x)}+\frac {b^2 \left (2 b c \left (b^2 c^4-4 a b d^2 c^2+6 a^2 d^4\right ) (c+d x)-3 \left (b^3 c^6-4 a b^2 d^2 c^4+6 a^2 b d^4 c^2+a^3 d^6\right )\right )}{a^3 \left (b c^2-a d^2\right )^4 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}-\frac {2}{a^3 c^3 x d}+\frac {1}{a^3 c^2 x^2 d^2}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 d \left (\frac {2 \left (4 b c^2-a d^2\right ) d^6}{c^3 \left (b c^2-a d^2\right )^4 \sqrt {c+d x}}+\frac {d^6}{3 c^2 \left (b c^2-a d^2\right )^3 (c+d x)^{3/2}}-\frac {b^2 \sqrt {c+d x} \left (b^2 c^4+6 a b d^2 c^2-b \left (b c^2+3 a d^2\right ) (c+d x) c+a^2 d^4\right ) d^2}{8 a^2 \left (b c^2-a d^2\right )^3 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )^2}+\frac {5 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 a^3 c^{7/2}}+\frac {b^2 \sqrt {c+d x} \left (6 b^3 c^6-7 a b^2 d^2 c^4-40 a^2 b d^4 c^2-2 b \left (3 b^2 c^4-4 a b d^2 c^2-11 a^2 d^4\right ) (c+d x) c-7 a^3 d^6\right )}{32 a^3 \left (b c^2-a d^2\right )^4 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}+\frac {b^2 \sqrt {c+d x} \left (b^3 c^6+2 a b^2 d^2 c^4-17 a^2 b d^4 c^2-b \left (b^2 c^4-a b d^2 c^2-8 a^2 d^4\right ) (c+d x) c-2 a^3 d^6\right )}{4 a^3 \left (b c^2-a d^2\right )^4 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}-\frac {b^{7/4} \left (12 b^2 c^4-30 \sqrt {a} b^{3/2} d c^3+29 a b d^2 c^2-20 a^{3/2} \sqrt {b} d^3 c+21 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{64 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2} \left (b c^2-a d^2\right )^2 d}-\frac {b^{7/4} \left (b^{5/2} c^5+\sqrt {a} b^2 d c^4-5 a b^{3/2} d^2 c^3-3 a^{3/2} b d^3 c^2+9 a^2 \sqrt {b} d^4 c+3 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{9/2} \left (\sqrt {b} c+\sqrt {a} d\right )^3 d}-\frac {b^{7/4} \left (2 b^{5/2} c^5-3 \sqrt {a} b^2 d c^4-8 a b^{3/2} d^2 c^3+9 a^{3/2} b d^3 c^2+2 a^2 \sqrt {b} d^4 c+6 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{8 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \left (b c^2-a d^2\right )^3 d}+\frac {b^{7/4} \left (12 b^2 c^4+30 \sqrt {a} b^{3/2} d c^3+29 a b d^2 c^2+20 a^{3/2} \sqrt {b} d^3 c+21 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{64 a^{7/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2} \left (b c^2-a d^2\right )^2 d}+\frac {b^{7/4} \left (2 b^{5/2} c^5+3 \sqrt {a} b^2 d c^4-8 a b^{3/2} d^2 c^3-9 a^{3/2} b d^3 c^2+2 a^2 \sqrt {b} d^4 c-6 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{8 a^{7/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} \left (b c^2-a d^2\right )^3 d}+\frac {b^{7/4} \left (b^{5/2} c^5-\sqrt {a} b^2 d c^4-5 a b^{3/2} d^2 c^3+3 a^{3/2} b d^3 c^2+9 a^2 \sqrt {b} d^4 c-3 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{2 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^3 \left (\sqrt {b} c+\sqrt {a} d\right )^{9/2} d}-\frac {\sqrt {c+d x}}{2 a^3 c^3 x d}\right )\)

Input:

Int[1/(x^2*(c + d*x)^(5/2)*(a - b*x^2)^3),x]
 

Output:

2*d*(d^6/(3*c^2*(b*c^2 - a*d^2)^3*(c + d*x)^(3/2)) + (2*d^6*(4*b*c^2 - a*d 
^2))/(c^3*(b*c^2 - a*d^2)^4*Sqrt[c + d*x]) - Sqrt[c + d*x]/(2*a^3*c^3*d*x) 
 - (b^2*d^2*Sqrt[c + d*x]*(b^2*c^4 + 6*a*b*c^2*d^2 + a^2*d^4 - b*c*(b*c^2 
+ 3*a*d^2)*(c + d*x)))/(8*a^2*(b*c^2 - a*d^2)^3*(b*c^2 - a*d^2 - 2*b*c*(c 
+ d*x) + b*(c + d*x)^2)^2) + (b^2*Sqrt[c + d*x]*(6*b^3*c^6 - 7*a*b^2*c^4*d 
^2 - 40*a^2*b*c^2*d^4 - 7*a^3*d^6 - 2*b*c*(3*b^2*c^4 - 4*a*b*c^2*d^2 - 11* 
a^2*d^4)*(c + d*x)))/(32*a^3*(b*c^2 - a*d^2)^4*(b*c^2 - a*d^2 - 2*b*c*(c + 
 d*x) + b*(c + d*x)^2)) + (b^2*Sqrt[c + d*x]*(b^3*c^6 + 2*a*b^2*c^4*d^2 - 
17*a^2*b*c^2*d^4 - 2*a^3*d^6 - b*c*(b^2*c^4 - a*b*c^2*d^2 - 8*a^2*d^4)*(c 
+ d*x)))/(4*a^3*(b*c^2 - a*d^2)^4*(b*c^2 - a*d^2 - 2*b*c*(c + d*x) + b*(c 
+ d*x)^2)) + (5*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*a^3*c^(7/2)) - (b^(7/4) 
*(12*b^2*c^4 - 30*Sqrt[a]*b^(3/2)*c^3*d + 29*a*b*c^2*d^2 - 20*a^(3/2)*Sqrt 
[b]*c*d^3 + 21*a^2*d^4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - S 
qrt[a]*d]])/(64*a^(7/2)*d*(Sqrt[b]*c - Sqrt[a]*d)^(5/2)*(b*c^2 - a*d^2)^2) 
 - (b^(7/4)*(b^(5/2)*c^5 + Sqrt[a]*b^2*c^4*d - 5*a*b^(3/2)*c^3*d^2 - 3*a^( 
3/2)*b*c^2*d^3 + 9*a^2*Sqrt[b]*c*d^4 + 3*a^(5/2)*d^5)*ArcTanh[(b^(1/4)*Sqr 
t[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^(7/2)*d*(Sqrt[b]*c - Sqrt[a 
]*d)^(9/2)*(Sqrt[b]*c + Sqrt[a]*d)^3) - (b^(7/4)*(2*b^(5/2)*c^5 - 3*Sqrt[a 
]*b^2*c^4*d - 8*a*b^(3/2)*c^3*d^2 + 9*a^(3/2)*b*c^2*d^3 + 2*a^2*Sqrt[b]*c* 
d^4 + 6*a^(5/2)*d^5)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - S...
 

Defintions of rubi rules used

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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1674

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && N 
eQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 2.67 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.40

