\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\) [475]

Optimal result

Integrand size = 35, antiderivative size = 446 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 (b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {4 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)} \] Output:

-2*(-a*e+b*d)^5*(-A*e+B*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(1/2)-2*( 
-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e^7/( 
b*x+a)+10/3*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(3/2)*((b*x+a) 
^2)^(1/2)/e^7/(b*x+a)-4*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(5 
/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+10/7*b^3*(-a*e+b*d)*(-A*b*e-2*B*a*e+3*B* 
b*d)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/9*b^4*(-A*b*e-5*B*a*e+6 
*B*b*d)*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/11*b^5*B*(e*x+d)^(11 
/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.09 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {(a+b x)^2} \left (693 a^5 e^5 (2 B d-A e+B e x)+1155 a^4 b e^4 \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+462 a^3 b^2 e^3 \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-198 a^2 b^3 e^2 \left (-7 A e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+B \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )+11 a b^4 e \left (9 A e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 B \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )+b^5 \left (11 A e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )-3 B \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )\right )}{693 e^7 (a+b x) \sqrt {d+e x}} \] Input:

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

Output:

(2*Sqrt[(a + b*x)^2]*(693*a^5*e^5*(2*B*d - A*e + B*e*x) + 1155*a^4*b*e^4*( 
3*A*e*(2*d + e*x) + B*(-8*d^2 - 4*d*e*x + e^2*x^2)) + 462*a^3*b^2*e^3*(5*A 
*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*B*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + 
e^3*x^3)) - 198*a^2*b^3*e^2*(-7*A*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^ 
3*x^3) + B*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^ 
4)) + 11*a*b^4*e*(9*A*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3* 
x^3 + 5*e^4*x^4) + 5*B*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^ 
3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)) + b^5*(11*A*e*(256*d^5 + 128*d^4*e*x - 
32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5) - 3*B*(1024*d^ 
6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d 
*e^5*x^5 - 21*e^6*x^6))))/(693*e^7*(a + b*x)*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.64, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1187, 27, 86, 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.

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

\(\Big \downarrow \) 1187

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 86

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B (d+e x)^{9/2} b^5}{e^6}+\frac {(-6 b B d+A b e+5 a B e) (d+e x)^{7/2} b^4}{e^6}-\frac {5 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{5/2} b^3}{e^6}+\frac {10 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{3/2} b^2}{e^6}-\frac {5 (b d-a e)^3 (-3 b B d+2 A b e+a B e) \sqrt {d+e x} b}{e^6}+\frac {(a e-b d)^4 (-6 b B d+5 A b e+a B e)}{e^6 \sqrt {d+e x}}+\frac {(a e-b d)^5 (A e-B d)}{e^6 (d+e x)^{3/2}}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b^4 (d+e x)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7}+\frac {10 b^3 (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7}-\frac {4 b^2 (d+e x)^{5/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7}+\frac {10 b (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7}-\frac {2 \sqrt {d+e x} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7}-\frac {2 (b d-a e)^5 (B d-A e)}{e^7 \sqrt {d+e x}}+\frac {2 b^5 B (d+e x)^{11/2}}{11 e^7}\right )}{a+b x}\)

Input:

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

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)^5*(B*d - A*e))/(e^7*Sqrt[d 
 + e*x]) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[d + e*x])/e^7 
 + (10*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(3/2))/(3*e^7 
) - (4*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2))/e^7 + 
(10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(7/2))/(7*e^7) - 
 (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(9/2))/(9*e^7) + (2*b^5*B*(d 
 + e*x)^(11/2))/(11*e^7)))/(a + b*x)
 

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 1187

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ 
IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 
 - 4*a*c, 0] &&  !IntegerQ[p]
 

