\(\int \frac {A+B x}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [376]

Optimal result

Integrand size = 33, antiderivative size = 376 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2 e^4 (B d-A e)}{3 (b d-a e)^6 (d+e x)^{3/2}}+\frac {2 e^4 (5 b B d-6 A b e+a B e)}{(b d-a e)^7 \sqrt {d+e x}}-\frac {b (A b-a B) \sqrt {d+e x}}{5 (b d-a e)^3 (a+b x)^5}-\frac {b (10 b B d-29 A b e+19 a B e) \sqrt {d+e x}}{40 (b d-a e)^4 (a+b x)^4}+\frac {b e (230 b B d-443 A b e+213 a B e) \sqrt {d+e x}}{240 (b d-a e)^5 (a+b x)^3}-\frac {b e^2 (518 b B d-827 A b e+309 a B e) \sqrt {d+e x}}{192 (b d-a e)^6 (a+b x)^2}+\frac {b e^3 (1030 b B d-1467 A b e+437 a B e) \sqrt {d+e x}}{128 (b d-a e)^7 (a+b x)}-\frac {231 \sqrt {b} e^4 (10 b B d-13 A b e+3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}} \] Output:

2/3*e^4*(-A*e+B*d)/(-a*e+b*d)^6/(e*x+d)^(3/2)+2*e^4*(-6*A*b*e+B*a*e+5*B*b* 
d)/(-a*e+b*d)^7/(e*x+d)^(1/2)-1/5*b*(A*b-B*a)*(e*x+d)^(1/2)/(-a*e+b*d)^3/( 
b*x+a)^5-1/40*b*(-29*A*b*e+19*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/(-a*e+b*d)^4/( 
b*x+a)^4+1/240*b*e*(-443*A*b*e+213*B*a*e+230*B*b*d)*(e*x+d)^(1/2)/(-a*e+b* 
d)^5/(b*x+a)^3-1/192*b*e^2*(-827*A*b*e+309*B*a*e+518*B*b*d)*(e*x+d)^(1/2)/ 
(-a*e+b*d)^6/(b*x+a)^2+1/128*b*e^3*(-1467*A*b*e+437*B*a*e+1030*B*b*d)*(e*x 
+d)^(1/2)/(-a*e+b*d)^7/(b*x+a)-231/128*b^(1/2)*e^4*(-13*A*b*e+3*B*a*e+10*B 
*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(15/2)
 

Mathematica [A] (verified)

Time = 8.15 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.88 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {-B \left (1280 a^6 e^5 (2 d+3 e x)+5 a^5 b e^4 \left (6625 d^2+11458 d e x+6369 e^2 x^2\right )+10 a^4 b^2 e^3 \left (1218 d^3+16657 d^2 e x+21340 d e^2 x^2+7821 e^3 x^3\right )+2 a^3 b^3 e^2 \left (-1778 d^4+27783 d^3 e x+188452 d^2 e^2 x^2+190443 d e^3 x^3+44352 e^4 x^4\right )+10 b^6 d x \left (-48 d^5+88 d^4 e x-198 d^3 e^2 x^2+693 d^2 e^3 x^3+4620 d e^4 x^4+3465 e^5 x^5\right )+2 a^2 b^4 e \left (416 d^5-8266 d^4 e x+30921 d^3 e^2 x^2+205359 d^2 e^3 x^3+180411 d e^4 x^4+24255 e^5 x^5\right )+a b^5 \left (-96 d^6+4016 d^5 e x-9196 d^4 e^2 x^2+33066 d^3 e^3 x^3+219219 d^2 e^4 x^4+175560 d e^5 x^5+10395 e^6 x^6\right )\right )+A \left (-1280 a^6 e^6+1280 a^5 b e^5 (19 d+13 e x)+5 a^4 b^2 e^4 \left (7119 d^2+38558 d e x+27599 e^2 x^2\right )+10 a^3 b^3 e^3 \left (-2107 d^3+7917 d^2 e x+46475 d e^2 x^2+33891 e^3 x^3\right )+2 a^2 b^4 e^2 \left (5012 d^4-11557 d^3 e x+42042 d^2 e^2 x^2+260403 d e^3 x^3+192192 e^4 x^4\right )+2 a b^5 e \left (-1464 d^5+2704 d^4 e x-6149 d^3 e^2 x^2+21879 d^2 e^3 x^3+141141 d e^4 x^4+105105 e^5 x^5\right )+b^6 \left (384 d^6-624 d^5 e x+1144 d^4 e^2 x^2-2574 d^3 e^3 x^3+9009 d^2 e^4 x^4+60060 d e^5 x^5+45045 e^6 x^6\right )\right )}{(-b d+a e)^7 (a+b x)^5 (d+e x)^{3/2}}-\frac {3465 \sqrt {b} e^4 (10 b B d-13 A b e+3 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{15/2}}}{1920} \] Input:

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

Output:

((-(B*(1280*a^6*e^5*(2*d + 3*e*x) + 5*a^5*b*e^4*(6625*d^2 + 11458*d*e*x + 
6369*e^2*x^2) + 10*a^4*b^2*e^3*(1218*d^3 + 16657*d^2*e*x + 21340*d*e^2*x^2 
 + 7821*e^3*x^3) + 2*a^3*b^3*e^2*(-1778*d^4 + 27783*d^3*e*x + 188452*d^2*e 
^2*x^2 + 190443*d*e^3*x^3 + 44352*e^4*x^4) + 10*b^6*d*x*(-48*d^5 + 88*d^4* 
e*x - 198*d^3*e^2*x^2 + 693*d^2*e^3*x^3 + 4620*d*e^4*x^4 + 3465*e^5*x^5) + 
 2*a^2*b^4*e*(416*d^5 - 8266*d^4*e*x + 30921*d^3*e^2*x^2 + 205359*d^2*e^3* 
x^3 + 180411*d*e^4*x^4 + 24255*e^5*x^5) + a*b^5*(-96*d^6 + 4016*d^5*e*x - 
9196*d^4*e^2*x^2 + 33066*d^3*e^3*x^3 + 219219*d^2*e^4*x^4 + 175560*d*e^5*x 
^5 + 10395*e^6*x^6))) + A*(-1280*a^6*e^6 + 1280*a^5*b*e^5*(19*d + 13*e*x) 
+ 5*a^4*b^2*e^4*(7119*d^2 + 38558*d*e*x + 27599*e^2*x^2) + 10*a^3*b^3*e^3* 
(-2107*d^3 + 7917*d^2*e*x + 46475*d*e^2*x^2 + 33891*e^3*x^3) + 2*a^2*b^4*e 
^2*(5012*d^4 - 11557*d^3*e*x + 42042*d^2*e^2*x^2 + 260403*d*e^3*x^3 + 1921 
92*e^4*x^4) + 2*a*b^5*e*(-1464*d^5 + 2704*d^4*e*x - 6149*d^3*e^2*x^2 + 218 
79*d^2*e^3*x^3 + 141141*d*e^4*x^4 + 105105*e^5*x^5) + b^6*(384*d^6 - 624*d 
^5*e*x + 1144*d^4*e^2*x^2 - 2574*d^3*e^3*x^3 + 9009*d^2*e^4*x^4 + 60060*d* 
e^5*x^5 + 45045*e^6*x^6)))/((-(b*d) + a*e)^7*(a + b*x)^5*(d + e*x)^(3/2)) 
- (3465*Sqrt[b]*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d 
 + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(15/2))/1920
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1184, 27, 87, 52, 52, 52, 52, 61, 61, 73, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {A+B x}{b^6 (a+b x)^6 (d+e x)^{5/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {A+B x}{(a+b x)^6 (d+e x)^{5/2}}dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \int \frac {1}{(a+b x)^5 (d+e x)^{5/2}}dx}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \left (-\frac {11 e \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}}dx}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \left (-\frac {11 e \left (-\frac {3 e \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(3 a B e-13 A b e+10 b B d) \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 b (b d-a e)}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

Input:

Int[(A + B*x)/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

-1/5*(A*b - a*B)/(b*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(3/2)) + ((10*b*B*d 
- 13*A*b*e + 3*a*B*e)*(-1/4*1/((b*d - a*e)*(a + b*x)^4*(d + e*x)^(3/2)) - 
(11*e*(-1/3*1/((b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)) - (3*e*(-1/2*1/((b 
*d - a*e)*(a + b*x)^2*(d + e*x)^(3/2)) - (7*e*(-(1/((b*d - a*e)*(a + b*x)* 
(d + e*x)^(3/2))) - (5*e*(2/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (b*(2/((b*d 
- a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b* 
d - a*e]])/(b*d - a*e)^(3/2)))/(b*d - a*e)))/(2*(b*d - a*e))))/(4*(b*d - a 
*e))))/(2*(b*d - a*e))))/(8*(b*d - a*e))))/(10*b*(b*d - a*e))
 

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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 1.63 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {b \left (\frac {\left (\frac {1467}{256} A \,b^{5} e -\frac {437}{256} B a \,b^{4} e -\frac {515}{128} B \,b^{5} d \right ) \left (e x +d \right )^{\frac {9}{2}}+\frac {b^{3} \left (9629 A a b \,e^{2}-9629 A \,b^{2} d e -2931 B \,e^{2} a^{2}-3767 B a b d e +6698 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3} e^{3}-\frac {1253}{15} A a \,b^{4} d \,e^{2}+\frac {1253}{30} A \,b^{5} d^{2} e -\frac {131}{10} B \,a^{3} b^{2} e^{3}-\frac {37}{15} B \,a^{2} b^{3} d \,e^{2}+\frac {1327}{30} B a \,b^{4} d^{2} e -\frac {86}{3} B \,b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {12131}{384} A \,a^{3} b^{2} e^{4}-\frac {12131}{128} A \,a^{2} b^{3} d \,e^{3}+\frac {12131}{128} A a \,b^{4} d^{2} e^{2}-\frac {12131}{384} A \,b^{5} d^{3} e -\frac {1327}{128} B \,e^{4} a^{4} b +\frac {3793}{384} B \,a^{3} b^{2} d \,e^{3}+\frac {4169}{128} B \,a^{2} b^{3} d^{2} e^{2}-\frac {6823}{128} B a \,b^{4} d^{3} e +\frac {4075}{192} B \,b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b \,e^{5}-\frac {2373}{64} A \,a^{3} b^{2} d \,e^{4}+\frac {7119}{128} A \,a^{2} b^{3} d^{2} e^{3}-\frac {2373}{64} A a \,b^{4} d^{3} e^{2}+\frac {2373}{256} A \,b^{5} d^{4} e -\frac {843}{256} B \,a^{5} e^{5}+\frac {921}{128} B \,a^{4} b d \,e^{4}+\frac {531}{128} B \,a^{3} b^{2} d^{2} e^{3}-\frac {363}{16} B \,a^{2} b^{3} d^{3} e^{2}+\frac {5277}{256} B a \,b^{4} d^{4} e -\frac {765}{128} B \,b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {231 \left (13 A b e -3 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{7}}-\frac {A e -B d}{3 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {3}{2}}}-\frac {-6 A b e +B a e +5 B b d}{\left (a e -b d \right )^{7} \sqrt {e x +d}}\right )\)	\(579\)
default	\(2 e^{4} \left (\frac {b \left (\frac {\left (\frac {1467}{256} A \,b^{5} e -\frac {437}{256} B a \,b^{4} e -\frac {515}{128} B \,b^{5} d \right ) \left (e x +d \right )^{\frac {9}{2}}+\frac {b^{3} \left (9629 A a b \,e^{2}-9629 A \,b^{2} d e -2931 B \,e^{2} a^{2}-3767 B a b d e +6698 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3} e^{3}-\frac {1253}{15} A a \,b^{4} d \,e^{2}+\frac {1253}{30} A \,b^{5} d^{2} e -\frac {131}{10} B \,a^{3} b^{2} e^{3}-\frac {37}{15} B \,a^{2} b^{3} d \,e^{2}+\frac {1327}{30} B a \,b^{4} d^{2} e -\frac {86}{3} B \,b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {12131}{384} A \,a^{3} b^{2} e^{4}-\frac {12131}{128} A \,a^{2} b^{3} d \,e^{3}+\frac {12131}{128} A a \,b^{4} d^{2} e^{2}-\frac {12131}{384} A \,b^{5} d^{3} e -\frac {1327}{128} B \,e^{4} a^{4} b +\frac {3793}{384} B \,a^{3} b^{2} d \,e^{3}+\frac {4169}{128} B \,a^{2} b^{3} d^{2} e^{2}-\frac {6823}{128} B