\(\int \frac {(A+B x) (d+e x)^{5/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [371]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F(-2)]
Giac [B] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 33, antiderivative size = 282 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {e (22 b B d+3 A b e-25 a B e) \sqrt {d+e x}}{48 b^4 (a+b x)^3}-\frac {e^2 (118 b B d+3 A b e-121 a B e) \sqrt {d+e x}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {e^3 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac {(2 b B d+A b e-3 a B e) (d+e x)^{3/2}}{8 b^3 (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b^2 (a+b x)^5}+\frac {e^4 (10 b B d-3 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}} \] Output:

-1/48*e*(3*A*b*e-25*B*a*e+22*B*b*d)*(e*x+d)^(1/2)/b^4/(b*x+a)^3-1/192*e^2* 
(3*A*b*e-121*B*a*e+118*B*b*d)*(e*x+d)^(1/2)/b^4/(-a*e+b*d)/(b*x+a)^2-1/128 
*e^3*(-3*A*b*e-7*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^4/(-a*e+b*d)^2/(b*x+a)-1/ 
8*(A*b*e-3*B*a*e+2*B*b*d)*(e*x+d)^(3/2)/b^3/(b*x+a)^4-1/5*(A*b-B*a)*(e*x+d 
)^(5/2)/b^2/(b*x+a)^5+1/128*e^4*(-3*A*b*e-7*B*a*e+10*B*b*d)*arctanh(b^(1/2 
)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(9/2)/(-a*e+b*d)^(5/2)
 

Mathematica [A] (verified)

Time = 5.69 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (B \left (105 a^5 e^4+10 a^4 b e^3 (-8 d+49 e x)+2 a^3 b^2 e^2 \left (-22 d^2-189 d e x+448 e^2 x^2\right )-2 a^2 b^3 e \left (16 d^3+102 d^2 e x+351 d e^2 x^2-395 e^3 x^3\right )+10 b^5 d x \left (48 d^3+136 d^2 e x+118 d e^2 x^2+15 e^3 x^3\right )+a b^4 \left (96 d^4-208 d^3 e x-1284 d^2 e^2 x^2-1790 d e^3 x^3-105 e^4 x^4\right )\right )+3 A b \left (15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+2 a^2 b^2 e^2 \left (4 d^2+23 d e x+64 e^2 x^2\right )-2 a b^3 e \left (88 d^3+256 d^2 e x+233 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (128 d^4+336 d^3 e x+248 d^2 e^2 x^2+10 d e^3 x^3-15 e^4 x^4\right )\right )\right )}{1920 b^4 (b d-a e)^2 (a+b x)^5}+\frac {e^4 (-10 b B d+3 A b e+7 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{9/2} (-b d+a e)^{5/2}} \] Input:

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

Output:

-1/1920*(Sqrt[d + e*x]*(B*(105*a^5*e^4 + 10*a^4*b*e^3*(-8*d + 49*e*x) + 2* 
a^3*b^2*e^2*(-22*d^2 - 189*d*e*x + 448*e^2*x^2) - 2*a^2*b^3*e*(16*d^3 + 10 
2*d^2*e*x + 351*d*e^2*x^2 - 395*e^3*x^3) + 10*b^5*d*x*(48*d^3 + 136*d^2*e* 
x + 118*d*e^2*x^2 + 15*e^3*x^3) + a*b^4*(96*d^4 - 208*d^3*e*x - 1284*d^2*e 
^2*x^2 - 1790*d*e^3*x^3 - 105*e^4*x^4)) + 3*A*b*(15*a^4*e^4 + 10*a^3*b*e^3 
*(d + 7*e*x) + 2*a^2*b^2*e^2*(4*d^2 + 23*d*e*x + 64*e^2*x^2) - 2*a*b^3*e*( 
88*d^3 + 256*d^2*e*x + 233*d*e^2*x^2 + 35*e^3*x^3) + b^4*(128*d^4 + 336*d^ 
3*e*x + 248*d^2*e^2*x^2 + 10*d*e^3*x^3 - 15*e^4*x^4))))/(b^4*(b*d - a*e)^2 
*(a + b*x)^5) + (e^4*(-10*b*B*d + 3*A*b*e + 7*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[ 
d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(9/2)*(-(b*d) + a*e)^(5/2))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1184, 27, 87, 51, 51, 51, 52, 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.

