Integrand size = 22, antiderivative size = 128 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{5/2}}{5 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{7/2}}{7 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{9/2}}{9 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4} \] Output:
-2/5*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(5/2)/e^4+2/7*(-a*e+b*d)*(-2*A*b*e-B* a*e+3*B*b*d)*(e*x+d)^(7/2)/e^4-2/9*b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(9/2 )/e^4+2/11*b^2*B*(e*x+d)^(11/2)/e^4
Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {2 (d+e x)^{5/2} \left (99 a^2 e^2 (-2 B d+7 A e+5 B e x)+22 a b e \left (9 A e (-2 d+5 e x)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+b^2 \left (11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \] Input:
Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^(3/2),x]
Output:
(2*(d + e*x)^(5/2)*(99*a^2*e^2*(-2*B*d + 7*A*e + 5*B*e*x) + 22*a*b*e*(9*A* e*(-2*d + 5*e*x) + B*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) + b^2*(11*A*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) - 3*B*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105* e^3*x^3))))/(3465*e^4)
Time = 0.27 (sec) , antiderivative size = 128, 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.
| \(\displaystyle \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b (d+e x)^{7/2} (2 a B e+A b e-3 b B d)}{e^3}+\frac {(d+e x)^{5/2} (a e-b d) (a B e+2 A b e-3 b B d)}{e^3}+\frac {(d+e x)^{3/2} (a e-b d)^2 (A e-B d)}{e^3}+\frac {b^2 B (d+e x)^{9/2}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac {2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac {2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4}\) |
Input:
Int[(a + b*x)^2*(A + B*x)*(d + e*x)^(3/2),x]
Output:
(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^4) + (2*(b*d - a*e)*(3 *b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(7/2))/(7*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(9/2))/(9*e^4) + (2*b^2*B*(d + e*x)^(11/2))/(11*e^4)
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]))))
Time = 0.67 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (\left (\frac {5 x^{2} \left (\frac {9 B x}{11}+A \right ) b^{2}}{9}+\frac {10 a x \left (\frac {7 B x}{9}+A \right ) b}{7}+a^{2} \left (\frac {5 B x}{7}+A \right )\right ) e^{3}-\frac {4 \left (\frac {5 x \left (\frac {21 B x}{22}+A \right ) b^{2}}{9}+a \left (\frac {10 B x}{9}+A \right ) b +\frac {a^{2} B}{2}\right ) d \,e^{2}}{7}+\frac {8 \left (\left (\frac {15 B x}{11}+A \right ) b +2 B a \right ) b \,d^{2} e}{63}-\frac {16 b^{2} B \,d^{3}}{231}\right )}{5 e^{4}}\) | \(117\) |
| derivativedivides | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 b \left (a e -d b \right ) B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -d b \right )^{2} B +2 b \left (a e -d b \right ) \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -d b \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(122\) |
| default | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 b \left (a e -d b \right ) B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -d b \right )^{2} B +2 b \left (a e -d b \right ) \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -d b \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(122\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (315 b^{2} B \,x^{3} e^{3}+385 A \,x^{2} b^{2} e^{3}+770 B \,x^{2} a b \,e^{3}-210 B \,x^{2} b^{2} d \,e^{2}+990 A x a b \,e^{3}-220 A x \,b^{2} d \,e^{2}+495 B x \,a^{2} e^{3}-440 B x a b d \,e^{2}+120 B x \,b^{2} d^{2} e +693 a^{2} A \,e^{3}-396 A a b d \,e^{2}+88 A \,b^{2} d^{2} e -198 B \,a^{2} d \,e^{2}+176 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{3465 e^{4}}\) | \(169\) |
| orering | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (315 b^{2} B \,x^{3} e^{3}+385 A \,x^{2} b^{2} e^{3}+770 B \,x^{2} a b \,e^{3}-210 B \,x^{2} b^{2} d \,e^{2}+990 A x a b \,e^{3}-220 A x \,b^{2} d \,e^{2}+495 B x \,a^{2} e^{3}-440 B x a b d \,e^{2}+120 B x \,b^{2} d^{2} e +693 a^{2} A \,e^{3}-396 A a b d \,e^{2}+88 A \,b^{2} d^{2} e -198 B \,a^{2} d \,e^{2}+176 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{3465 e^{4}}\) | \(169\) |