method	result	size
default	\(2 d^{7} \left (\frac {b^{2} \left (\frac {\left (\frac {43}{16} a^{2} c \,d^{4} b^{2}+\frac {1}{2} a \,c^{3} d^{2} b^{3}-\frac {7}{16} b^{4} c^{5}\right ) \left (d x +c \right )^{\frac {7}{2}}-\frac {b \left (23 a^{3} d^{6}+348 a^{2} b \,c^{2} d^{4}+23 a \,b^{2} c^{4} d^{2}-42 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32}-\frac {b c \left (26 a^{3} d^{6}-215 a^{2} b \,c^{2} d^{4}-8 a \,b^{2} c^{4} d^{2}+21 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16}+\left (\frac {27}{32} a^{4} d^{8}+\frac {173}{32} a^{3} b \,c^{2} d^{6}-\frac {205}{32} a^{2} b^{2} c^{4} d^{4}-\frac {9}{32} a \,b^{3} c^{6} d^{2}+\frac {7}{16} b^{4} c^{8}\right ) \sqrt {d x +c}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {5 b \left (\frac {\left (33 a^{3} d^{6}+58 a^{2} b \,c^{2} d^{4}-47 a \,b^{2} c^{4} d^{2}+12 b^{3} c^{6}+94 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}-48 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+10 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-33 a^{3} d^{6}-58 a^{2} b \,c^{2} d^{4}+47 a \,b^{2} c^{4} d^{2}-12 b^{3} c^{6}+94 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}-48 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+10 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}\right )}{a^{3} d^{6} \left (a \,d^{2}-b \,c^{2}\right )^{4}}-\frac {1}{3 c^{2} \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 a \,d^{2}-8 b \,c^{2}}{c^{3} \left (a \,d^{2}-b \,c^{2}\right )^{4} \sqrt {d x +c}}+\frac {-\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{6} c^{3}}\right )\)	\(676\)
risch	\(-\frac {\sqrt {d x +c}}{a^{3} c^{3} x}-\frac {d \left (-\frac {2 c^{3} b^{2} \left (\frac {\left (\frac {43}{16} a^{2} c \,d^{4} b^{2}+\frac {1}{2} a \,c^{3} d^{2} b^{3}-\frac {7}{16} b^{4} c^{5}\right ) \left (d x +c \right )^{\frac {7}{2}}-\frac {b \left (23 a^{3} d^{6}+348 a^{2} b \,c^{2} d^{4}+23 a \,b^{2} c^{4} d^{2}-42 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32}-\frac {b c \left (26 a^{3} d^{6}-215 a^{2} b \,c^{2} d^{4}-8 a \,b^{2} c^{4} d^{2}+21 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16}+\left (\frac {27}{32} a^{4} d^{8}+\frac {173}{32} a^{3} b \,c^{2} d^{6}-\frac {205}{32} a^{2} b^{2} c^{4} d^{4}-\frac {9}{32} a \,b^{3} c^{6} d^{2}+\frac {7}{16} b^{4} c^{8}\right ) \sqrt {d x +c}}{\left (b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {5 b \left (\frac {\left (33 a^{3} d^{6}+58 a^{2} b \,c^{2} d^{4}-47 a \,b^{2} c^{4} d^{2}+12 b^{3} c^{6}+94 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}-48 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+10 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-33 a^{3} d^{6}-58 a^{2} b \,c^{2} d^{4}+47 a \,b^{2} c^{4} d^{2}-12 b^{3} c^{6}+94 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}-48 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+10 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{4}}+\frac {2 a^{3} c \,d^{6}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 a^{3} d^{6} \left (a \,d^{2}-4 b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{4} \sqrt {d x +c}}-\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{a^{3} c^{3}}\)	\(676\)
derivativedivides	\(-2 d^{7} \left (-\frac {b^{2} \left (\frac {\left (\frac {43}{16} a^{2} c \,d^{4} b^{2}+\frac {1}{2} a \,c^{3} d^{2} b^{3}-\frac {7}{16} b^{4} c^{5}\right ) \left (d x +c \right )^{\frac {7}{2}}-\frac {b \left (23 a^{3} d^{6}+348 a^{2} b \,c^{2} d^{4}+23 a \,b^{2} c^{4} d^{2}-42 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32}-\frac {b c \left (26 a^{3} d^{6}-215 a^{2} b \,c^{2} d^{4}-8 a \,b^{2} c^{4} d^{2}+21 b^{3} c^{6}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16}+\left (\frac {27}{32} a^{4} d^{8}+\frac {173}{32} a^{3} b \,c^{2} d^{6}-\frac {205}{32} a^{2} b^{2} c^{4} d^{4}-\frac {9}{32} a \,b^{3} c^{6} d^{2}+\frac {7}{16} b^{4} c^{8}\right ) \sqrt {d x +c}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {5 b \left (\frac {\left (33 a^{3} d^{6}+58 a^{2} b \,c^{2} d^{4}-47 a \,b^{2} c^{4} d^{2}+12 b^{3} c^{6}+94 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}-48 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+10 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-33 a^{3} d^{6}-58 a^{2} b \,c^{2} d^{4}+47 a \,b^{2} c^{4} d^{2}-12 b^{3} c^{6}+94 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}-48 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+10 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32}\right )}{a^{3} d^{6} \left (a \,d^{2}-b \,c^{2}\right )^{4}}-\frac {-2 a \,d^{2}+8 b \,c^{2}}{c^{3} \left (a \,d^{2}-b \,c^{2}\right )^{4} \sqrt {d x +c}}+\frac {1}{3 c^{2} \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (d x +c \right )^{\frac {3}{2}}}-\frac {-\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{6} c^{3}}\right )\)	\(678\)
pseudoelliptic	\(-\frac {-\frac {15 d \left (d x +c \right )^{\frac {3}{2}} x \,b^{3} \left (-b \,x^{2}+a \right )^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (\frac {5 c^{\frac {17}{2}} b^{2}}{6}-4 a \,c^{\frac {13}{2}} b \,d^{2}+\frac {47 a^{2} c^{\frac {9}{2}} d^{4}}{6}\right ) \sqrt {a b \,d^{2}}+\frac {29 a^{2} c^{\frac {11}{2}} b \,d^{4}}{6}+\frac {11 a^{3} c^{\frac {7}{2}} d^{6}}{4}+\left (b \,c^{2}-\frac {47 a \,d^{2}}{12}\right ) b^{2} c^{\frac {15}{2}}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{8}+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-\frac {15 d x \sqrt {d x +c}\, b^{3} \left (\frac {\left (-47 a^{2} d^{5} x \,c^{\frac {9}{2}}-47 a^{2} c^{\frac {11}{2}} d^{4}-5 \left (d x +c \right ) b \left (b \,c^{2}-\frac {24 a \,d^{2}}{5}\right ) c^{\frac {13}{2}}\right ) \sqrt {a b \,d^{2}}}{6}+\frac {11 c^{\frac {9}{2}} a^{3} d^{6}}{4}+\frac {29 a^{2} c^{\frac {11}{2}} b \,d^{5} x}{6}+\frac {11 a^{3} d^{7} x \,c^{\frac {7}{2}}}{4}+\left (c^{3} b^{2} \left (d x +c \right )-\frac {47 b c \,d^{2} \left (d x +c \right ) a}{12}+\frac {29 a^{2} d^{4}}{6}\right ) b \,c^{\frac {13}{2}}\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{8}+\left (-5 d x \left (a \,d^{2}-b \,c^{2}\right )^{4} \left (d x +c \right )^{\frac {3}{2}} \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )-4 d^{6} b \,a^{2} \left (-\frac {5}{32} b^{3} x^{6}+\frac {7}{4} a \,b^{2} x^{4}-\frac {85}{32} a^{2} b \,x^{2}+a^{3}\right ) c^{\frac {9}{2}}+\frac {165 d^{5} x \left (\frac {2}{3} b^{2} x^{4}-\frac {263}{165} a b \,x^{2}+a^{2}\right ) b^{2} a^{2} c^{\frac {11}{2}}}{16}-\frac {74 d^{7} x \left (\frac {1115}{1184} b^{2} x^{4}-\frac {2287}{1184} a b \,x^{2}+a^{2}\right ) b \,a^{3} c^{\frac {7}{2}}}{3}+a^{3} d^{8} \left (-20 b \,x^{2}+a \right ) \left (-b \,x^{2}+a \right )^{2} c^{\frac {5}{2}}+\frac {20 a^{4} x \left (-b \,x^{2}+a \right )^{2} d^{9} c^{\frac {3}{2}}}{3}+5 a^{4} d^{10} x^{2} \left (-b \,x^{2}+a \right )^{2} \sqrt {c}+b^{2} \left (\frac {15 c^{2} x^{4} \left (d x +c \right )^{2} b^{4}}{8}-\frac {25 \left (d x +c \right )^{2} x^{2} a \left (\frac {8}{5} d^{2} x^{2}+\frac {1}{2} c d x +c^{2}\right ) b^{3}}{8}+a^{2} \left (\frac {399}{16} c \,d^{3} x^{3}+\frac {69}{16} c^{3} d x +\frac {113}{8} d^{2} c^{2} x^{2}+\frac {163}{8} d^{4} x^{4}+c^{4}\right ) b^{2}-4 d^{2} a^{3} \left (\frac {177}{32} d^{2} x^{2}+\frac {113}{32} c d x +c^{2}\right ) b +6 a^{4} d^{4}\right ) c^{\frac {13}{2}}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\right )}{c^{\frac {7}{2}} \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d x +c \right )^{\frac {3}{2}} \left (-b \,x^{2}+a \right )^{2} \left (a \,d^{2}-b \,c^{2}\right )^{4} x \,a^{3}}\)	\(830\)

