Integrand size = 22, antiderivative size = 173 \[ \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{3/2}}{3 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{5/2}}{5 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{7/2}}{7 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{9/2}}{9 e^5}+\frac {2 b^3 B (d+e x)^{11/2}}{11 e^5} \] Output:
2/3*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^(3/2)/e^5-2/5*(-a*e+b*d)^2*(-3*A*b*e-B *a*e+4*B*b*d)*(e*x+d)^(5/2)/e^5+6/7*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e *x+d)^(7/2)/e^5-2/9*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(9/2)/e^5+2/11*b^ 3*B*(e*x+d)^(11/2)/e^5
Time = 0.14 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.31 \[ \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx=\frac {2 (d+e x)^{3/2} \left (231 a^3 e^3 (-2 B d+5 A e+3 B e x)+99 a^2 b e^2 \left (7 A e (-2 d+3 e x)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-33 a b^2 e \left (-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )\right )+b^3 \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )\right )}{3465 e^5} \] Input:
Integrate[(a + b*x)^3*(A + B*x)*Sqrt[d + e*x],x]
Output:
(2*(d + e*x)^(3/2)*(231*a^3*e^3*(-2*B*d + 5*A*e + 3*B*e*x) + 99*a^2*b*e^2* (7*A*e*(-2*d + 3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 33*a*b^2*e*(- 3*A*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*(16*d^3 - 24*d^2*e*x + 30*d*e^2* x^2 - 35*e^3*x^3)) + b^3*(11*A*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35 *e^3*x^3) + B*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 3 15*e^4*x^4))))/(3465*e^5)
Time = 0.32 (sec) , antiderivative size = 173, 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.
| \(\displaystyle \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^2 (d+e x)^{7/2} (3 a B e+A b e-4 b B d)}{e^4}-\frac {3 b (d+e x)^{5/2} (b d-a e) (a B e+A b e-2 b B d)}{e^4}+\frac {(d+e x)^{3/2} (a e-b d)^2 (a B e+3 A b e-4 b B d)}{e^4}+\frac {\sqrt {d+e x} (a e-b d)^3 (A e-B d)}{e^4}+\frac {b^3 B (d+e x)^{9/2}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^2 (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac {6 b (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac {2 (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5}+\frac {2 (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5}+\frac {2 b^3 B (d+e x)^{11/2}}{11 e^5}\) |
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)*(d + e*x)^(3/2))/(3*e^5) - (2*(b*d - a*e)^2*( 4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^5) + (6*b*(b*d - a*e)*(2* b*B*d - A*b*e - a*B*e)*(d + e*x)^(7/2))/(7*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(9/2))/(9*e^5) + (2*b^3*B*(d + e*x)^(11/2))/(11*e^5)
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 = 0.70 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (a e -d b \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (a e -d b \right )^{2} b B +3 \left (a e -d b \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -d b \right )^{3} B +3 \left (a e -d b \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -d b \right )^{3} \left (A e -B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(171\) |
| default | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (a e -d b \right ) b^{2} B +b^{3} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (a e -d b \right )^{2} b B +3 \left (a e -d b \right ) b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -d b \right )^{3} B +3 \left (a e -d b \right )^{2} b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -d b \right )^{3} \left (A e -B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(171\) |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {x^{3} \left (\frac {9 B x}{11}+A \right ) b^{3}}{3}+\frac {9 a \,x^{2} \left (\frac {7 B x}{9}+A \right ) b^{2}}{7}+\frac {9 a^{2} x \left (\frac {5 B x}{7}+A \right ) b}{5}+a^{3} \left (\frac {3 B x}{5}+A \right )\right ) e^{4}-\frac {6 \left (\frac {5 \left (\frac {28 B x}{33}+A \right ) x^{2} b^{3}}{21}+\frac {6 a \left (\frac {5 B x}{6}+A \right ) x \,b^{2}}{7}+a^{2} \left (\frac {6 B x}{7}+A \right ) b +\frac {a^{3} B}{3}\right ) d \,e^{3}}{5}+\frac {24 \left (\frac {\left (\frac {10 B x}{11}+A \right ) x \,b^{2}}{3}+a \left (B x +A \right ) b +a^{2} B \right ) b \,d^{2} e^{2}}{35}-\frac {16 b^{2} \left (\left (\frac {12 B x}{11}+A \right ) b +3 B a \right ) d^{3} e}{105}+\frac {128 b^{3} B \,d^{4}}{1155}\right )}{3 e^{5}}\) | \(188\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 B \,x^{4} b^{3} e^{4}+385 A \,x^{3} b^{3} e^{4}+1155 B \,x^{3} a \,b^{2} e^{4}-280 B \,x^{3} b^{3} d \,e^{3}+1485 A \,x^{2} a \,b^{2} e^{4}-330 A \,x^{2} b^{3} d \,e^{3}+1485 B \,x^{2} a^{2} b \,e^{4}-990 B \,x^{2} a \,b^{2} d \,e^{3}+240 B \,x^{2} b^{3} d^{2} e^{2}+2079 A x \,a^{2} b \,e^{4}-1188 A x a \,b^{2} d \,e^{3}+264 A x \,b^{3} d^{2} e^{2}+693 B x \,a^{3} e^{4}-1188 B x \,a^{2} b d \,e^{3}+792 B x a \,b^{2} d^{2} e^{2}-192 B x \,b^{3} d^{3} e +1155 a^{3} A \,e^{4}-1386 A \,a^{2} b d \,e^{3}+792 A a \,b^{2} d^{2} e^{2}-176 A \,b^{3} d^{3} e -462 B \,a^{3} d \,e^{3}+792 B \,a^{2} b \,d^{2} e^{2}-528 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{3465 e^{5}}\) | \(301\) |
| orering | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 B \,x^{4} b^{3} e^{4}+385 A \,x^{3} b^{3} e^{4}+1155 B \,x^{3} a \,b^{2} e^{4}-280 B \,x^{3} b^{3} d \,e^{3}+1485 A \,x^{2} a \,b^{2} e^{4}-330 A \,x^{2} b^{3} d \,e^{3}+1485 B \,x^{2} a^{2} b \,e^{4}-990 B \,x^{2} a \,b^{2} d \,e^{3}+240 B \,x^{2} b^{3} d^{2} e^{2}+2079 A x \,a^{2} b \,e^{4}-1188 A x a \,b^{2} d \,e^{3}+264 A x \,b^{3} d^{2} e^{2}+693 B x \,a^{3} e^{4}-1188 B x \,a^{2} b d \,e^{3}+792 B x a \,b^{2} d^{2} e^{2}-192 B x \,b^{3} d^{3} e +1155 a^{3} A \,e^{4}-1386 A \,a^{2} b d \,e^{3}+792 A a \,b^{2} d^{2} e^{2}-176 A \,b^{3} d^{3} e -462 B \,a^{3} d \,e^{3}+792 B \,a^{2} b \,d^{2} e^{2}-528 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{3465 e^{5}}\) | \(301\) |
| trager | \(\frac {2 \left (315 b^{3} B \,e^{5} x^{5}+385 A \,b^{3} e^{5} x^{4}+1155 B a \,b^{2} e^{5} x^{4}+35 b^{3} B d \,e^{4} x^{4}+1485 A a \,b^{2} e^{5} x^{3}+55 A \,b^{3} d \,e^{4} x^{3}+1485 B \,a^{2} b \,e^{5} x^{3}+165 B a \,b^{2} d \,e^{4} x^{3}-40 b^{3} B \,d^{2} e^{3} x^{3}+2079 A \,a^{2} b \,e^{5} x^{2}+297 A a \,b^{2} d \,e^{4} x^{2}-66 A \,b^{3} d^{2} e^{3} x^{2}+693 B \,a^{3} e^{5} x^{2}+297 B \,a^{2} b d \,e^{4} x^{2}-198 B a \,b^{2} d^{2} e^{3} x^{2}+48 b^{3} B \,d^{3} e^{2} x^{2}+1155 a^{3} A \,e^{5} x +693 A \,a^{2} b d \,e^{4} x -396 A a \,b^{2} d^{2} e^{3} x +88 A \,b^{3} d^{3} e^{2} x +231 B \,a^{3} d \,e^{4} x -396 B \,a^{2} b \,d^{2} e^{3} x +264 B a \,b^{2} d^{3} e^{2} x -64 b^{3} B \,d^{4} e x +1155 a^{3} A d \,e^{4}-1386 A \,a^{2} b \,d^{2} e^{3}+792 A a \,b^{2} d^{3} e^{2}-176 A \,b^{3} d^{4} e -462 B \,a^{3} d^{2} e^{3}+792 B \,a^{2} b \,d^{3} e^{2}-528 B a \,b^{2} d^{4} e +128 b^{3} B \,d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(421\) |
| risch | \(\frac {2 \left (315 b^{3} B \,e^{5} x^{5}+385 A \,b^{3} e^{5} x^{4}+1155 B a \,b^{2} e^{5} x^{4}+35 b^{3} B d \,e^{4} x^{4}+1485 A a \,b^{2} e^{5} x^{3}+55 A \,b^{3} d \,e^{4} x^{3}+1485 B \,a^{2} b \,e^{5} x^{3}+165 B a \,b^{2} d \,e^{4} x^{3}-40 b^{3} B \,d^{2} e^{3} x^{3}+2079 A \,a^{2} b \,e^{5} x^{2}+297 A a \,b^{2} d \,e^{4} x^{2}-66 A \,b^{3} d^{2} e^{3} x^{2}+693 B \,a^{3} e^{5} x^{2}+297 B \,a^{2} b d \,e^{4} x^{2}-198 B a \,b^{2} d^{2} e^{3} x^{2}+48 b^{3} B \,d^{3} e^{2} x^{2}+1155 a^{3} A \,e^{5} x +693 A \,a^{2} b d \,e^{4} x -396 A a \,b^{2} d^{2} e^{3} x +88 A \,b^{3} d^{3} e^{2} x +231 B \,a^{3} d \,e^{4} x -396 B \,a^{2} b \,d^{2} e^{3} x +264 B a \,b^{2} d^{3} e^{2} x -64 b^{3} B \,d^{4} e x +1155 a^{3} A d \,e^{4}-1386 A \,a^{2} b \,d^{2} e^{3}+792 A a \,b^{2} d^{3} e^{2}-176 A \,b^{3} d^{4} e -462 B \,a^{3} d^{2} e^{3}+792 B \,a^{2} b \,d^{3} e^{2}-528 B a \,b^{2} d^{4} e +128 b^{3} B \,d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(421\) |
Input:
int((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/e^5*(1/11*b^3*B*(e*x+d)^(11/2)+1/9*(3*(a*e-b*d)*b^2*B+b^3*(A*e-B*d))*(e* x+d)^(9/2)+1/7*(3*(a*e-b*d)^2*b*B+3*(a*e-b*d)*b^2*(A*e-B*d))*(e*x+d)^(7/2) +1/5*((a*e-b*d)^3*B+3*(a*e-b*d)^2*b*(A*e-B*d))*(e*x+d)^(5/2)+1/3*(a*e-b*d) ^3*(A*e-B*d)*(e*x+d)^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (153) = 306\).