| trager | \(\frac {2 \left (315 B \,b^{2} e^{5} x^{5}+385 A \,b^{2} e^{5} x^{4}+770 B a b \,e^{5} x^{4}+420 B \,b^{2} d \,e^{4} x^{4}+990 A a b \,e^{5} x^{3}+550 A \,b^{2} d \,e^{4} x^{3}+495 B \,a^{2} e^{5} x^{3}+1100 B a b d \,e^{4} x^{3}+15 B \,b^{2} d^{2} e^{3} x^{3}+693 A \,a^{2} e^{5} x^{2}+1584 A a b d \,e^{4} x^{2}+33 A \,b^{2} d^{2} e^{3} x^{2}+792 B \,a^{2} d \,e^{4} x^{2}+66 B a b \,d^{2} e^{3} x^{2}-18 B \,b^{2} d^{3} e^{2} x^{2}+1386 A \,a^{2} d \,e^{4} x +198 A a b \,d^{2} e^{3} x -44 A \,b^{2} d^{3} e^{2} x +99 B \,a^{2} d^{2} e^{3} x -88 B a b \,d^{3} e^{2} x +24 B \,b^{2} d^{4} e x +693 A \,a^{2} d^{2} e^{3}-396 A a b \,d^{3} e^{2}+88 A \,b^{2} d^{4} e -198 B \,a^{2} d^{3} e^{2}+176 B a b \,d^{4} e -48 B \,b^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\) | \(341\) |
| risch | \(\frac {2 \left (315 B \,b^{2} e^{5} x^{5}+385 A \,b^{2} e^{5} x^{4}+770 B a b \,e^{5} x^{4}+420 B \,b^{2} d \,e^{4} x^{4}+990 A a b \,e^{5} x^{3}+550 A \,b^{2} d \,e^{4} x^{3}+495 B \,a^{2} e^{5} x^{3}+1100 B a b d \,e^{4} x^{3}+15 B \,b^{2} d^{2} e^{3} x^{3}+693 A \,a^{2} e^{5} x^{2}+1584 A a b d \,e^{4} x^{2}+33 A \,b^{2} d^{2} e^{3} x^{2}+792 B \,a^{2} d \,e^{4} x^{2}+66 B a b \,d^{2} e^{3} x^{2}-18 B \,b^{2} d^{3} e^{2} x^{2}+1386 A \,a^{2} d \,e^{4} x +198 A a b \,d^{2} e^{3} x -44 A \,b^{2} d^{3} e^{2} x +99 B \,a^{2} d^{2} e^{3} x -88 B a b \,d^{3} e^{2} x +24 B \,b^{2} d^{4} e x +693 A \,a^{2} d^{2} e^{3}-396 A a b \,d^{3} e^{2}+88 A \,b^{2} d^{4} e -198 B \,a^{2} d^{3} e^{2}+176 B a b \,d^{4} e -48 B \,b^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\) | \(341\) |
Input:
int((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/5*(e*x+d)^(5/2)*((5/9*x^2*(9/11*B*x+A)*b^2+10/7*a*x*(7/9*B*x+A)*b+a^2*(5 /7*B*x+A))*e^3-4/7*(5/9*x*(21/22*B*x+A)*b^2+a*(10/9*B*x+A)*b+1/2*a^2*B)*d* e^2+8/63*((15/11*B*x+A)*b+2*B*a)*b*d^2*e-16/231*b^2*B*d^3)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (112) = 224\).
Time = 0.10 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.26 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {2 \, {\left (315 \, B b^{2} e^{5} x^{5} - 48 \, B b^{2} d^{5} + 693 \, A a^{2} d^{2} e^{3} + 88 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e - 198 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2} + 35 \, {\left (12 \, B b^{2} d e^{4} + 11 \, {\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{4} + 5 \, {\left (3 \, B b^{2} d^{2} e^{3} + 110 \, {\left (2 \, B a b + A b^{2}\right )} d e^{4} + 99 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{3} - 3 \, {\left (6 \, B b^{2} d^{3} e^{2} - 231 \, A a^{2} e^{5} - 11 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} - 264 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{2} + {\left (24 \, B b^{2} d^{4} e + 1386 \, A a^{2} d e^{4} - 44 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 99 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{4}} \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x, algorithm="fricas")
Output:
2/3465*(315*B*b^2*e^5*x^5 - 48*B*b^2*d^5 + 693*A*a^2*d^2*e^3 + 88*(2*B*a*b + A*b^2)*d^4*e - 198*(B*a^2 + 2*A*a*b)*d^3*e^2 + 35*(12*B*b^2*d*e^4 + 11* (2*B*a*b + A*b^2)*e^5)*x^4 + 5*(3*B*b^2*d^2*e^3 + 110*(2*B*a*b + A*b^2)*d* e^4 + 99*(B*a^2 + 2*A*a*b)*e^5)*x^3 - 3*(6*B*b^2*d^3*e^2 - 231*A*a^2*e^5 - 11*(2*B*a*b + A*b^2)*d^2*e^3 - 264*(B*a^2 + 2*A*a*b)*d*e^4)*x^2 + (24*B*b ^2*d^4*e + 1386*A*a^2*d*e^4 - 44*(2*B*a*b + A*b^2)*d^3*e^2 + 99*(B*a^2 + 2 *A*a*b)*d^2*e^3)*x)*sqrt(e*x + d)/e^4
Time = 1.11 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.01 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\begin {cases} \frac {2 \left (\frac {B b^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{3}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (A b^{2} e + 2 B a b e - 3 B b^{2} d\right )}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} - 4 B a b d e + 3 B b^{2} d^{2}\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A a^{2} e^{3} - 2 A a b d e^{2} + A b^{2} d^{2} e - B a^{2} d e^{2} + 2 B a b d^{2} e - B b^{2} d^{3}\right )}{5 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (A a^{2} x + \frac {B b^{2} x^{4}}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**2*(B*x+A)*(e*x+d)**(3/2),x)