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

\(\Big \downarrow \) 1184

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-7 a B e-3 A b e+10 b B d) \int \frac {(d+e x)^{5/2}}{(a+b x)^5}dx}{10 b (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\)

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

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)*(d + e*x)^(7/2))/(b*(b*d - a*e)*(a + b*x)^5) + ((10*b*B* 
d - 3*A*b*e - 7*a*B*e)*(-1/4*(d + e*x)^(5/2)/(b*(a + b*x)^4) + (5*e*(-1/3* 
(d + e*x)^(3/2)/(b*(a + b*x)^3) + (e*(-1/2*Sqrt[d + e*x]/(b*(a + b*x)^2) + 
 (e*(-(Sqrt[d + e*x]/((b*d - a*e)*(a + b*x))) + (e*ArcTanh[(Sqrt[b]*Sqrt[d 
 + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^(3/2))))/(4*b)))/(2*b)))/( 
8*b)))/(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 51
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

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 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 = 2.60 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.07

method result size
derivativedivides \(2 e^{4} \left (\frac {\frac {\left (3 A b e +7 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {9}{2}}}{256 e^{2} a^{2}-512 a b d e +256 b^{2} d^{2}}+\frac {\left (21 A b e -79 B a e +58 B b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{384 b \left (a e -b d \right )}-\frac {\left (3 A b e +7 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 b^{2}}-\frac {7 \left (a e -b d \right ) \left (3 A b e +7 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b^{3}}-\frac {\left (3 A b e +7 B a e -10 B b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}}{256 b^{4}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {\left (3 A b e +7 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b^{4} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {b \left (a e -b d \right )}}\right )\) \(303\)
default \(2 e^{4} \left (\frac {\frac {\left (3 A b e +7 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {9}{2}}}{256 e^{2} a^{2}-512 a b d e +256 b^{2} d^{2}}+\frac {\left (21 A b e -79 B a e +58 B b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{384 b \left (a e -b d \right )}-\frac {\left (3 A b e +7 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 b^{2}}-\frac {7 \left (a e -b d \right ) \left (3 A b e +7 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b^{3}}-\frac {\left (3 A b e +7 B a e -10 B b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}}{256 b^{4}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {\left (3 A b e +7 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b^{4} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {b \left (a e -b d \right )}}\right )\) \(303\)
pseudoelliptic \(-\frac {3 \left (-e^{4} \left (\left (A e -\frac {10 B d}{3}\right ) b +\frac {7 B a e}{3}\right ) \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )+\left (\left (-A \,e^{4} x^{4}+\frac {2 d \,x^{3} \left (5 B x +A \right ) e^{3}}{3}+\frac {248 d^{2} \left (\frac {295 B x}{186}+A \right ) x^{2} e^{2}}{15}+\frac {112 \left (\frac {85 B x}{63}+A \right ) d^{3} x e}{5}+\frac {128 d^{4} \left (\frac {5 B x}{4}+A \right )}{15}\right ) b^{5}-\frac {176 \left (\frac {35 \left (\frac {B x}{2}+A \right ) x^{3} e^{4}}{88}+\frac {233 d \left (\frac {895 B x}{699}+A \right ) x^{2} e^{3}}{88}+\frac {32 d^{2} x \left (\frac {107 B x}{128}+A \right ) e^{2}}{11}+d^{3} \left (\frac {13 B x}{33}+A \right ) e -\frac {2 B \,d^{4}}{11}\right ) a \,b^{4}}{15}+\frac {8 e \left (\left (\frac {395}{12} B \,x^{3}+16 A \,x^{2}\right ) e^{3}+\frac {23 \left (-\frac {117 B x}{23}+A \right ) d x \,e^{2}}{4}+d^{2} \left (-\frac {17 B x}{2}+A \right ) e -\frac {4 B \,d^{3}}{3}\right ) a^{2} b^{3}}{15}+\frac {2 e^{2} \left (7 x \left (\frac {64 B x}{15}+A \right ) e^{2}+d \left (-\frac {63 B x}{5}+A \right ) e -\frac {22 B \,d^{2}}{15}\right ) a^{3} b^{2}}{3}+e^{3} \left (\left (A +\frac {98 B x}{9}\right ) e -\frac {16 B d}{9}\right ) a^{4} b +\frac {7 B \,a^{5} e^{4}}{3}\right ) \sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}\right )}{128 \sqrt {b \left (a e -b d \right )}\, \left (b x +a \right )^{5} \left (a e -b d \right )^{2} b^{4}}\) \(371\)