Time = 0.09 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.04 \[ \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx=\frac {2 \, {\left (315 \, B b^{3} e^{5} x^{5} + 128 \, B b^{3} d^{5} + 1155 \, A a^{3} d e^{4} - 176 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 792 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} - 462 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3} + 35 \, {\left (B b^{3} d e^{4} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{2} e^{3} - 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} - 297 \, {\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{3} e^{2} - 22 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 99 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{4} + 231 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{4} e - 1155 \, A a^{3} e^{5} - 88 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 396 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} - 231 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \] Input:
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x, algorithm="fricas")
Output:
2/3465*(315*B*b^3*e^5*x^5 + 128*B*b^3*d^5 + 1155*A*a^3*d*e^4 - 176*(3*B*a* b^2 + A*b^3)*d^4*e + 792*(B*a^2*b + A*a*b^2)*d^3*e^2 - 462*(B*a^3 + 3*A*a^ 2*b)*d^2*e^3 + 35*(B*b^3*d*e^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*x^4 - 5*(8*B* b^3*d^2*e^3 - 11*(3*B*a*b^2 + A*b^3)*d*e^4 - 297*(B*a^2*b + A*a*b^2)*e^5)* x^3 + 3*(16*B*b^3*d^3*e^2 - 22*(3*B*a*b^2 + A*b^3)*d^2*e^3 + 99*(B*a^2*b + A*a*b^2)*d*e^4 + 231*(B*a^3 + 3*A*a^2*b)*e^5)*x^2 - (64*B*b^3*d^4*e - 115 5*A*a^3*e^5 - 88*(3*B*a*b^2 + A*b^3)*d^3*e^2 + 396*(B*a^2*b + A*a*b^2)*d^2 *e^3 - 231*(B*a^3 + 3*A*a^2*b)*d*e^4)*x)*sqrt(e*x + d)/e^5
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (170) = 340\).
Time = 1.12 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.45 \[ \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx=\begin {cases} \frac {2 \left (\frac {B b^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (A b^{3} e + 3 B a b^{2} e - 4 B b^{3} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 A a b^{2} e^{2} - 3 A b^{3} d e + 3 B a^{2} b e^{2} - 9 B a b^{2} d e + 6 B b^{3} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 A a^{2} b e^{3} - 6 A a b^{2} d e^{2} + 3 A b^{3} d^{2} e + B a^{3} e^{3} - 6 B a^{2} b d e^{2} + 9 B a b^{2} d^{2} e - 4 B b^{3} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{3} e^{4} - 3 A a^{2} b d e^{3} + 3 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - B a^{3} d e^{3} + 3 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e + B b^{3} d^{4}\right )}{3 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (A a^{3} x + \frac {B b^{3} x^{5}}{5} + \frac {x^{4} \left (A b^{3} + 3 B a b^{2}\right )}{4} + \frac {x^{3} \cdot \left (3 A a b^{2} + 3 B a^{2} b\right )}{3} + \frac {x^{2} \cdot \left (3 A a^{2} b + B a^{3}\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(1/2),x)
Output:
Piecewise((2*(B*b**3*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(9/2)*(A*b** 3*e + 3*B*a*b**2*e - 4*B*b**3*d)/(9*e**4) + (d + e*x)**(7/2)*(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)/(7* e**4) + (d + e*x)**(5/2)*(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)/(5*e**4) + (d + e*x)**(3/2)*(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)/(3*e**4))/e, Ne(e, 0)), (sqrt(d)*(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), True))
Time = 0.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.53 \[ \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B b^{3} - 385 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 1485 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \] Input:
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x, algorithm="maxima")
Output:
2/3465*(315*(e*x + d)^(11/2)*B*b^3 - 385*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)* e)*(e*x + d)^(9/2) + 1485*(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)^(7/2) - 693*(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 )^(5/2) + 1155*(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)*(e*x + d)^(3/2))/e^5
Leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (153) = 306\).