Output:
Piecewise((2*(B*b**2*(d + e*x)**(11/2)/(11*e**3) + (d + e*x)**(9/2)*(A*b** 2*e + 2*B*a*b*e - 3*B*b**2*d)/(9*e**3) + (d + e*x)**(7/2)*(2*A*a*b*e**2 - 2*A*b**2*d*e + B*a**2*e**2 - 4*B*a*b*d*e + 3*B*b**2*d**2)/(7*e**3) + (d + e*x)**(5/2)*(A*a**2*e**3 - 2*A*a*b*d*e**2 + A*b**2*d**2*e - B*a**2*d*e**2 + 2*B*a*b*d**2*e - B*b**2*d**3)/(5*e**3))/e, Ne(e, 0)), (d**(3/2)*(A*a**2* x + B*b**2*x**4/4 + x**3*(A*b**2 + 2*B*a*b)/3 + x**2*(2*A*a*b + B*a**2)/2) , True))
Time = 0.04 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.24 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B b^{2} - 385 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{3465 \, e^{4}} \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x, algorithm="maxima")
Output:
2/3465*(315*(e*x + d)^(11/2)*B*b^2 - 385*(3*B*b^2*d - (2*B*a*b + A*b^2)*e) *(e*x + d)^(9/2) + 495*(3*B*b^2*d^2 - 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2 *A*a*b)*e^2)*(e*x + d)^(7/2) - 693*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b + A*b ^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2)*(e*x + d)^(5/2))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (112) = 224\).
Time = 0.13 (sec) , antiderivative size = 854, normalized size of antiderivative = 6.67 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x, algorithm="giac")
Output:
2/3465*(3465*sqrt(e*x + d)*A*a^2*d^2 + 2310*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^2*d + 1155*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^2*d^2/e + 2310*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a*b*d^2/e + 231*(3*(e*x + d) ^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^2 + 462*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a*b*d^2/e^2 + 231*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*b^ 2*d^2/e^2 + 462*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^2*d/e + 924*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqr t(e*x + d)*d^2)*A*a*b*d/e + 99*(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*b^2*d^2/e^3 + 396*(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*d/e^2 + 198*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 3 5*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*b^2*d/e^2 + 99*(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^2/e + 198*(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*a*b/e + 22*(35*(e*x + d)^(9/2) - 18 0*(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^2*d/e^3 + 22*(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*a*b/e^2 + 11*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d...
Time = 0.98 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{9\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{7\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \] Input:
int((A + B*x)*(a + b*x)^2*(d + e*x)^(3/2),x)
Output:
((d + e*x)^(9/2)*(2*A*b^2*e - 6*B*b^2*d + 4*B*a*b*e))/(9*e^4) + (2*B*b^2*( d + e*x)^(11/2))/(11*e^4) + (2*(a*e - b*d)*(d + e*x)^(7/2)*(2*A*b*e + B*a* e - 3*B*b*d))/(7*e^4) + (2*(A*e - B*d)*(a*e - b*d)^2*(d + e*x)^(5/2))/(5*e ^4)
Time = 0.15 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.77 \[ \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx=\frac {2 \sqrt {e x +d}\, \left (105 b^{3} e^{5} x^{5}+385 a \,b^{2} e^{5} x^{4}+140 b^{3} d \,e^{4} x^{4}+495 a^{2} b \,e^{5} x^{3}+550 a \,b^{2} d \,e^{4} x^{3}+5 b^{3} d^{2} e^{3} x^{3}+231 a^{3} e^{5} x^{2}+792 a^{2} b d \,e^{4} x^{2}+33 a \,b^{2} d^{2} e^{3} x^{2}-6 b^{3} d^{3} e^{2} x^{2}+462 a^{3} d \,e^{4} x +99 a^{2} b \,d^{2} e^{3} x -44 a \,b^{2} d^{3} e^{2} x +8 b^{3} d^{4} e x +231 a^{3} d^{2} e^{3}-198 a^{2} b \,d^{3} e^{2}+88 a \,b^{2} d^{4} e -16 b^{3} d^{5}\right )}{1155 e^{4}} \] Input:
int((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x)
Output:
(2*sqrt(d + e*x)*(231*a**3*d**2*e**3 + 462*a**3*d*e**4*x + 231*a**3*e**5*x **2 - 198*a**2*b*d**3*e**2 + 99*a**2*b*d**2*e**3*x + 792*a**2*b*d*e**4*x** 2 + 495*a**2*b*e**5*x**3 + 88*a*b**2*d**4*e - 44*a*b**2*d**3*e**2*x + 33*a *b**2*d**2*e**3*x**2 + 550*a*b**2*d*e**4*x**3 + 385*a*b**2*e**5*x**4 - 16* b**3*d**5 + 8*b**3*d**4*e*x - 6*b**3*d**3*e**2*x**2 + 5*b**3*d**2*e**3*x** 3 + 140*b**3*d*e**4*x**4 + 105*b**3*e**5*x**5))/(1155*e**4)