Input:

int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2*d^7*(b^2/a^3/d^6/(a*d^2-b*c^2)^4*(((43/16*a^2*c*d^4*b^2+1/2*a*c^3*d^2*b^ 
3-7/16*b^4*c^5)*(d*x+c)^(7/2)-1/32*b*(23*a^3*d^6+348*a^2*b*c^2*d^4+23*a*b^ 
2*c^4*d^2-42*b^3*c^6)*(d*x+c)^(5/2)-1/16*b*c*(26*a^3*d^6-215*a^2*b*c^2*d^4 
-8*a*b^2*c^4*d^2+21*b^3*c^6)*(d*x+c)^(3/2)+(27/32*a^4*d^8+173/32*a^3*b*c^2 
*d^6-205/32*a^2*b^2*c^4*d^4-9/32*a*b^3*c^6*d^2+7/16*b^4*c^8)*(d*x+c)^(1/2) 
)/(-b*(d*x+c)^2+2*b*c*(d*x+c)+a*d^2-b*c^2)^2+5/32*b*(1/2*(33*a^3*d^6+58*a^ 
2*b*c^2*d^4-47*a*b^2*c^4*d^2+12*b^3*c^6+94*(a*b*d^2)^(1/2)*a^2*c*d^4-48*(a 
*b*d^2)^(1/2)*a*b*c^3*d^2+10*(a*b*d^2)^(1/2)*b^2*c^5)/(a*b*d^2)^(1/2)/((-b 
*c+(a*b*d^2)^(1/2))*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2) 
)*b)^(1/2))-1/2*(-33*a^3*d^6-58*a^2*b*c^2*d^4+47*a*b^2*c^4*d^2-12*b^3*c^6+ 
94*(a*b*d^2)^(1/2)*a^2*c*d^4-48*(a*b*d^2)^(1/2)*a*b*c^3*d^2+10*(a*b*d^2)^( 
1/2)*b^2*c^5)/(a*b*d^2)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*arctanh(b*(d 
*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))))-1/3/c^2/(a*d^2-b*c^2)^3/(d* 
x+c)^(3/2)-(2*a*d^2-8*b*c^2)/c^3/(a*d^2-b*c^2)^4/(d*x+c)^(1/2)+1/a^3/d^6/c 
^3*(-1/2*(d*x+c)^(1/2)/d/x+5/2/c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x**2/(d*x+c)**(5/2)/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )}^{3} {\left (d x + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:

integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-integrate(1/((b*x^2 - a)^3*(d*x + c)^(5/2)*x^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2747 vs. \(2 (412) = 824\).

Time = 0.62 (sec) , antiderivative size = 2747, normalized size of antiderivative = 5.69 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

5/32*(2*(a^3*b^4*c^8*d - 4*a^4*b^3*c^6*d^3 + 6*a^5*b^2*c^4*d^5 - 4*a^6*b*c 
^2*d^7 + a^7*d^9)^2*(5*b^3*c^5*d - 24*a*b^2*c^3*d^3 + 47*a^2*b*c*d^5)*sqrt 
(-b^2*c - sqrt(a*b)*b*d)*abs(b) - (22*sqrt(a*b)*a^2*b^7*c^14*d - 183*sqrt( 
a*b)*a^3*b^6*c^12*d^3 + 664*sqrt(a*b)*a^4*b^5*c^10*d^5 - 1233*sqrt(a*b)*a^ 
5*b^4*c^8*d^7 + 1182*sqrt(a*b)*a^6*b^3*c^6*d^9 - 505*sqrt(a*b)*a^7*b^2*c^4 
*d^11 + 20*sqrt(a*b)*a^8*b*c^2*d^13 + 33*sqrt(a*b)*a^9*d^15)*sqrt(-b^2*c - 
 sqrt(a*b)*b*d)*abs(a^3*b^4*c^8*d - 4*a^4*b^3*c^6*d^3 + 6*a^5*b^2*c^4*d^5 
- 4*a^6*b*c^2*d^7 + a^7*d^9)*abs(b) + (12*a^5*b^12*c^23*d - 143*a^6*b^11*c 
^21*d^3 + 770*a^7*b^10*c^19*d^5 - 2419*a^8*b^9*c^17*d^7 + 4832*a^9*b^8*c^1 
5*d^9 - 6286*a^10*b^7*c^13*d^11 + 5180*a^11*b^6*c^11*d^13 - 2350*a^12*b^5* 
c^9*d^15 + 164*a^13*b^4*c^7*d^17 + 413*a^14*b^3*c^5*d^19 - 206*a^15*b^2*c^ 
3*d^21 + 33*a^16*b*c*d^23)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b))*arctan(sqr 
t(d*x + c)/sqrt(-(a^3*b^5*c^9 - 4*a^4*b^4*c^7*d^2 + 6*a^5*b^3*c^5*d^4 - 4* 
a^6*b^2*c^3*d^6 + a^7*b*c*d^8 + sqrt((a^3*b^5*c^9 - 4*a^4*b^4*c^7*d^2 + 6* 
a^5*b^3*c^5*d^4 - 4*a^6*b^2*c^3*d^6 + a^7*b*c*d^8)^2 - (a^3*b^5*c^10 - 5*a 
^4*b^4*c^8*d^2 + 10*a^5*b^3*c^6*d^4 - 10*a^6*b^2*c^4*d^6 + 5*a^7*b*c^2*d^8 
 - a^8*d^10)*(a^3*b^5*c^8 - 4*a^4*b^4*c^6*d^2 + 6*a^5*b^3*c^4*d^4 - 4*a^6* 
b^2*c^2*d^6 + a^7*b*d^8)))/(a^3*b^5*c^8 - 4*a^4*b^4*c^6*d^2 + 6*a^5*b^3*c^ 
4*d^4 - 4*a^6*b^2*c^2*d^6 + a^7*b*d^8)))/((sqrt(a*b)*a^5*b^9*c^18 - 9*sqrt 
(a*b)*a^6*b^8*c^16*d^2 + 36*sqrt(a*b)*a^7*b^7*c^14*d^4 - 84*sqrt(a*b)*a...
 

