\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{\sqrt {d+e x}} \, dx\) [466]

Optimal result

Integrand size = 35, antiderivative size = 306 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 (b d-a e)^3 (B d-A e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} \left (105 a^3 e^3 (-2 B d+3 A e+B e x)+63 a^2 b e^2 \left (5 A e (-2 d+e x)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-9 a b^2 e \left (-7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )+3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )+b^3 \left (9 A e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{315 e^5 (a+b x)} \] Input:

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

Output:

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(105*a^3*e^3*(-2*B*d + 3*A*e + B*e*x) + 
 63*a^2*b*e^2*(5*A*e*(-2*d + e*x) + B*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) - 9*a 
*b^2*e*(-7*A*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 3*B*(16*d^3 - 8*d^2*e*x + 6 
*d*e^2*x^2 - 5*e^3*x^3)) + b^3*(9*A*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 
 5*e^3*x^3) + B*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35 
*e^4*x^4))))/(315*e^5*(a + b*x))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.65, 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.

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

\(\Big \downarrow \) 1187

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[d + e*x] 
)/e^5 - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(3/2))/(3*e 
^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^5) 
- (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(7*e^5) + (2*b^3*B*( 
d + e*x)^(9/2))/(9*e^5)))/(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 = 1.85 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.04

method	result	size
gosper	\(\frac {2 \sqrt {e x +d}\, \left (35 B \,b^{3} x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 A \,a^{3} e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 e^{5} \left (b x +a \right )^{3}}\)	\(317\)
default	\(\frac {2 \sqrt {e x +d}\, \left (35 B \,b^{3} x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 A \,a^{3} e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 e^{5} \left (b x +a \right )^{3}}\)	\(317\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (35 B \,b^{3} x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 A \,a^{3} e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \sqrt {e x +d}}{315 \left (b x +a \right ) e^{5}}\)	\(317\)
orering	\(\frac {2 \left (35 B \,b^{3} x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 A \,a^{3} e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \sqrt {e x +d}\, \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{315 e^{5} \left (b x +a \right )^{3}}\)	\(326\)

Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

2/315*(e*x+d)^(1/2)*(35*B*b^3*e^4*x^4+45*A*b^3*e^4*x^3+135*B*a*b^2*e^4*x^3 
-40*B*b^3*d*e^3*x^3+189*A*a*b^2*e^4*x^2-54*A*b^3*d*e^3*x^2+189*B*a^2*b*e^4 
*x^2-162*B*a*b^2*d*e^3*x^2+48*B*b^3*d^2*e^2*x^2+315*A*a^2*b*e^4*x-252*A*a* 
b^2*d*e^3*x+72*A*b^3*d^2*e^2*x+105*B*a^3*e^4*x-252*B*a^2*b*d*e^3*x+216*B*a 
*b^2*d^2*e^2*x-64*B*b^3*d^3*e*x+315*A*a^3*e^4-630*A*a^2*b*d*e^3+504*A*a*b^ 
2*d^2*e^2-144*A*b^3*d^3*e-210*B*a^3*d*e^3+504*B*a^2*b*d^2*e^2-432*B*a*b^2* 
d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (35 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} + 315 \, A a^{3} e^{4} - 144 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 504 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 210 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \, {\left (8 \, B b^{3} d e^{3} - 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{2} e^{2} - 18 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 63 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - {\left (64 \, B b^{3} d^{3} e - 72 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{5}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x, algorithm=" 
fricas")
 

Output:

2/315*(35*B*b^3*e^4*x^4 + 128*B*b^3*d^4 + 315*A*a^3*e^4 - 144*(3*B*a*b^2 + 
 A*b^3)*d^3*e + 504*(B*a^2*b + A*a*b^2)*d^2*e^2 - 210*(B*a^3 + 3*A*a^2*b)* 
d*e^3 - 5*(8*B*b^3*d*e^3 - 9*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 3*(16*B*b^3*d^ 
2*e^2 - 18*(3*B*a*b^2 + A*b^3)*d*e^3 + 63*(B*a^2*b + A*a*b^2)*e^4)*x^2 - ( 
64*B*b^3*d^3*e - 72*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 252*(B*a^2*b + A*a*b^2)* 
d*e^3 - 105*(B*a^3 + 3*A*a^2*b)*e^4)*x)*sqrt(e*x + d)/e^5
 

Sympy [F]

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

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(1/2),x)
 

Output:

Integral((A + B*x)*((a + b*x)**2)**(3/2)/sqrt(d + e*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.25 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (5 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 56 \, a b^{2} d^{3} e - 70 \, a^{2} b d^{2} e^{2} + 35 \, a^{3} d e^{3} - {\left (b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} + {\left (2 \, b^{3} d^{2} e^{2} - 7 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} - {\left (8 \, b^{3} d^{3} e - 28 \, a b^{2} d^{2} e^{2} + 35 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} A}{35 \, \sqrt {e x + d} e^{4}} + \frac {2 \, {\left (35 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 432 \, a b^{2} d^{4} e + 504 \, a^{2} b d^{3} e^{2} - 210 \, a^{3} d^{2} e^{3} - 5 \, {\left (b^{3} d e^{4} - 27 \, a b^{2} e^{5}\right )} x^{4} + {\left (8 \, b^{3} d^{2} e^{3} - 27 \, a b^{2} d e^{4} + 189 \, a^{2} b e^{5}\right )} x^{3} - {\left (16 \, b^{3} d^{3} e^{2} - 54 \, a b^{2} d^{2} e^{3} + 63 \, a^{2} b d e^{4} - 105 \, a^{3} e^{5}\right )} x^{2} + {\left (64 \, b^{3} d^{4} e - 216 \, a b^{2} d^{3} e^{2} + 252 \, a^{2} b d^{2} e^{3} - 105 \, a^{3} d e^{4}\right )} x\right )} B}{315 \, \sqrt {e x + d} e^{5}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x, algorithm=" 
maxima")
 

Output:

2/35*(5*b^3*e^4*x^4 - 16*b^3*d^4 + 56*a*b^2*d^3*e - 70*a^2*b*d^2*e^2 + 35* 
a^3*d*e^3 - (b^3*d*e^3 - 21*a*b^2*e^4)*x^3 + (2*b^3*d^2*e^2 - 7*a*b^2*d*e^ 
3 + 35*a^2*b*e^4)*x^2 - (8*b^3*d^3*e - 28*a*b^2*d^2*e^2 + 35*a^2*b*d*e^3 - 
 35*a^3*e^4)*x)*A/(sqrt(e*x + d)*e^4) + 2/315*(35*b^3*e^5*x^5 + 128*b^3*d^ 
5 - 432*a*b^2*d^4*e + 504*a^2*b*d^3*e^2 - 210*a^3*d^2*e^3 - 5*(b^3*d*e^4 - 
 27*a*b^2*e^5)*x^4 + (8*b^3*d^2*e^3 - 27*a*b^2*d*e^4 + 189*a^2*b*e^5)*x^3 
- (16*b^3*d^3*e^2 - 54*a*b^2*d^2*e^3 + 63*a^2*b*d*e^4 - 105*a^3*e^5)*x^2 + 
 (64*b^3*d^4*e - 216*a*b^2*d^3*e^2 + 252*a^2*b*d^2*e^3 - 105*a^3*d*e^4)*x) 
*B/(sqrt(e*x + d)*e^5)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.23 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} A a^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {105 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} B a^{3} \mathrm {sgn}\left (b x + a\right )}{e} + \frac {315 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} A a^{2} b \mathrm {sgn}\left (b x + a\right )}{e} + \frac {63 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} B a^{2} b \mathrm {sgn}\left (b x + a\right )}{e^{2}} + \frac {63 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} A a b^{2} \mathrm {sgn}\left (b x + a\right )}{e^{2}} + \frac {27 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} B a b^{2} \mathrm {sgn}\left (b x + a\right )}{e^{3}} + \frac {9 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} A b^{3} \mathrm {sgn}\left (b x + a\right )}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} B b^{3} \mathrm {sgn}\left (b x + a\right )}{e^{4}}\right )}}{315 \, e} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x, algorithm=" 
giac")
 

Output:

2/315*(315*sqrt(e*x + d)*A*a^3*sgn(b*x + a) + 105*((e*x + d)^(3/2) - 3*sqr 
t(e*x + d)*d)*B*a^3*sgn(b*x + a)/e + 315*((e*x + d)^(3/2) - 3*sqrt(e*x + d 
)*d)*A*a^2*b*sgn(b*x + a)/e + 63*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d 
 + 15*sqrt(e*x + d)*d^2)*B*a^2*b*sgn(b*x + a)/e^2 + 63*(3*(e*x + d)^(5/2) 
- 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a*b^2*sgn(b*x + a)/e^2 + 
27*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35 
*sqrt(e*x + d)*d^3)*B*a*b^2*sgn(b*x + a)/e^3 + 9*(5*(e*x + d)^(7/2) - 21*( 
e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*b^3*sg 
n(b*x + a)/e^3 + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + 
d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*B*b^3*sgn( 
b*x + a)/e^4)/e
 

Mupad [B] (verification not implemented)