rule 2009

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

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.53

method	result	size
risch	\(\frac {2 \left (63 b^{5} B \,x^{5} e^{5}+77 A \,b^{5} e^{5} x^{4}+385 B a \,b^{4} e^{5} x^{4}-147 B \,b^{5} d \,e^{4} x^{4}+495 A a \,b^{4} e^{5} x^{3}-187 A \,b^{5} d \,e^{4} x^{3}+990 B \,a^{2} b^{3} e^{5} x^{3}-935 B a \,b^{4} d \,e^{4} x^{3}+267 B \,b^{5} d^{2} e^{3} x^{3}+1386 A \,a^{2} b^{3} e^{5} x^{2}-1287 A a \,b^{4} d \,e^{4} x^{2}+363 A \,b^{5} d^{2} e^{3} x^{2}+1386 B \,a^{3} b^{2} e^{5} x^{2}-2574 B \,a^{2} b^{3} d \,e^{4} x^{2}+1815 B a \,b^{4} d^{2} e^{3} x^{2}-459 B \,b^{5} d^{3} e^{2} x^{2}+2310 A \,a^{3} b^{2} e^{5} x -4158 A \,a^{2} b^{3} d \,e^{4} x +2871 A a \,b^{4} d^{2} e^{3} x -715 A \,b^{5} d^{3} e^{2} x +1155 B \,a^{4} b \,e^{5} x -4158 B \,a^{3} b^{2} d \,e^{4} x +5742 B \,a^{2} b^{3} d^{2} e^{3} x -3575 B a \,b^{4} d^{3} e^{2} x +843 B \,b^{5} d^{4} e x +3465 A \,a^{4} b \,e^{5}-11550 A \,a^{3} b^{2} d \,e^{4}+15246 A \,a^{2} b^{3} d^{2} e^{3}-9207 A a \,b^{4} d^{3} e^{2}+2123 A \,b^{5} d^{4} e +693 B \,a^{5} e^{5}-5775 B \,a^{4} b d \,e^{4}+15246 B \,a^{3} b^{2} d^{2} e^{3}-18414 B \,a^{2} b^{3} d^{3} e^{2}+10615 B a \,b^{4} d^{4} e -2379 B \,b^{5} d^{5}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{693 e^{7} \left (b x +a \right )}-\frac {2 \left (A \,a^{5} e^{6}-5 A \,a^{4} b d \,e^{5}+10 A \,a^{3} b^{2} d^{2} e^{4}-10 A \,a^{2} b^{3} d^{3} e^{3}+5 A a \,b^{4} d^{4} e^{2}-A \,b^{5} d^{5} e -B \,a^{5} d \,e^{5}+5 B \,a^{4} b \,d^{2} e^{4}-10 B \,a^{3} b^{2} d^{3} e^{3}+10 B \,a^{2} b^{3} d^{4} e^{2}-5 B a \,b^{4} d^{5} e +b^{5} B \,d^{6}\right ) \sqrt {\left (b x +a \right )^{2}}}{e^{7} \sqrt {e x +d}\, \left (b x +a \right )}\)	\(681\)
gosper	\(-\frac {2 \left (-63 B \,b^{5} e^{6} x^{6}-77 A \,b^{5} e^{6} x^{5}-385 B a \,b^{4} e^{6} x^{5}+84 B \,b^{5} d \,e^{5} x^{5}-495 A a \,b^{4} e^{6} x^{4}+110 A \,b^{5} d \,e^{5} x^{4}-990 B \,a^{2} b^{3} e^{6} x^{4}+550 B a \,b^{4} d \,e^{5} x^{4}-120 B \,b^{5} d^{2} e^{4} x^{4}-1386 A \,a^{2} b^{3} e^{6} x^{3}+792 A a \,b^{4} d \,e^{5} x^{3}-176 A \,b^{5} d^{2} e^{4} x^{3}-1386 B \,a^{3} b^{2} e^{6} x^{3}+1584 B \,a^{2} b^{3} d \,e^{5} x^{3}-880 B a \,b^{4} d^{2} e^{4} x^{3}+192 B \,b^{5} d^{3} e^{3} x^{3}-2310 A \,a^{3} b^{2} e^{6} x^{2}+2772 A \,a^{2} b^{3} d \,e^{5} x^{2}-1584 A a \,b^{4} d^{2} e^{4} x^{2}+352 A \,b^{5} d^{3} e^{3} x^{2}-1155 B \,a^{4} b \,e^{6} x^{2}+2772 B \,a^{3} b^{2} d \,e^{5} x^{2}-3168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1760 B a \,b^{4} d^{3} e^{3} x^{2}-384 B \,b^{5} d^{4} e^{2} x^{2}-3465 A \,a^{4} b \,e^{6} x +9240 A \,a^{3} b^{2} d \,e^{5} x -11088 A \,a^{2} b^{3} d^{2} e^{4} x +6336 A a \,b^{4} d^{3} e^{3} x -1408 A \,b^{5} d^{4} e^{2} x -693 B \,a^{5} e^{6} x +4620 B \,a^{4} b d \,e^{5} x -11088 B \,a^{3} b^{2} d^{2} e^{4} x +12672 B \,a^{2} b^{3} d^{3} e^{3} x -7040 B a \,b^{4} d^{4} e^{2} x +1536 B \,b^{5} d^{5} e x +693 A \,a^{5} e^{6}-6930 A \,a^{4} b d \,e^{5}+18480 A \,a^{3} b^{2} d^{2} e^{4}-22176 A \,a^{2} b^{3} d^{3} e^{3}+12672 A a \,b^{4} d^{4} e^{2}-2816 A \,b^{5} d^{5} e -1386 B \,a^{5} d \,e^{5}+9240 B \,a^{4} b \,d^{2} e^{4}-22176 B \,a^{3} b^{2} d^{3} e^{3}+25344 B \,a^{2} b^{3} d^{4} e^{2}-14080 B a \,b^{4} d^{5} e +3072 b^{5} B \,d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{693 \sqrt {e x +d}\, e^{7} \left (b x +a \right )^{5}}\)	\(689\)