Time = 0.13 (sec) , antiderivative size = 761, normalized size of antiderivative = 4.40 \[ \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x, algorithm="giac")
Output:
2/3465*(3465*sqrt(e*x + d)*A*a^3*d + 1155*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^3 + 1155*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^3*d/e + 3465* ((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^2*b*d/e + 693*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^2*b*d/e^2 + 693*(3*(e* x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a*b^2*d/e^2 + 231*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B* a^3/e + 693*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d ^2)*A*a^2*b/e + 297*(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*d/e^3 + 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)*A *b^3*d/e^3 + 297*(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*b/e^2 + 297*(5*(e*x + d)^(7/2) - 2 1*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*a*b ^2/e^2 + 11*(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*d/e^4 + 3 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*a*b^2/e^3 + 11*(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)*A*b^3/e^3 + 5*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2...
Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.89 \[ \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{9\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{5\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{7\,e^5} \] Input:
int((A + B*x)*(a + b*x)^3*(d + e*x)^(1/2),x)
Output:
((d + e*x)^(9/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(9*e^5) + (2*(a*e - b*d)^2*(d + e*x)^(5/2)*(3*A*b*e + B*a*e - 4*B*b*d))/(5*e^5) + (2*B*b^3*( d + e*x)^(11/2))/(11*e^5) + (2*(A*e - B*d)*(a*e - b*d)^3*(d + e*x)^(3/2))/ (3*e^5) + (6*b*(a*e - b*d)*(d + e*x)^(7/2)*(A*b*e + B*a*e - 2*B*b*d))/(7*e ^5)
Time = 0.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.47 \[ \int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx=\frac {2 \sqrt {e x +d}\, \left (315 b^{4} e^{5} x^{5}+1540 a \,b^{3} e^{5} x^{4}+35 b^{4} d \,e^{4} x^{4}+2970 a^{2} b^{2} e^{5} x^{3}+220 a \,b^{3} d \,e^{4} x^{3}-40 b^{4} d^{2} e^{3} x^{3}+2772 a^{3} b \,e^{5} x^{2}+594 a^{2} b^{2} d \,e^{4} x^{2}-264 a \,b^{3} d^{2} e^{3} x^{2}+48 b^{4} d^{3} e^{2} x^{2}+1155 a^{4} e^{5} x +924 a^{3} b d \,e^{4} x -792 a^{2} b^{2} d^{2} e^{3} x +352 a \,b^{3} d^{3} e^{2} x -64 b^{4} d^{4} e x +1155 a^{4} d \,e^{4}-1848 a^{3} b \,d^{2} e^{3}+1584 a^{2} b^{2} d^{3} e^{2}-704 a \,b^{3} d^{4} e +128 b^{4} d^{5}\right )}{3465 e^{5}} \] Input:
int((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x)
Output:
(2*sqrt(d + e*x)*(1155*a**4*d*e**4 + 1155*a**4*e**5*x - 1848*a**3*b*d**2*e **3 + 924*a**3*b*d*e**4*x + 2772*a**3*b*e**5*x**2 + 1584*a**2*b**2*d**3*e* *2 - 792*a**2*b**2*d**2*e**3*x + 594*a**2*b**2*d*e**4*x**2 + 2970*a**2*b** 2*e**5*x**3 - 704*a*b**3*d**4*e + 352*a*b**3*d**3*e**2*x - 264*a*b**3*d**2 *e**3*x**2 + 220*a*b**3*d*e**4*x**3 + 1540*a*b**3*e**5*x**4 + 128*b**4*d** 5 - 64*b**4*d**4*e*x + 48*b**4*d**3*e**2*x**2 - 40*b**4*d**2*e**3*x**3 + 3 5*b**4*d*e**4*x**4 + 315*b**4*e**5*x**5))/(3465*e**5)