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*((1/256*(3*A*b*e+7*B*a*e-10*B*b*d)/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+ 
d)^(9/2)+1/384*(21*A*b*e-79*B*a*e+58*B*b*d)/b/(a*e-b*d)*(e*x+d)^(7/2)-1/30 
*(3*A*b*e+7*B*a*e-10*B*b*d)/b^2*(e*x+d)^(5/2)-7/384/b^3*(a*e-b*d)*(3*A*b*e 
+7*B*a*e-10*B*b*d)*(e*x+d)^(3/2)-1/256*(3*A*b*e+7*B*a*e-10*B*b*d)*(a^2*e^2 
-2*a*b*d*e+b^2*d^2)/b^4*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^5+1/256*(3*A*b* 
e+7*B*a*e-10*B*b*d)/b^4/(a^2*e^2-2*a*b*d*e+b^2*d^2)/(b*(a*e-b*d))^(1/2)*ar 
ctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1112 vs. \(2 (250) = 500\).

Time = 0.20 (sec) , antiderivative size = 2238, normalized size of antiderivative = 7.94 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] 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:

[-1/3840*(15*(10*B*a^5*b*d*e^4 - (7*B*a^6 + 3*A*a^5*b)*e^5 + (10*B*b^6*d*e 
^4 - (7*B*a*b^5 + 3*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (7*B*a^2*b^4 + 
 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (7*B*a^3*b^3 + 3*A*a^2*b^4 
)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (7*B*a^4*b^2 + 3*A*a^3*b^3)*e^5)*x^2 
 + 5*(10*B*a^4*b^2*d*e^4 - (7*B*a^5*b + 3*A*a^4*b^2)*e^5)*x)*sqrt(b^2*d - 
a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b* 
x + a)) + 2*(96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(8*B*a^2*b^5 + 57*A*a*b^6)*d^ 
4*e - 12*(B*a^3*b^4 - 46*A*a^2*b^5)*d^3*e^2 - 6*(6*B*a^4*b^3 - A*a^3*b^4)* 
d^2*e^3 + 5*(37*B*a^5*b^2 + 3*A*a^4*b^3)*d*e^4 - 15*(7*B*a^6*b + 3*A*a^5*b 
^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (17*B*a*b^6 + 3*A*b^7)*d*e^4 + (7*B*a^2*b 
^5 + 3*A*a*b^6)*e^5)*x^4 + 10*(118*B*b^7*d^3*e^2 - 3*(99*B*a*b^6 - A*b^7)* 
d^2*e^3 + 6*(43*B*a^2*b^5 - 4*A*a*b^6)*d*e^4 - (79*B*a^3*b^4 - 21*A*a^2*b^ 
5)*e^5)*x^3 + 2*(680*B*b^7*d^4*e - 2*(661*B*a*b^6 - 186*A*b^7)*d^3*e^2 + 3 
*(97*B*a^2*b^5 - 357*A*a*b^6)*d^2*e^3 + (799*B*a^3*b^4 + 891*A*a^2*b^5)*d* 
e^4 - 64*(7*B*a^4*b^3 + 3*A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5 - 8*(43*B 
*a*b^6 - 63*A*b^7)*d^4*e + 2*(B*a^2*b^5 - 636*A*a*b^6)*d^3*e^2 - 3*(29*B*a 
^3*b^4 - 279*A*a^2*b^5)*d^2*e^3 + 2*(217*B*a^4*b^3 + 18*A*a^3*b^4)*d*e^4 - 
 35*(7*B*a^5*b^2 + 3*A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^3 - 3*a^ 
6*b^7*d^2*e + 3*a^7*b^6*d*e^2 - a^8*b^5*e^3 + (b^13*d^3 - 3*a*b^12*d^2*e + 
 3*a^2*b^11*d*e^2 - a^3*b^10*e^3)*x^5 + 5*(a*b^12*d^3 - 3*a^2*b^11*d^2*...
 

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. 803 vs. \(2 (250) = 500\).

Time = 0.23 (sec) , antiderivative size = 803, normalized size of antiderivative = 2.85 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] 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:

-1/128*(10*B*b*d*e^4 - 7*B*a*e^5 - 3*A*b*e^5)*arctan(sqrt(e*x + d)*b/sqrt( 
-b^2*d + a*b*e))/((b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*sqrt(-b^2*d + a*b* 
e)) - 1/1920*(150*(e*x + d)^(9/2)*B*b^5*d*e^4 + 580*(e*x + d)^(7/2)*B*b^5* 
d^2*e^4 - 1280*(e*x + d)^(5/2)*B*b^5*d^3*e^4 + 700*(e*x + d)^(3/2)*B*b^5*d 
^4*e^4 - 150*sqrt(e*x + d)*B*b^5*d^5*e^4 - 105*(e*x + d)^(9/2)*B*a*b^4*e^5 
 - 45*(e*x + d)^(9/2)*A*b^5*e^5 - 1370*(e*x + d)^(7/2)*B*a*b^4*d*e^5 + 210 
*(e*x + d)^(7/2)*A*b^5*d*e^5 + 3456*(e*x + d)^(5/2)*B*a*b^4*d^2*e^5 + 384* 
(e*x + d)^(5/2)*A*b^5*d^2*e^5 - 2590*(e*x + d)^(3/2)*B*a*b^4*d^3*e^5 - 210 
*(e*x + d)^(3/2)*A*b^5*d^3*e^5 + 705*sqrt(e*x + d)*B*a*b^4*d^4*e^5 + 45*sq 
rt(e*x + d)*A*b^5*d^4*e^5 + 790*(e*x + d)^(7/2)*B*a^2*b^3*e^6 - 210*(e*x + 
 d)^(7/2)*A*a*b^4*e^6 - 3072*(e*x + d)^(5/2)*B*a^2*b^3*d*e^6 - 768*(e*x + 
d)^(5/2)*A*a*b^4*d*e^6 + 3570*(e*x + d)^(3/2)*B*a^2*b^3*d^2*e^6 + 630*(e*x 
 + d)^(3/2)*A*a*b^4*d^2*e^6 - 1320*sqrt(e*x + d)*B*a^2*b^3*d^3*e^6 - 180*s 
qrt(e*x + d)*A*a*b^4*d^3*e^6 + 896*(e*x + d)^(5/2)*B*a^3*b^2*e^7 + 384*(e* 
x + d)^(5/2)*A*a^2*b^3*e^7 - 2170*(e*x + d)^(3/2)*B*a^3*b^2*d*e^7 - 630*(e 
*x + d)^(3/2)*A*a^2*b^3*d*e^7 + 1230*sqrt(e*x + d)*B*a^3*b^2*d^2*e^7 + 270 
*sqrt(e*x + d)*A*a^2*b^3*d^2*e^7 + 490*(e*x + d)^(3/2)*B*a^4*b*e^8 + 210*( 
e*x + d)^(3/2)*A*a^3*b^2*e^8 - 570*sqrt(e*x + d)*B*a^4*b*d*e^8 - 180*sqrt( 
e*x + d)*A*a^3*b^2*d*e^8 + 105*sqrt(e*x + d)*B*a^5*e^9 + 45*sqrt(e*x + d)* 
A*a^4*b*e^9)/((b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*((e*x + d)*b - b*d ...
 

Mupad [B] (verification not implemented)

Time = 11.41 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.03 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (3\,A\,b\,e+7\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (3\,A\,b\,e+7\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{9/2}\,{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {{\left (d+e\,x\right )}^{5/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,b^2}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^2}+\frac {7\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^3}-\frac {{\left (d+e\,x\right )}^{7/2}\,\left (21\,A\,b\,e^5-79\,B\,a\,e^5+58\,B\,b\,d\,e^4\right )}{192\,b\,\left (a\,e-b\,d\right )}+\frac {\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,b^4}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \] Input:

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

Output:

(e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(3*A*b*e + 7*B*a*e - 10*B*b*d))/((a 
*e - b*d)^(1/2)*(3*A*b*e^5 + 7*B*a*e^5 - 10*B*b*d*e^4)))*(3*A*b*e + 7*B*a* 
e - 10*B*b*d))/(128*b^(9/2)*(a*e - b*d)^(5/2)) - (((d + e*x)^(5/2)*(3*A*b* 
e^5 + 7*B*a*e^5 - 10*B*b*d*e^4))/(15*b^2) - ((d + e*x)^(9/2)*(3*A*b*e^5 + 
7*B*a*e^5 - 10*B*b*d*e^4))/(128*(a*e - b*d)^2) + (7*(a*e - b*d)*(d + e*x)^ 
(3/2)*(3*A*b*e^5 + 7*B*a*e^5 - 10*B*b*d*e^4))/(192*b^3) - ((d + e*x)^(7/2) 
*(21*A*b*e^5 - 79*B*a*e^5 + 58*B*b*d*e^4))/(192*b*(a*e - b*d)) + ((d + e*x 
)^(1/2)*(a^2*e^2 + b^2*d^2 - 2*a*b*d*e)*(3*A*b*e^5 + 7*B*a*e^5 - 10*B*b*d* 
e^4))/(128*b^4))/((d + e*x)*(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) - (d + e*x)^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) + b^5*(d + e*x)^5 - (5*b^5*d - 
5*a*b^4*e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d^2 + 10* 
a^2*b^3*e^2 - 20*a*b^4*d*e) - 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)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.34 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {15 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{4} e^{4}+60 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{3} b \,e^{4} x +90 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{2} b^{2} e^{4} x^{2}+60 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a \,b^{3} e^{4} x^{3}+15 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) b^{4} e^{4} x^{4}-15 \sqrt {e x +d}\, a^{4} b \,e^{4}+5 \sqrt {e x +d}\, a^{3} b^{2} d \,e^{3}-55 \sqrt {e x +d}\, a^{3} b^{2} e^{4} x +2 \sqrt {e x +d}\, a^{2} b^{3} d^{2} e^{2}+19 \sqrt {e x +d}\, a^{2} b^{3} d \,e^{3} x -73 \sqrt {e x +d}\, a^{2} b^{3} e^{4} x^{2}+56 \sqrt {e x +d}\, a \,b^{4} d^{3} e +172 \sqrt {e x +d}\, a \,b^{4} d^{2} e^{2} x +191 \sqrt {e x +d}\, a \,b^{4} d \,e^{3} x^{2}+15 \sqrt {e x +d}\, a \,b^{4} e^{4} x^{3}-48 \sqrt {e x +d}\, b^{5} d^{4}-136 \sqrt {e x +d}\, b^{5} d^{3} e x -118 \sqrt {e x +d}\, b^{5} d^{2} e^{2} x^{2}-15 \sqrt {e x +d}\, b^{5} d \,e^{3} x^{3}}{192 b^{4} \left (a^{2} b^{4} e^{2} x^{4}-2 a \,b^{5} d e \,x^{4}+b^{6} d^{2} x^{4}+4 a^{3} b^{3} e^{2} x^{3}-8 a^{2} b^{4} d e \,x^{3}+4 a \,b^{5} d^{2} x^{3}+6 a^{4} b^{2} e^{2} x^{2}-12 a^{3} b^{3} d e \,x^{2}+6 a^{2} b^{4} d^{2} x^{2}+4 a^{5} b \,e^{2} x -8 a^{4} b^{2} d e x +4 a^{3} b^{3} d^{2} x +a^{6} e^{2}-2 a^{5} b d e +a^{4} b^{2} d^{2}\right )} \] Input:

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

Output:

(15*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d 
)))*a**4*e**4 + 60*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b) 
*sqrt(a*e - b*d)))*a**3*b*e**4*x + 90*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d 
 + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2*b**2*e**4*x**2 + 60*sqrt(b)*sqr 
t(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**3*e**4 
*x**3 + 15*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a* 
e - b*d)))*b**4*e**4*x**4 - 15*sqrt(d + e*x)*a**4*b*e**4 + 5*sqrt(d + e*x) 
*a**3*b**2*d*e**3 - 55*sqrt(d + e*x)*a**3*b**2*e**4*x + 2*sqrt(d + e*x)*a* 
*2*b**3*d**2*e**2 + 19*sqrt(d + e*x)*a**2*b**3*d*e**3*x - 73*sqrt(d + e*x) 
*a**2*b**3*e**4*x**2 + 56*sqrt(d + e*x)*a*b**4*d**3*e + 172*sqrt(d + e*x)* 
a*b**4*d**2*e**2*x + 191*sqrt(d + e*x)*a*b**4*d*e**3*x**2 + 15*sqrt(d + e* 
x)*a*b**4*e**4*x**3 - 48*sqrt(d + e*x)*b**5*d**4 - 136*sqrt(d + e*x)*b**5* 
d**3*e*x - 118*sqrt(d + e*x)*b**5*d**2*e**2*x**2 - 15*sqrt(d + e*x)*b**5*d 
*e**3*x**3)/(192*b**4*(a**6*e**2 - 2*a**5*b*d*e + 4*a**5*b*e**2*x + a**4*b 
**2*d**2 - 8*a**4*b**2*d*e*x + 6*a**4*b**2*e**2*x**2 + 4*a**3*b**3*d**2*x 
- 12*a**3*b**3*d*e*x**2 + 4*a**3*b**3*e**2*x**3 + 6*a**2*b**4*d**2*x**2 - 
8*a**2*b**4*d*e*x**3 + a**2*b**4*e**2*x**4 + 4*a*b**5*d**2*x**3 - 2*a*b**5 
*d*e*x**4 + b**6*d**2*x**4))