default	\(-\frac {2 \left (-63 B \,b^{5} e^{6} x^{6}-77 A \,b^{5} e^{6} x^{5}-385 B a \,b^{4} e^{6} x^{5}+84 B \,b^{5} d \,e^{5} x^{5}-495 A a \,b^{4} e^{6} x^{4}+110 A \,b^{5} d \,e^{5} x^{4}-990 B \,a^{2} b^{3} e^{6} x^{4}+550 B a \,b^{4} d \,e^{5} x^{4}-120 B \,b^{5} d^{2} e^{4} x^{4}-1386 A \,a^{2} b^{3} e^{6} x^{3}+792 A a \,b^{4} d \,e^{5} x^{3}-176 A \,b^{5} d^{2} e^{4} x^{3}-1386 B \,a^{3} b^{2} e^{6} x^{3}+1584 B \,a^{2} b^{3} d \,e^{5} x^{3}-880 B a \,b^{4} d^{2} e^{4} x^{3}+192 B \,b^{5} d^{3} e^{3} x^{3}-2310 A \,a^{3} b^{2} e^{6} x^{2}+2772 A \,a^{2} b^{3} d \,e^{5} x^{2}-1584 A a \,b^{4} d^{2} e^{4} x^{2}+352 A \,b^{5} d^{3} e^{3} x^{2}-1155 B \,a^{4} b \,e^{6} x^{2}+2772 B \,a^{3} b^{2} d \,e^{5} x^{2}-3168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1760 B a \,b^{4} d^{3} e^{3} x^{2}-384 B \,b^{5} d^{4} e^{2} x^{2}-3465 A \,a^{4} b \,e^{6} x +9240 A \,a^{3} b^{2} d \,e^{5} x -11088 A \,a^{2} b^{3} d^{2} e^{4} x +6336 A a \,b^{4} d^{3} e^{3} x -1408 A \,b^{5} d^{4} e^{2} x -693 B \,a^{5} e^{6} x +4620 B \,a^{4} b d \,e^{5} x -11088 B \,a^{3} b^{2} d^{2} e^{4} x +12672 B \,a^{2} b^{3} d^{3} e^{3} x -7040 B a \,b^{4} d^{4} e^{2} x +1536 B \,b^{5} d^{5} e x +693 A \,a^{5} e^{6}-6930 A \,a^{4} b d \,e^{5}+18480 A \,a^{3} b^{2} d^{2} e^{4}-22176 A \,a^{2} b^{3} d^{3} e^{3}+12672 A a \,b^{4} d^{4} e^{2}-2816 A \,b^{5} d^{5} e -1386 B \,a^{5} d \,e^{5}+9240 B \,a^{4} b \,d^{2} e^{4}-22176 B \,a^{3} b^{2} d^{3} e^{3}+25344 B \,a^{2} b^{3} d^{4} e^{2}-14080 B a \,b^{4} d^{5} e +3072 b^{5} B \,d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{693 \sqrt {e x +d}\, e^{7} \left (b x +a \right )^{5}}\)	\(689\)
orering	\(-\frac {2 \left (-63 B \,b^{5} e^{6} x^{6}-77 A \,b^{5} e^{6} x^{5}-385 B a \,b^{4} e^{6} x^{5}+84 B \,b^{5} d \,e^{5} x^{5}-495 A a \,b^{4} e^{6} x^{4}+110 A \,b^{5} d \,e^{5} x^{4}-990 B \,a^{2} b^{3} e^{6} x^{4}+550 B a \,b^{4} d \,e^{5} x^{4}-120 B \,b^{5} d^{2} e^{4} x^{4}-1386 A \,a^{2} b^{3} e^{6} x^{3}+792 A a \,b^{4} d \,e^{5} x^{3}-176 A \,b^{5} d^{2} e^{4} x^{3}-1386 B \,a^{3} b^{2} e^{6} x^{3}+1584 B \,a^{2} b^{3} d \,e^{5} x^{3}-880 B a \,b^{4} d^{2} e^{4} x^{3}+192 B \,b^{5} d^{3} e^{3} x^{3}-2310 A \,a^{3} b^{2} e^{6} x^{2}+2772 A \,a^{2} b^{3} d \,e^{5} x^{2}-1584 A a \,b^{4} d^{2} e^{4} x^{2}+352 A \,b^{5} d^{3} e^{3} x^{2}-1155 B \,a^{4} b \,e^{6} x^{2}+2772 B \,a^{3} b^{2} d \,e^{5} x^{2}-3168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1760 B a \,b^{4} d^{3} e^{3} x^{2}-384 B \,b^{5} d^{4} e^{2} x^{2}-3465 A \,a^{4} b \,e^{6} x +9240 A \,a^{3} b^{2} d \,e^{5} x -11088 A \,a^{2} b^{3} d^{2} e^{4} x +6336 A a \,b^{4} d^{3} e^{3} x -1408 A \,b^{5} d^{4} e^{2} x -693 B \,a^{5} e^{6} x +4620 B \,a^{4} b d \,e^{5} x -11088 B \,a^{3} b^{2} d^{2} e^{4} x +12672 B \,a^{2} b^{3} d^{3} e^{3} x -7040 B a \,b^{4} d^{4} e^{2} x +1536 B \,b^{5} d^{5} e x +693 A \,a^{5} e^{6}-6930 A \,a^{4} b d \,e^{5}+18480 A \,a^{3} b^{2} d^{2} e^{4}-22176 A \,a^{2} b^{3} d^{3} e^{3}+12672 A a \,b^{4} d^{4} e^{2}-2816 A \,b^{5} d^{5} e -1386 B \,a^{5} d \,e^{5}+9240 B \,a^{4} b \,d^{2} e^{4}-22176 B \,a^{3} b^{2} d^{3} e^{3}+25344 B \,a^{2} b^{3} d^{4} e^{2}-14080 B a \,b^{4} d^{5} e +3072 b^{5} B \,d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{693 e^{7} \left (b x +a \right )^{5} \sqrt {e x +d}}\)	\(698\)