Mupad [B] (verification not implemented)

Time = 21.59 (sec) , antiderivative size = 50085, normalized size of antiderivative = 103.70 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/(x^2*(a - b*x^2)^3*(c + d*x)^(5/2)),x)
 

Output:

atan((((c + d*x)^(1/2)*(27179089920000*a^17*b^43*c^91*d^8 - 91049951232000 
0*a^18*b^42*c^89*d^10 + 14737672765440000*a^19*b^41*c^87*d^12 - 1534280912 
07680000*a^20*b^40*c^85*d^14 + 1155164338913280000*a^21*b^39*c^83*d^16 - 6 
717977222184960000*a^22*b^38*c^81*d^18 + 31562375555973120000*a^23*b^37*c^ 
79*d^20 - 124185892633968640000*a^24*b^36*c^77*d^22 + 42283823810347008000 
0*a^25*b^35*c^75*d^24 - 1284815761157652480000*a^26*b^34*c^73*d^26 + 35745 
33764850647040000*a^27*b^33*c^71*d^28 - 9242487355657420800000*a^28*b^32*c 
^69*d^30 + 22235345040054681600000*a^29*b^31*c^67*d^32 - 49304060237787955 
200000*a^30*b^30*c^65*d^34 + 99436064518058803200000*a^31*b^29*c^63*d^36 - 
 180257214868881408000000*a^32*b^28*c^61*d^38 + 291243876105053798400000*a 
^33*b^27*c^59*d^40 - 417157087469292748800000*a^34*b^26*c^57*d^42 + 527922 
989534910873600000*a^35*b^25*c^55*d^44 - 588938344479955353600000*a^36*b^2 
4*c^53*d^46 + 577987691546876313600000*a^37*b^23*c^51*d^48 - 4978798911788 
67916800000*a^38*b^22*c^49*d^50 + 375324478809086361600000*a^39*b^21*c^47* 
d^52 - 246629041826247475200000*a^40*b^20*c^45*d^54 + 14051226245842206720 
0000*a^41*b^19*c^43*d^56 - 68911047564979077120000*a^42*b^18*c^41*d^58 + 2 
8809532178741329920000*a^43*b^17*c^39*d^60 - 10130734188739952640000*a^44* 
b^16*c^37*d^62 + 2940053847984046080000*a^45*b^15*c^35*d^64 - 684403452338 
503680000*a^46*b^14*c^33*d^66 + 121918373361090560000*a^47*b^13*c^31*d^68 
- 15135521526251520000*a^48*b^12*c^29*d^70 + 984221452861440000*a^49*b^...
 

Reduce [F]

\[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^3} \, dx=\int \frac {1}{x^{2} \left (d x +c \right )^{\frac {5}{2}} \left (-b \,x^{2}+a \right )^{3}}d x \] Input:

int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^3,x)
 

Output:

int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^3,x)