\(\int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx\) [143]

Optimal result

Integrand size = 22, antiderivative size = 171 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx=\frac {2 (b d-a e)^3 (B d-A e) \sqrt {d+e x}}{e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2}}{3 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2}}{5 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{7/2}}{7 e^5}+\frac {2 b^3 B (d+e x)^{9/2}}{9 e^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (105 a^3 e^3 (-2 B d+3 A e+B e x)+63 a^2 b e^2 \left (5 A e (-2 d+e x)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-9 a b^2 e \left (-7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )+3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )+b^3 \left (9 A e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{315 e^5} \] Input:

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

Output:

(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*(1 
28*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)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {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.

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(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)
 

Defintions of rubi rules used

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 2009

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

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (a e -d b \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 \left (a e -d b \right )^{2} b B +3 \left (a e -d b \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a e -d b \right )^{3} B +3 \left (a e -d b \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a e -d b \right )^{3} \left (A e -B d \right ) \sqrt {e x +d}}{e^{5}}\)	\(170\)
default	\(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (a e -d b \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 \left (a e -d b \right )^{2} b B +3 \left (a e -d b \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a e -d b \right )^{3} B +3 \left (a e -d b \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a e -d b \right )^{3} \left (A e -B d \right ) \sqrt {e x +d}}{e^{5}}\)	\(170\)
pseudoelliptic	\(\frac {2 \left (\left (\frac {x^{3} \left (\frac {7 B x}{9}+A \right ) b^{3}}{7}+\frac {3 a \,x^{2} \left (\frac {5 B x}{7}+A \right ) b^{2}}{5}+a^{2} x \left (\frac {3 B x}{5}+A \right ) b +a^{3} \left (\frac {B x}{3}+A \right )\right ) e^{4}-2 \left (\left (\frac {4}{63} B \,x^{3}+\frac {3}{35} A \,x^{2}\right ) b^{3}+\frac {2 \left (\frac {9 B x}{14}+A \right ) a x \,b^{2}}{5}+a^{2} \left (\frac {2 B x}{5}+A \right ) b +\frac {a^{3} B}{3}\right ) d \,e^{3}+\frac {8 b \left (\frac {\left (\frac {2 B x}{3}+A \right ) x \,b^{2}}{7}+a \left (\frac {3 B x}{7}+A \right ) b +a^{2} B \right ) d^{2} e^{2}}{5}-\frac {16 b^{2} \left (\left (\frac {4 B x}{9}+A \right ) b +3 B a \right ) d^{3} e}{35}+\frac {128 b^{3} B \,d^{4}}{315}\right ) \sqrt {e x +d}}{e^{5}}\)	\(191\)
gosper	\(\frac {2 \sqrt {e x +d}\, \left (35 B \,x^{4} b^{3} e^{4}+45 A \,x^{3} b^{3} e^{4}+135 B \,x^{3} a \,b^{2} e^{4}-40 B \,x^{3} b^{3} d \,e^{3}+189 A \,x^{2} a \,b^{2} e^{4}-54 A \,x^{2} b^{3} d \,e^{3}+189 B \,x^{2} a^{2} b \,e^{4}-162 B \,x^{2} a \,b^{2} d \,e^{3}+48 B \,x^{2} b^{3} d^{2} e^{2}+315 A x \,a^{2} b \,e^{4}-252 A x a \,b^{2} d \,e^{3}+72 A x \,b^{3} d^{2} e^{2}+105 B x \,a^{3} e^{4}-252 B x \,a^{2} b d \,e^{3}+216 B x a \,b^{2} d^{2} e^{2}-64 B x \,b^{3} d^{3} e +315 a^{3} A \,e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{315 e^{5}}\)	\(301\)
trager	\(\frac {2 \sqrt {e x +d}\, \left (35 B \,x^{4} b^{3} e^{4}+45 A \,x^{3} b^{3} e^{4}+135 B \,x^{3} a \,b^{2} e^{4}-40 B \,x^{3} b^{3} d \,e^{3}+189 A \,x^{2} a \,b^{2} e^{4}-54 A \,x^{2} b^{3} d \,e^{3}+189 B \,x^{2} a^{2} b \,e^{4}-162 B \,x^{2} a \,b^{2} d \,e^{3}+48 B \,x^{2} b^{3} d^{2} e^{2}+315 A x \,a^{2} b \,e^{4}-252 A x a \,b^{2} d \,e^{3}+72 A x \,b^{3} d^{2} e^{2}+105 B x \,a^{3} e^{4}-252 B x \,a^{2} b d \,e^{3}+216 B x a \,b^{2} d^{2} e^{2}-64 B x \,b^{3} d^{3} e +315 a^{3} A \,e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{315 e^{5}}\)	\(301\)
risch	\(\frac {2 \sqrt {e x +d}\, \left (35 B \,x^{4} b^{3} e^{4}+45 A \,x^{3} b^{3} e^{4}+135 B \,x^{3} a \,b^{2} e^{4}-40 B \,x^{3} b^{3} d \,e^{3}+189 A \,x^{2} a \,b^{2} e^{4}-54 A \,x^{2} b^{3} d \,e^{3}+189 B \,x^{2} a^{2} b \,e^{4}-162 B \,x^{2} a \,b^{2} d \,e^{3}+48 B \,x^{2} b^{3} d^{2} e^{2}+315 A x \,a^{2} b \,e^{4}-252 A x a \,b^{2} d \,e^{3}+72 A x \,b^{3} d^{2} e^{2}+105 B x \,a^{3} e^{4}-252 B x \,a^{2} b d \,e^{3}+216 B x a \,b^{2} d^{2} e^{2}-64 B x \,b^{3} d^{3} e +315 a^{3} A \,e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{315 e^{5}}\)	\(301\)
orering	\(\frac {2 \sqrt {e x +d}\, \left (35 B \,x^{4} b^{3} e^{4}+45 A \,x^{3} b^{3} e^{4}+135 B \,x^{3} a \,b^{2} e^{4}-40 B \,x^{3} b^{3} d \,e^{3}+189 A \,x^{2} a \,b^{2} e^{4}-54 A \,x^{2} b^{3} d \,e^{3}+189 B \,x^{2} a^{2} b \,e^{4}-162 B \,x^{2} a \,b^{2} d \,e^{3}+48 B \,x^{2} b^{3} d^{2} e^{2}+315 A x \,a^{2} b \,e^{4}-252 A x a \,b^{2} d \,e^{3}+72 A x \,b^{3} d^{2} e^{2}+105 B x \,a^{3} e^{4}-252 B x \,a^{2} b d \,e^{3}+216 B x a \,b^{2} d^{2} e^{2}-64 B x \,b^{3} d^{3} e +315 a^{3} A \,e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{315 e^{5}}\)	\(301\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (35 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} + 315 \, A a^{3} e^{4} - 144 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 504 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 210 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \, {\left (8 \, B b^{3} d e^{3} - 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{2} e^{2} - 18 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 63 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - {\left (64 \, B b^{3} d^{3} e - 72 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{5}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (168) = 336\).