a \,b^{4} d^{3} e +\frac {4075}{192} B \,b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b \,e^{5}-\frac {2373}{64} A \,a^{3} b^{2} d \,e^{4}+\frac {7119}{128} A \,a^{2} b^{3} d^{2} e^{3}-\frac {2373}{64} A a \,b^{4} d^{3} e^{2}+\frac {2373}{256} A \,b^{5} d^{4} e -\frac {843}{256} B \,a^{5} e^{5}+\frac {921}{128} B \,a^{4} b d \,e^{4}+\frac {531}{128} B \,a^{3} b^{2} d^{2} e^{3}-\frac {363}{16} B \,a^{2} b^{3} d^{3} e^{2}+\frac {5277}{256} B a \,b^{4} d^{4} e -\frac {765}{128} B \,b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {231 \left (13 A b e -3 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{7}}-\frac {A e -B d}{3 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {3}{2}}}-\frac {-6 A b e +B a e +5 B b d}{\left (a e -b d \right )^{7} \sqrt {e x +d}}\right )\)	\(579\)
pseudoelliptic	\(-\frac {2 \left (-\frac {9009 b \,e^{4} \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{5} \left (\left (A e -\frac {10 B d}{13}\right ) b -\frac {3 B a e}{13}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256}+\left (\left (-\frac {9009 A \,e^{6} x^{6}}{256}-\frac {3003 d \left (-\frac {15 B x}{26}+A \right ) x^{5} e^{5}}{64}-\frac {9009 d^{2} x^{4} \left (-\frac {200 B x}{39}+A \right ) e^{4}}{1280}+\frac {1287 d^{3} x^{3} \left (\frac {35 B x}{13}+A \right ) e^{3}}{640}-\frac {143 \left (\frac {45 B x}{26}+A \right ) d^{4} x^{2} e^{2}}{160}+\frac {39 d^{5} \left (\frac {55 B x}{39}+A \right ) x e}{80}-\frac {3 d^{6} \left (\frac {5 B x}{4}+A \right )}{10}\right ) b^{6}+\frac {183 \left (-\frac {35035 x^{5} \left (-\frac {9 B x}{182}+A \right ) e^{6}}{488}-\frac {47047 d \left (-\frac {380 B x}{611}+A \right ) x^{4} e^{5}}{488}-\frac {7293 d^{2} x^{3} \left (-\frac {511 B x}{102}+A \right ) e^{4}}{488}+\frac {6149 d^{3} x^{2} \left (\frac {1503 B x}{559}+A \right ) e^{3}}{1464}-\frac {338 \left (\frac {2299 B x}{1352}+A \right ) d^{4} x \,e^{2}}{183}+d^{5} \left (\frac {251 B x}{183}+A \right ) e -\frac {2 d^{6} B}{61}\right ) a \,b^{5}}{80}-\frac {1253 e \left (\frac {6864 \left (-\frac {105 B x}{832}+A \right ) x^{4} e^{5}}{179}+\frac {260403 d \left (-\frac {5467 B x}{7891}+A \right ) x^{3} e^{4}}{5012}+\frac {3003 \left (-\frac {127 B x}{26}+A \right ) d^{2} x^{2} e^{3}}{358}-\frac {1651 d^{3} x \left (\frac {30921 B x}{11557}+A \right ) e^{2}}{716}+d^{4} \left (\frac {4133 B x}{2506}+A \right ) e -\frac {104 B \,d^{5}}{1253}\right ) a^{2} b^{4}}{160}+\frac {2107 \left (-\frac {33891 x^{3} \left (-\frac {1344 B x}{5135}+A \right ) e^{4}}{2107}-\frac {46475 d \left (-\frac {17313 B x}{21125}+A \right ) x^{2} e^{3}}{2107}-\frac {1131 \left (-\frac {188452 B x}{39585}+A \right ) d^{2} x \,e^{2}}{301}+d^{3} \left (\frac {567 B x}{215}+A \right ) e -\frac {254 B \,d^{4}}{1505}\right ) e^{2} a^{3} b^{3}}{128}-\frac {7119 \left (\frac {27599 \left (-\frac {1422 B x}{2509}+A \right ) x^{2} e^{3}}{7119}+\frac {38558 d \left (-\frac {21340 B x}{19279}+A \right ) x \,e^{2}}{7119}+d^{2} \left (-\frac {33314 B x}{7119}+A \right ) e -\frac {116 B \,d^{3}}{339}\right ) e^{3} a^{4} b^{2}}{256}-19 \left (\frac {13 \left (-\frac {6369 B x}{3328}+A \right ) x \,e^{2}}{19}+d \left (-\frac {5729 B x}{2432}+A \right ) e -\frac {6625 B \,d^{2}}{4864}\right ) e^{4} a^{5} b +e^{5} \left (e \left (3 B x +A \right )+2 B d \right ) a^{6}\right ) \sqrt {b \left (a e -b d \right )}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \sqrt {b \left (a e -b d \right )}\, \left (b x +a \right )^{5} \left (a e -b d \right )^{7}}\)	\(592\)