Time = 12.53 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {-420\,B\,a^3\,d^2\,e^3+630\,A\,a^3\,d\,e^4+1008\,B\,a^2\,b\,d^3\,e^2-1260\,A\,a^2\,b\,d^2\,e^3-864\,B\,a\,b^2\,d^4\,e+1008\,A\,a\,b^2\,d^3\,e^2+256\,B\,b^3\,d^5-288\,A\,b^3\,d^4\,e}{315\,b\,e^5}+\frac {2\,B\,b^2\,x^5}{9}+\frac {x^3\,\left (378\,B\,a^2\,b\,e^5-54\,B\,a\,b^2\,d\,e^4+378\,A\,a\,b^2\,e^5+16\,B\,b^3\,d^2\,e^3-18\,A\,b^3\,d\,e^4\right )}{315\,b\,e^5}+\frac {x\,\left (-210\,B\,a^3\,d\,e^4+630\,A\,a^3\,e^5+504\,B\,a^2\,b\,d^2\,e^3-630\,A\,a^2\,b\,d\,e^4-432\,B\,a\,b^2\,d^3\,e^2+504\,A\,a\,b^2\,d^2\,e^3+128\,B\,b^3\,d^4\,e-144\,A\,b^3\,d^3\,e^2\right )}{315\,b\,e^5}+\frac {x^2\,\left (210\,B\,a^3\,e^5-126\,B\,a^2\,b\,d\,e^4+630\,A\,a^2\,b\,e^5+108\,B\,a\,b^2\,d^2\,e^3-126\,A\,a\,b^2\,d\,e^4-32\,B\,b^3\,d^3\,e^2+36\,A\,b^3\,d^2\,e^3\right )}{315\,b\,e^5}+\frac {2\,b\,x^4\,\left (9\,A\,b\,e+27\,B\,a\,e-B\,b\,d\right )}{63\,e}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \] Input:

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(d + e*x)^(1/2),x)
 

Output:

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((256*B*b^3*d^5 + 630*A*a^3*d*e^4 - 288*A 
*b^3*d^4*e - 420*B*a^3*d^2*e^3 + 1008*A*a*b^2*d^3*e^2 - 1260*A*a^2*b*d^2*e 
^3 + 1008*B*a^2*b*d^3*e^2 - 864*B*a*b^2*d^4*e)/(315*b*e^5) + (2*B*b^2*x^5) 
/9 + (x^3*(378*A*a*b^2*e^5 + 378*B*a^2*b*e^5 - 18*A*b^3*d*e^4 + 16*B*b^3*d 
^2*e^3 - 54*B*a*b^2*d*e^4))/(315*b*e^5) + (x*(630*A*a^3*e^5 - 210*B*a^3*d* 
e^4 + 128*B*b^3*d^4*e - 144*A*b^3*d^3*e^2 + 504*A*a*b^2*d^2*e^3 - 432*B*a* 
b^2*d^3*e^2 + 504*B*a^2*b*d^2*e^3 - 630*A*a^2*b*d*e^4))/(315*b*e^5) + (x^2 
*(210*B*a^3*e^5 + 630*A*a^2*b*e^5 + 36*A*b^3*d^2*e^3 - 32*B*b^3*d^3*e^2 + 
108*B*a*b^2*d^2*e^3 - 126*A*a*b^2*d*e^4 - 126*B*a^2*b*d*e^4))/(315*b*e^5) 
+ (2*b*x^4*(9*A*b*e + 27*B*a*e - B*b*d))/(63*e)))/(x*(d + e*x)^(1/2) + (a* 
(d + e*x)^(1/2))/b)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.60 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {e x +d}\, \left (35 b^{4} e^{4} x^{4}+180 a \,b^{3} e^{4} x^{3}-40 b^{4} d \,e^{3} x^{3}+378 a^{2} b^{2} e^{4} x^{2}-216 a \,b^{3} d \,e^{3} x^{2}+48 b^{4} d^{2} e^{2} x^{2}+420 a^{3} b \,e^{4} x -504 a^{2} b^{2} d \,e^{3} x +288 a \,b^{3} d^{2} e^{2} x -64 b^{4} d^{3} e x +315 a^{4} e^{4}-840 a^{3} b d \,e^{3}+1008 a^{2} b^{2} d^{2} e^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{315 e^{5}} \] Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x)
 

Output:

(2*sqrt(d + e*x)*(315*a**4*e**4 - 840*a**3*b*d*e**3 + 420*a**3*b*e**4*x + 
1008*a**2*b**2*d**2*e**2 - 504*a**2*b**2*d*e**3*x + 378*a**2*b**2*e**4*x** 
2 - 576*a*b**3*d**3*e + 288*a*b**3*d**2*e**2*x - 216*a*b**3*d*e**3*x**2 + 
180*a*b**3*e**4*x**3 + 128*b**4*d**4 - 64*b**4*d**3*e*x + 48*b**4*d**2*e** 
2*x**2 - 40*b**4*d*e**3*x**3 + 35*b**4*e**4*x**4))/(315*e**5)