Time = 1.06 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.47 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {B b^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{3} e + 3 B a b^{2} e - 4 B b^{3} d\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 A a b^{2} e^{2} - 3 A b^{3} d e + 3 B a^{2} b e^{2} - 9 B a b^{2} d e + 6 B b^{3} d^{2}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 A a^{2} b e^{3} - 6 A a b^{2} d e^{2} + 3 A b^{3} d^{2} e + B a^{3} e^{3} - 6 B a^{2} b d e^{2} + 9 B a b^{2} d^{2} e - 4 B b^{3} d^{3}\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (A a^{3} e^{4} - 3 A a^{2} b d e^{3} + 3 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - B a^{3} d e^{3} + 3 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e + B b^{3} d^{4}\right )}{e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {A a^{3} x + \frac {B b^{3} x^{5}}{5} + \frac {x^{4} \left (A b^{3} + 3 B a b^{2}\right )}{4} + \frac {x^{3} \cdot \left (3 A a b^{2} + 3 B a^{2} b\right )}{3} + \frac {x^{2} \cdot \left (3 A a^{2} b + B a^{3}\right )}{2}}{\sqrt {d}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*(B*b**3*(d + e*x)**(9/2)/(9*e**4) + (d + e*x)**(7/2)*(A*b**3* 
e + 3*B*a*b**2*e - 4*B*b**3*d)/(7*e**4) + (d + e*x)**(5/2)*(3*A*a*b**2*e** 
2 - 3*A*b**3*d*e + 3*B*a**2*b*e**2 - 9*B*a*b**2*d*e + 6*B*b**3*d**2)/(5*e* 
*4) + (d + e*x)**(3/2)*(3*A*a**2*b*e**3 - 6*A*a*b**2*d*e**2 + 3*A*b**3*d** 
2*e + B*a**3*e**3 - 6*B*a**2*b*d*e**2 + 9*B*a*b**2*d**2*e - 4*B*b**3*d**3) 
/(3*e**4) + sqrt(d + e*x)*(A*a**3*e**4 - 3*A*a**2*b*d*e**3 + 3*A*a*b**2*d* 
*2*e**2 - A*b**3*d**3*e - B*a**3*d*e**3 + 3*B*a**2*b*d**2*e**2 - 3*B*a*b** 
2*d**3*e + B*b**3*d**4)/e**4)/e, Ne(e, 0)), ((A*a**3*x + B*b**3*x**5/5 + x 
**4*(A*b**3 + 3*B*a*b**2)/4 + x**3*(3*A*a*b**2 + 3*B*a**2*b)/3 + x**2*(3*A 
*a**2*b + B*a**3)/2)/sqrt(d), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.55 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{3} - 45 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \sqrt {e x + d}\right )}}{315 \, e^{5}} \] Input:

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

Output:

2/315*(35*(e*x + d)^(9/2)*B*b^3 - 45*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*( 
e*x + d)^(7/2) + 189*(2*B*b^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A 
*a*b^2)*e^2)*(e*x + d)^(5/2) - 105*(4*B*b^3*d^3 - 3*(3*B*a*b^2 + A*b^3)*d^ 
2*e + 6*(B*a^2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d)^(3/ 
2) + 315*(B*b^3*d^4 + A*a^3*e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + 
 A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3)*sqrt(e*x + d))/e^5
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (153) = 306\).

Time = 0.12 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} A a^{3} + \frac {105 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} B a^{3}}{e} + \frac {315 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} A a^{2} b}{e} + \frac {63 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} B a^{2} b}{e^{2}} + \frac {63 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} A a b^{2}}{e^{2}} + \frac {27 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} B a b^{2}}{e^{3}} + \frac {9 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} A b^{3}}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} B b^{3}}{e^{4}}\right )}}{315 \, e} \] Input:

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

Output:

2/315*(315*sqrt(e*x + d)*A*a^3 + 105*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d) 
*B*a^3/e + 315*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^2*b/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/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/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/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/e^3 + 
(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 42 
0*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*B*b^3/e^4)/e
 

Mupad [B] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{7\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{3\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{5\,e^5} \] Input:

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

Output:

((d + e*x)^(7/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(7*e^5) + (2*(a*e 
- b*d)^2*(d + e*x)^(3/2)*(3*A*b*e + B*a*e - 4*B*b*d))/(3*e^5) + (2*B*b^3*( 
d + e*x)^(9/2))/(9*e^5) + (2*(A*e - B*d)*(a*e - b*d)^3*(d + e*x)^(1/2))/e^ 
5 + (6*b*(a*e - b*d)*(d + e*x)^(5/2)*(A*b*e + B*a*e - 2*B*b*d))/(5*e^5)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {e x +d}\, \left (35 b^{4} e^{4} x^{4}+180 a \,b^{3} e^{4} x^{3}-40 b^{4} d \,e^{3} x^{3}+378 a^{2} b^{2} e^{4} x^{2}-216 a \,b^{3} d \,e^{3} x^{2}+48 b^{4} d^{2} e^{2} x^{2}+420 a^{3} b \,e^{4} x -504 a^{2} b^{2} d \,e^{3} x +288 a \,b^{3} d^{2} e^{2} x -64 b^{4} d^{3} e x +315 a^{4} e^{4}-840 a^{3} b d \,e^{3}+1008 a^{2} b^{2} d^{2} e^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{315 e^{5}} \] Input:

int((b*x+a)^3*(B*x+A)/(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)