Input:

int((B*x+A)/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^4*(1/(a*e-b*d)^7*b*(((1467/256*A*b^5*e-437/256*B*a*b^4*e-515/128*B*b^5 
*d)*(e*x+d)^(9/2)+1/384*b^3*(9629*A*a*b*e^2-9629*A*b^2*d*e-2931*B*a^2*e^2- 
3767*B*a*b*d*e+6698*B*b^2*d^2)*(e*x+d)^(7/2)+(1253/30*A*a^2*b^3*e^3-1253/1 
5*A*a*b^4*d*e^2+1253/30*A*b^5*d^2*e-131/10*B*a^3*b^2*e^3-37/15*B*a^2*b^3*d 
*e^2+1327/30*B*a*b^4*d^2*e-86/3*B*b^5*d^3)*(e*x+d)^(5/2)+(12131/384*A*a^3* 
b^2*e^4-12131/128*A*a^2*b^3*d*e^3+12131/128*A*a*b^4*d^2*e^2-12131/384*A*b^ 
5*d^3*e-1327/128*B*e^4*a^4*b+3793/384*B*a^3*b^2*d*e^3+4169/128*B*a^2*b^3*d 
^2*e^2-6823/128*B*a*b^4*d^3*e+4075/192*B*b^5*d^4)*(e*x+d)^(3/2)+(2373/256* 
A*a^4*b*e^5-2373/64*A*a^3*b^2*d*e^4+7119/128*A*a^2*b^3*d^2*e^3-2373/64*A*a 
*b^4*d^3*e^2+2373/256*A*b^5*d^4*e-843/256*B*a^5*e^5+921/128*B*a^4*b*d*e^4+ 
531/128*B*a^3*b^2*d^2*e^3-363/16*B*a^2*b^3*d^3*e^2+5277/256*B*a*b^4*d^4*e- 
765/128*B*b^5*d^5)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^5+231/256*(13*A*b*e- 
3*B*a*e-10*B*b*d)/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d)) 
^(1/2)))-1/3*(A*e-B*d)/(a*e-b*d)^6/(e*x+d)^(3/2)-1/(a*e-b*d)^7*(-6*A*b*e+B 
*a*e+5*B*b*d)/(e*x+d)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2307 vs. \(2 (338) = 676\).

Time = 1.72 (sec) , antiderivative size = 4644, normalized size of antiderivative = 12.35 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fric 
as")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxi 
ma")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1119 vs. \(2 (338) = 676\).