Input:

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

Output:

2/693*(63*B*b^5*e^5*x^5+77*A*b^5*e^5*x^4+385*B*a*b^4*e^5*x^4-147*B*b^5*d*e 
^4*x^4+495*A*a*b^4*e^5*x^3-187*A*b^5*d*e^4*x^3+990*B*a^2*b^3*e^5*x^3-935*B 
*a*b^4*d*e^4*x^3+267*B*b^5*d^2*e^3*x^3+1386*A*a^2*b^3*e^5*x^2-1287*A*a*b^4 
*d*e^4*x^2+363*A*b^5*d^2*e^3*x^2+1386*B*a^3*b^2*e^5*x^2-2574*B*a^2*b^3*d*e 
^4*x^2+1815*B*a*b^4*d^2*e^3*x^2-459*B*b^5*d^3*e^2*x^2+2310*A*a^3*b^2*e^5*x 
-4158*A*a^2*b^3*d*e^4*x+2871*A*a*b^4*d^2*e^3*x-715*A*b^5*d^3*e^2*x+1155*B* 
a^4*b*e^5*x-4158*B*a^3*b^2*d*e^4*x+5742*B*a^2*b^3*d^2*e^3*x-3575*B*a*b^4*d 
^3*e^2*x+843*B*b^5*d^4*e*x+3465*A*a^4*b*e^5-11550*A*a^3*b^2*d*e^4+15246*A* 
a^2*b^3*d^2*e^3-9207*A*a*b^4*d^3*e^2+2123*A*b^5*d^4*e+693*B*a^5*e^5-5775*B 
*a^4*b*d*e^4+15246*B*a^3*b^2*d^2*e^3-18414*B*a^2*b^3*d^3*e^2+10615*B*a*b^4 
*d^4*e-2379*B*b^5*d^5)*(e*x+d)^(1/2)/e^7*((b*x+a)^2)^(1/2)/(b*x+a)-2*(A*a^ 
5*e^6-5*A*a^4*b*d*e^5+10*A*a^3*b^2*d^2*e^4-10*A*a^2*b^3*d^3*e^3+5*A*a*b^4* 
d^4*e^2-A*b^5*d^5*e-B*a^5*d*e^5+5*B*a^4*b*d^2*e^4-10*B*a^3*b^2*d^3*e^3+10* 
B*a^2*b^3*d^4*e^2-5*B*a*b^4*d^5*e+B*b^5*d^6)/e^7/(e*x+d)^(1/2)*((b*x+a)^2) 
^(1/2)/(b*x+a)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.28 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (63 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 693 \, A a^{5} e^{6} + 2816 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 12672 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 22176 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 9240 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 7 \, {\left (12 \, B b^{5} d e^{5} - 11 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \, {\left (24 \, B b^{5} d^{2} e^{4} - 22 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 99 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 2 \, {\left (96 \, B b^{5} d^{3} e^{3} - 88 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 396 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 693 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + {\left (384 \, B b^{5} d^{4} e^{2} - 352 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 1584 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 2772 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 1155 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - {\left (1536 \, B b^{5} d^{5} e - 1408 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 6336 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 11088 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 4620 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 693 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{693 \, {\left (e^{8} x + d e^{7}\right )}} \] Input:

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

Output:

2/693*(63*B*b^5*e^6*x^6 - 3072*B*b^5*d^6 - 693*A*a^5*e^6 + 2816*(5*B*a*b^4 
 + A*b^5)*d^5*e - 12672*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 22176*(B*a^3*b^2 
 + A*a^2*b^3)*d^3*e^3 - 9240*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 1386*(B*a^5 
 + 5*A*a^4*b)*d*e^5 - 7*(12*B*b^5*d*e^5 - 11*(5*B*a*b^4 + A*b^5)*e^6)*x^5 
+ 5*(24*B*b^5*d^2*e^4 - 22*(5*B*a*b^4 + A*b^5)*d*e^5 + 99*(2*B*a^2*b^3 + A 
*a*b^4)*e^6)*x^4 - 2*(96*B*b^5*d^3*e^3 - 88*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 
396*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - 693*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 
 (384*B*b^5*d^4*e^2 - 352*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 1584*(2*B*a^2*b^3 
+ A*a*b^4)*d^2*e^4 - 2772*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 1155*(B*a^4*b + 
2*A*a^3*b^2)*e^6)*x^2 - (1536*B*b^5*d^5*e - 1408*(5*B*a*b^4 + A*b^5)*d^4*e 
^2 + 6336*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 11088*(B*a^3*b^2 + A*a^2*b^3)* 
d^2*e^4 + 4620*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - 693*(B*a^5 + 5*A*a^4*b)*e^6 
)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} A}{63 \, \sqrt {e x + d} e^{6}} + \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 14080 \, a b^{4} d^{5} e - 25344 \, a^{2} b^{3} d^{4} e^{2} + 22176 \, a^{3} b^{2} d^{3} e^{3} - 9240 \, a^{4} b d^{2} e^{4} + 1386 \, a^{5} d e^{5} - 7 \, {\left (12 \, b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 10 \, {\left (12 \, b^{5} d^{2} e^{4} - 55 \, a b^{4} d e^{5} + 99 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (96 \, b^{5} d^{3} e^{3} - 440 \, a b^{4} d^{2} e^{4} + 792 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (384 \, b^{5} d^{4} e^{2} - 1760 \, a b^{4} d^{3} e^{3} + 3168 \, a^{2} b^{3} d^{2} e^{4} - 2772 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (1536 \, b^{5} d^{5} e - 7040 \, a b^{4} d^{4} e^{2} + 12672 \, a^{2} b^{3} d^{3} e^{3} - 11088 \, a^{3} b^{2} d^{2} e^{4} + 4620 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} B}{693 \, \sqrt {e x + d} e^{7}} \] Input:

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

Output:

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^ 
2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 
 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*x^ 
3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2 
*e^5)*x^2 + (128*b^5*d^4*e - 576*a*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 84 
0*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*A/(sqrt(e*x + d)*e^6) + 2/693*(63*b^5* 
e^6*x^6 - 3072*b^5*d^6 + 14080*a*b^4*d^5*e - 25344*a^2*b^3*d^4*e^2 + 22176 
*a^3*b^2*d^3*e^3 - 9240*a^4*b*d^2*e^4 + 1386*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 
 55*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 55*a*b^4*d*e^5 + 99*a^2*b^3*e^6) 
*x^4 - 2*(96*b^5*d^3*e^3 - 440*a*b^4*d^2*e^4 + 792*a^2*b^3*d*e^5 - 693*a^3 
*b^2*e^6)*x^3 + (384*b^5*d^4*e^2 - 1760*a*b^4*d^3*e^3 + 3168*a^2*b^3*d^2*e 
^4 - 2772*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (1536*b^5*d^5*e - 7040*a*b 
^4*d^4*e^2 + 12672*a^2*b^3*d^3*e^3 - 11088*a^3*b^2*d^2*e^4 + 4620*a^4*b*d* 
e^5 - 693*a^5*e^6)*x)*B/(sqrt(e*x + d)*e^7)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1133 vs. \(2 (347) = 694\).

Time = 0.23 (sec) , antiderivative size = 1133, normalized size of antiderivative = 2.54 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2*(B*b^5*d^6*sgn(b*x + a) - 5*B*a*b^4*d^5*e*sgn(b*x + a) - A*b^5*d^5*e*sg 
n(b*x + a) + 10*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 5*A*a*b^4*d^4*e^2*sgn(b*x 
 + a) - 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^3*e^3*sgn(b*x + 
 a) + 5*B*a^4*b*d^2*e^4*sgn(b*x + a) + 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) - 
 B*a^5*d*e^5*sgn(b*x + a) - 5*A*a^4*b*d*e^5*sgn(b*x + a) + A*a^5*e^6*sgn(b 
*x + a))/(sqrt(e*x + d)*e^7) + 2/693*(63*(e*x + d)^(11/2)*B*b^5*e^70*sgn(b 
*x + a) - 462*(e*x + d)^(9/2)*B*b^5*d*e^70*sgn(b*x + a) + 1485*(e*x + d)^( 
7/2)*B*b^5*d^2*e^70*sgn(b*x + a) - 2772*(e*x + d)^(5/2)*B*b^5*d^3*e^70*sgn 
(b*x + a) + 3465*(e*x + d)^(3/2)*B*b^5*d^4*e^70*sgn(b*x + a) - 4158*sqrt(e 
*x + d)*B*b^5*d^5*e^70*sgn(b*x + a) + 385*(e*x + d)^(9/2)*B*a*b^4*e^71*sgn 
(b*x + a) + 77*(e*x + d)^(9/2)*A*b^5*e^71*sgn(b*x + a) - 2475*(e*x + d)^(7 
/2)*B*a*b^4*d*e^71*sgn(b*x + a) - 495*(e*x + d)^(7/2)*A*b^5*d*e^71*sgn(b*x 
 + a) + 6930*(e*x + d)^(5/2)*B*a*b^4*d^2*e^71*sgn(b*x + a) + 1386*(e*x + d 
)^(5/2)*A*b^5*d^2*e^71*sgn(b*x + a) - 11550*(e*x + d)^(3/2)*B*a*b^4*d^3*e^ 
71*sgn(b*x + a) - 2310*(e*x + d)^(3/2)*A*b^5*d^3*e^71*sgn(b*x + a) + 17325 
*sqrt(e*x + d)*B*a*b^4*d^4*e^71*sgn(b*x + a) + 3465*sqrt(e*x + d)*A*b^5*d^ 
4*e^71*sgn(b*x + a) + 990*(e*x + d)^(7/2)*B*a^2*b^3*e^72*sgn(b*x + a) + 49 
5*(e*x + d)^(7/2)*A*a*b^4*e^72*sgn(b*x + a) - 5544*(e*x + d)^(5/2)*B*a^2*b 
^3*d*e^72*sgn(b*x + a) - 2772*(e*x + d)^(5/2)*A*a*b^4*d*e^72*sgn(b*x + a) 
+ 13860*(e*x + d)^(3/2)*B*a^2*b^3*d^2*e^72*sgn(b*x + a) + 6930*(e*x + d...
 