Time = 0.25 (sec) , antiderivative size = 1119, normalized size of antiderivative = 2.98 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

231/128*(10*B*b^2*d*e^4 + 3*B*a*b*e^5 - 13*A*b^2*e^5)*arctan(sqrt(e*x + d) 
*b/sqrt(-b^2*d + a*b*e))/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 
35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e 
^6 - a^7*e^7)*sqrt(-b^2*d + a*b*e)) + 2/3*(15*(e*x + d)*B*b*d*e^4 + B*b*d^ 
2*e^4 + 3*(e*x + d)*B*a*e^5 - 18*(e*x + d)*A*b*e^5 - B*a*d*e^5 - A*b*d*e^5 
 + A*a*e^6)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^ 
4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7) 
*(e*x + d)^(3/2)) + 1/1920*(15450*(e*x + d)^(9/2)*B*b^6*d*e^4 - 66980*(e*x 
 + d)^(7/2)*B*b^6*d^2*e^4 + 110080*(e*x + d)^(5/2)*B*b^6*d^3*e^4 - 81500*( 
e*x + d)^(3/2)*B*b^6*d^4*e^4 + 22950*sqrt(e*x + d)*B*b^6*d^5*e^4 + 6555*(e 
*x + d)^(9/2)*B*a*b^5*e^5 - 22005*(e*x + d)^(9/2)*A*b^6*e^5 + 37670*(e*x + 
 d)^(7/2)*B*a*b^5*d*e^5 + 96290*(e*x + d)^(7/2)*A*b^6*d*e^5 - 169856*(e*x 
+ d)^(5/2)*B*a*b^5*d^2*e^5 - 160384*(e*x + d)^(5/2)*A*b^6*d^2*e^5 + 204690 
*(e*x + d)^(3/2)*B*a*b^5*d^3*e^5 + 121310*(e*x + d)^(3/2)*A*b^6*d^3*e^5 - 
79155*sqrt(e*x + d)*B*a*b^5*d^4*e^5 - 35595*sqrt(e*x + d)*A*b^6*d^4*e^5 + 
29310*(e*x + d)^(7/2)*B*a^2*b^4*e^6 - 96290*(e*x + d)^(7/2)*A*a*b^5*e^6 + 
9472*(e*x + d)^(5/2)*B*a^2*b^4*d*e^6 + 320768*(e*x + d)^(5/2)*A*a*b^5*d*e^ 
6 - 125070*(e*x + d)^(3/2)*B*a^2*b^4*d^2*e^6 - 363930*(e*x + d)^(3/2)*A*a* 
b^5*d^2*e^6 + 87120*sqrt(e*x + d)*B*a^2*b^4*d^3*e^6 + 142380*sqrt(e*x + d) 
*A*a*b^5*d^3*e^6 + 50304*(e*x + d)^(5/2)*B*a^3*b^3*e^7 - 160384*(e*x + ...
 

Mupad [B] (verification not implemented)

Time = 12.39 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\frac {2\,\left (A\,e^5-B\,d\,e^4\right )}{3\,\left (a\,e-b\,d\right )}+\frac {2123\,{\left (d+e\,x\right )}^2\,\left (-13\,A\,b^2\,e^5+10\,B\,d\,b^2\,e^4+3\,B\,a\,b\,e^5\right )}{384\,{\left (a\,e-b\,d\right )}^3}+\frac {869\,{\left (d+e\,x\right )}^3\,\left (-13\,A\,b^3\,e^5+10\,B\,d\,b^3\,e^4+3\,B\,a\,b^2\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {539\,{\left (d+e\,x\right )}^5\,\left (-13\,A\,b^5\,e^5+10\,B\,d\,b^5\,e^4+3\,B\,a\,b^4\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,\left (d+e\,x\right )\,\left (3\,B\,a\,e^5-13\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {77\,b^3\,{\left (d+e\,x\right )}^4\,\left (3\,B\,a\,e^5-13\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{5\,{\left (a\,e-b\,d\right )}^5}+\frac {231\,b^5\,{\left (d+e\,x\right )}^6\,\left (3\,B\,a\,e^5-13\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^7}}{{\left (d+e\,x\right )}^{3/2}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{7/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{5/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{13/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{11/2}+{\left (d+e\,x\right )}^{9/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}-\frac {231\,\sqrt {b}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (3\,B\,a\,e-13\,A\,b\,e+10\,B\,b\,d\right )\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{{\left (a\,e-b\,d\right )}^{15/2}\,\left (3\,B\,a\,e^5-13\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}\right )\,\left (3\,B\,a\,e-13\,A\,b\,e+10\,B\,b\,d\right )}{128\,{\left (a\,e-b\,d\right )}^{15/2}} \] Input:

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

Output:

- ((2*(A*e^5 - B*d*e^4))/(3*(a*e - b*d)) + (2123*(d + e*x)^2*(3*B*a*b*e^5 
- 13*A*b^2*e^5 + 10*B*b^2*d*e^4))/(384*(a*e - b*d)^3) + (869*(d + e*x)^3*( 
3*B*a*b^2*e^5 - 13*A*b^3*e^5 + 10*B*b^3*d*e^4))/(64*(a*e - b*d)^4) + (539* 
(d + e*x)^5*(3*B*a*b^4*e^5 - 13*A*b^5*e^5 + 10*B*b^5*d*e^4))/(64*(a*e - b* 
d)^6) + (2*(d + e*x)*(3*B*a*e^5 - 13*A*b*e^5 + 10*B*b*d*e^4))/(3*(a*e - b* 
d)^2) + (77*b^3*(d + e*x)^4*(3*B*a*e^5 - 13*A*b*e^5 + 10*B*b*d*e^4))/(5*(a 
*e - b*d)^5) + (231*b^5*(d + e*x)^6*(3*B*a*e^5 - 13*A*b*e^5 + 10*B*b*d*e^4 
))/(128*(a*e - b*d)^7))/((d + e*x)^(3/2)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d 
^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) - (d + e*x)^( 
7/2)*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + ( 
d + e*x)^(5/2)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^ 
2*e^2 - 20*a*b^4*d^3*e) + b^5*(d + e*x)^(13/2) - (5*b^5*d - 5*a*b^4*e)*(d 
+ e*x)^(11/2) + (d + e*x)^(9/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d* 
e)) - (231*b^(1/2)*e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(3*B*a*e - 13*A*b 
*e + 10*B*b*d)*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^ 
3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^ 
6))/((a*e - b*d)^(15/2)*(3*B*a*e^5 - 13*A*b*e^5 + 10*B*b*d*e^4)))*(3*B*a*e 
 - 13*A*b*e + 10*B*b*d))/(128*(a*e - b*d)^(15/2))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1661, normalized size of antiderivative = 4.42 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(3465*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b 
)*sqrt(a*e - b*d)))*a**4*b*d*e**4 + 3465*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - 
b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**4*b*e**5*x + 138 
60*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*s 
qrt(a*e - b*d)))*a**3*b**2*d*e**4*x + 13860*sqrt(b)*sqrt(d + e*x)*sqrt(a*e 
 - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**3*b**2*e**5*x 
**2 + 20790*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/( 
sqrt(b)*sqrt(a*e - b*d)))*a**2*b**3*d*e**4*x**2 + 20790*sqrt(b)*sqrt(d + e 
*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2 
*b**3*e**5*x**3 + 13860*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d 
 + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**4*d*e**4*x**3 + 13860*sqrt(b)*s 
qrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b* 
d)))*a*b**4*e**5*x**4 + 3465*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((s 
qrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*b**5*d*e**4*x**4 + 3465*sqrt(b) 
*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - 
b*d)))*b**5*e**5*x**5 - 128*a**6*e**6 + 2176*a**5*b*d*e**5 + 1408*a**5*b*e 
**6*x + 247*a**4*b**2*d**2*e**4 + 11374*a**4*b**2*d*e**5*x + 9207*a**4*b** 
2*e**6*x**2 - 3325*a**3*b**3*d**3*e**3 - 8987*a**3*b**3*d**2*e**4*x + 1376 
1*a**3*b**3*d*e**5*x**2 + 16863*a**3*b**3*e**6*x**3 + 1358*a**2*b**4*d**4* 
e**2 - 4543*a**2*b**4*d**3*e**3*x - 20295*a**2*b**4*d**2*e**4*x**2 + 23...