Mupad [B] (verification not implemented)

Time = 13.13 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {x^2\,\left (2310\,B\,a^4\,b\,e^6-5544\,B\,a^3\,b^2\,d\,e^5+4620\,A\,a^3\,b^2\,e^6+6336\,B\,a^2\,b^3\,d^2\,e^4-5544\,A\,a^2\,b^3\,d\,e^5-3520\,B\,a\,b^4\,d^3\,e^3+3168\,A\,a\,b^4\,d^2\,e^4+768\,B\,b^5\,d^4\,e^2-704\,A\,b^5\,d^3\,e^3\right )}{693\,b\,e^7}-\frac {-2772\,B\,a^5\,d\,e^5+1386\,A\,a^5\,e^6+18480\,B\,a^4\,b\,d^2\,e^4-13860\,A\,a^4\,b\,d\,e^5-44352\,B\,a^3\,b^2\,d^3\,e^3+36960\,A\,a^3\,b^2\,d^2\,e^4+50688\,B\,a^2\,b^3\,d^4\,e^2-44352\,A\,a^2\,b^3\,d^3\,e^3-28160\,B\,a\,b^4\,d^5\,e+25344\,A\,a\,b^4\,d^4\,e^2+6144\,B\,b^5\,d^6-5632\,A\,b^5\,d^5\,e}{693\,b\,e^7}+\frac {x^3\,\left (2772\,B\,a^3\,b^2\,e^6-3168\,B\,a^2\,b^3\,d\,e^5+2772\,A\,a^2\,b^3\,e^6+1760\,B\,a\,b^4\,d^2\,e^4-1584\,A\,a\,b^4\,d\,e^5-384\,B\,b^5\,d^3\,e^3+352\,A\,b^5\,d^2\,e^4\right )}{693\,b\,e^7}+\frac {2\,b^3\,x^5\,\left (11\,A\,b\,e+55\,B\,a\,e-12\,B\,b\,d\right )}{99\,e^2}+\frac {x\,\left (1386\,B\,a^5\,e^6-9240\,B\,a^4\,b\,d\,e^5+6930\,A\,a^4\,b\,e^6+22176\,B\,a^3\,b^2\,d^2\,e^4-18480\,A\,a^3\,b^2\,d\,e^5-25344\,B\,a^2\,b^3\,d^3\,e^3+22176\,A\,a^2\,b^3\,d^2\,e^4+14080\,B\,a\,b^4\,d^4\,e^2-12672\,A\,a\,b^4\,d^3\,e^3-3072\,B\,b^5\,d^5\,e+2816\,A\,b^5\,d^4\,e^2\right )}{693\,b\,e^7}+\frac {10\,b^2\,x^4\,\left (198\,B\,a^2\,e^2-110\,B\,a\,b\,d\,e+99\,A\,a\,b\,e^2+24\,B\,b^2\,d^2-22\,A\,b^2\,d\,e\right )}{693\,e^3}+\frac {2\,B\,b^4\,x^6}{11\,e}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \] Input:

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

Output:

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((x^2*(2310*B*a^4*b*e^6 + 4620*A*a^3*b^2* 
e^6 - 704*A*b^5*d^3*e^3 + 768*B*b^5*d^4*e^2 + 3168*A*a*b^4*d^2*e^4 - 5544* 
A*a^2*b^3*d*e^5 - 3520*B*a*b^4*d^3*e^3 - 5544*B*a^3*b^2*d*e^5 + 6336*B*a^2 
*b^3*d^2*e^4))/(693*b*e^7) - (1386*A*a^5*e^6 + 6144*B*b^5*d^6 - 5632*A*b^5 
*d^5*e - 2772*B*a^5*d*e^5 + 25344*A*a*b^4*d^4*e^2 + 18480*B*a^4*b*d^2*e^4 
- 44352*A*a^2*b^3*d^3*e^3 + 36960*A*a^3*b^2*d^2*e^4 + 50688*B*a^2*b^3*d^4* 
e^2 - 44352*B*a^3*b^2*d^3*e^3 - 13860*A*a^4*b*d*e^5 - 28160*B*a*b^4*d^5*e) 
/(693*b*e^7) + (x^3*(2772*A*a^2*b^3*e^6 + 2772*B*a^3*b^2*e^6 + 352*A*b^5*d 
^2*e^4 - 384*B*b^5*d^3*e^3 + 1760*B*a*b^4*d^2*e^4 - 3168*B*a^2*b^3*d*e^5 - 
 1584*A*a*b^4*d*e^5))/(693*b*e^7) + (2*b^3*x^5*(11*A*b*e + 55*B*a*e - 12*B 
*b*d))/(99*e^2) + (x*(1386*B*a^5*e^6 + 6930*A*a^4*b*e^6 - 3072*B*b^5*d^5*e 
 + 2816*A*b^5*d^4*e^2 - 12672*A*a*b^4*d^3*e^3 - 18480*A*a^3*b^2*d*e^5 + 14 
080*B*a*b^4*d^4*e^2 + 22176*A*a^2*b^3*d^2*e^4 - 25344*B*a^2*b^3*d^3*e^3 + 
22176*B*a^3*b^2*d^2*e^4 - 9240*B*a^4*b*d*e^5))/(693*b*e^7) + (10*b^2*x^4*( 
198*B*a^2*e^2 + 24*B*b^2*d^2 + 99*A*a*b*e^2 - 22*A*b^2*d*e - 110*B*a*b*d*e 
))/(693*e^3) + (2*B*b^4*x^6)/(11*e)))/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1 
/2))/b)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.85 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {24 a^{5} b d \,e^{5}+12 a^{5} b \,e^{6} x -80 a^{4} b^{2} d^{2} e^{4}+10 a^{4} b^{2} e^{6} x^{2}+128 a^{3} b^{3} d^{3} e^{3}+8 a^{3} b^{3} e^{6} x^{3}-\frac {768}{7} a^{2} b^{4} d^{4} e^{2}+\frac {1024}{21} a \,b^{5} d^{5} e +\frac {4}{3} a \,b^{5} e^{6} x^{5}-\frac {1024}{231} b^{6} d^{5} e x +\frac {256}{231} b^{6} d^{4} e^{2} x^{2}-\frac {128}{231} b^{6} d^{3} e^{3} x^{3}-\frac {8}{33} b^{6} d \,e^{5} x^{5}+\frac {2}{11} b^{6} e^{6} x^{6}-40 a^{4} b^{2} d \,e^{5} x +64 a^{3} b^{3} d^{2} e^{4} x -16 a^{3} b^{3} d \,e^{5} x^{2}-\frac {384}{7} a^{2} b^{4} d^{3} e^{3} x +\frac {96}{7} a^{2} b^{4} d^{2} e^{4} x^{2}-\frac {48}{7} a^{2} b^{4} d \,e^{5} x^{3}+\frac {512}{21} a \,b^{5} d^{4} e^{2} x -\frac {128}{21} a \,b^{5} d^{3} e^{3} x^{2}+\frac {64}{21} a \,b^{5} d^{2} e^{4} x^{3}+\frac {30}{7} a^{2} b^{4} e^{6} x^{4}+\frac {80}{231} b^{6} d^{2} e^{4} x^{4}-\frac {40}{21} a \,b^{5} d \,e^{5} x^{4}-2 a^{6} e^{6}-\frac {2048}{231} b^{6} d^{6}}{\sqrt {e x +d}\, e^{7}} \] Input:

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

Output:

(2*( - 231*a**6*e**6 + 2772*a**5*b*d*e**5 + 1386*a**5*b*e**6*x - 9240*a**4 
*b**2*d**2*e**4 - 4620*a**4*b**2*d*e**5*x + 1155*a**4*b**2*e**6*x**2 + 147 
84*a**3*b**3*d**3*e**3 + 7392*a**3*b**3*d**2*e**4*x - 1848*a**3*b**3*d*e** 
5*x**2 + 924*a**3*b**3*e**6*x**3 - 12672*a**2*b**4*d**4*e**2 - 6336*a**2*b 
**4*d**3*e**3*x + 1584*a**2*b**4*d**2*e**4*x**2 - 792*a**2*b**4*d*e**5*x** 
3 + 495*a**2*b**4*e**6*x**4 + 5632*a*b**5*d**5*e + 2816*a*b**5*d**4*e**2*x 
 - 704*a*b**5*d**3*e**3*x**2 + 352*a*b**5*d**2*e**4*x**3 - 220*a*b**5*d*e* 
*5*x**4 + 154*a*b**5*e**6*x**5 - 1024*b**6*d**6 - 512*b**6*d**5*e*x + 128* 
b**6*d**4*e**2*x**2 - 64*b**6*d**3*e**3*x**3 + 40*b**6*d**2*e**4*x**4 - 28 
*b**6*d*e**5*x**5 + 21*b**6*e**6*x**6))/(231*sqrt(d + e*x)*e**7)