Integrand size = 22, antiderivative size = 167 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2 (b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^{7/2}}+\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{5 e^5 (d+e x)^{5/2}}-\frac {2 b (b d-a e) (2 b B d-A b e-a B e)}{e^5 (d+e x)^{3/2}}+\frac {2 b^2 (4 b B d-A b e-3 a B e)}{e^5 \sqrt {d+e x}}+\frac {2 b^3 B \sqrt {d+e x}}{e^5} \] Output:
-2/7*(-a*e+b*d)^3*(-A*e+B*d)/e^5/(e*x+d)^(7/2)+2/5*(-a*e+b*d)^2*(-3*A*b*e- B*a*e+4*B*b*d)/e^5/(e*x+d)^(5/2)-2*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)/e^5 /(e*x+d)^(3/2)+2*b^2*(-A*b*e-3*B*a*e+4*B*b*d)/e^5/(e*x+d)^(1/2)+2*b^3*B*(e *x+d)^(1/2)/e^5
Time = 0.16 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2 \left (a^3 e^3 (2 B d+5 A e+7 B e x)+a^2 b e^2 \left (3 A e (2 d+7 e x)+B \left (8 d^2+28 d e x+35 e^2 x^2\right )\right )+a b^2 e \left (A e \left (8 d^2+28 d e x+35 e^2 x^2\right )+3 B \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )\right )+b^3 \left (A e \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )-B \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{35 e^5 (d+e x)^{7/2}} \] Input:
Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^(9/2),x]
Output:
(-2*(a^3*e^3*(2*B*d + 5*A*e + 7*B*e*x) + a^2*b*e^2*(3*A*e*(2*d + 7*e*x) + B*(8*d^2 + 28*d*e*x + 35*e^2*x^2)) + a*b^2*e*(A*e*(8*d^2 + 28*d*e*x + 35*e ^2*x^2) + 3*B*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3)) + b^3*(A* e*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3) - B*(128*d^4 + 448*d^3 *e*x + 560*d^2*e^2*x^2 + 280*d*e^3*x^3 + 35*e^4*x^4))))/(35*e^5*(d + e*x)^ (7/2))
Time = 0.32 (sec) , antiderivative size = 167, 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.
| \(\displaystyle \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^2 (3 a B e+A b e-4 b B d)}{e^4 (d+e x)^{3/2}}-\frac {3 b (b d-a e) (a B e+A b e-2 b B d)}{e^4 (d+e x)^{5/2}}+\frac {(a e-b d)^2 (a B e+3 A b e-4 b B d)}{e^4 (d+e x)^{7/2}}+\frac {(a e-b d)^3 (A e-B d)}{e^4 (d+e x)^{9/2}}+\frac {b^3 B}{e^4 \sqrt {d+e x}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b^2 (-3 a B e-A b e+4 b B d)}{e^5 \sqrt {d+e x}}-\frac {2 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (d+e x)^{3/2}}+\frac {2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5 (d+e x)^{5/2}}-\frac {2 (b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^{7/2}}+\frac {2 b^3 B \sqrt {d+e x}}{e^5}\) |
Input:
Int[((a + b*x)^3*(A + B*x))/(d + e*x)^(9/2),x]
Output:
(-2*(b*d - a*e)^3*(B*d - A*e))/(7*e^5*(d + e*x)^(7/2)) + (2*(b*d - a*e)^2* (4*b*B*d - 3*A*b*e - a*B*e))/(5*e^5*(d + e*x)^(5/2)) - (2*b*(b*d - a*e)*(2 *b*B*d - A*b*e - a*B*e))/(e^5*(d + e*x)^(3/2)) + (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e))/(e^5*Sqrt[d + e*x]) + (2*b^3*B*Sqrt[d + e*x])/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 = 0.34 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(\frac {\left (-70 x^{3} \left (-B x +A \right ) b^{3}-70 a \,x^{2} \left (3 B x +A \right ) b^{2}-42 a^{2} x \left (\frac {5 B x}{3}+A \right ) b -10 a^{3} \left (\frac {7 B x}{5}+A \right )\right ) e^{4}-12 \left (\frac {35 x^{2} \left (-4 B x +A \right ) b^{3}}{3}+\frac {14 a x \left (\frac {15 B x}{2}+A \right ) b^{2}}{3}+a^{2} \left (\frac {14 B x}{3}+A \right ) b +\frac {a^{3} B}{3}\right ) d \,e^{3}-16 b \left (\left (-70 x^{2} B +7 x A \right ) b^{2}+a \left (21 B x +A \right ) b +a^{2} B \right ) d^{2} e^{2}-32 \left (\left (-28 B x +A \right ) b +3 B a \right ) b^{2} d^{3} e +256 b^{3} B \,d^{4}}{35 \left (e x +d \right )^{\frac {7}{2}} e^{5}}\) | \(193\) |
| derivativedivides | \(\frac {2 b^{3} B \sqrt {e x +d}-\frac {2 b^{2} \left (A b e +3 B a e -4 B b d \right )}{\sqrt {e x +d}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(270\) |
| default | \(\frac {2 b^{3} B \sqrt {e x +d}-\frac {2 b^{2} \left (A b e +3 B a e -4 B b d \right )}{\sqrt {e x +d}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\) | \(270\) |
| gosper | \(-\frac {2 \left (-35 B \,x^{4} b^{3} e^{4}+35 A \,x^{3} b^{3} e^{4}+105 B \,x^{3} a \,b^{2} e^{4}-280 B \,x^{3} b^{3} d \,e^{3}+35 A \,x^{2} a \,b^{2} e^{4}+70 A \,x^{2} b^{3} d \,e^{3}+35 B \,x^{2} a^{2} b \,e^{4}+210 B \,x^{2} a \,b^{2} d \,e^{3}-560 B \,x^{2} b^{3} d^{2} e^{2}+21 A x \,a^{2} b \,e^{4}+28 A x a \,b^{2} d \,e^{3}+56 A x \,b^{3} d^{2} e^{2}+7 B x \,a^{3} e^{4}+28 B x \,a^{2} b d \,e^{3}+168 B x a \,b^{2} d^{2} e^{2}-448 B x \,b^{3} d^{3} e +5 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+8 A a \,b^{2} d^{2} e^{2}+16 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+8 B \,a^{2} b \,d^{2} e^{2}+48 B a \,b^{2} d^{3} e -128 b^{3} B \,d^{4}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{5}}\) | \(301\) |
| trager | \(-\frac {2 \left (-35 B \,x^{4} b^{3} e^{4}+35 A \,x^{3} b^{3} e^{4}+105 B \,x^{3} a \,b^{2} e^{4}-280 B \,x^{3} b^{3} d \,e^{3}+35 A \,x^{2} a \,b^{2} e^{4}+70 A \,x^{2} b^{3} d \,e^{3}+35 B \,x^{2} a^{2} b \,e^{4}+210 B \,x^{2} a \,b^{2} d \,e^{3}-560 B \,x^{2} b^{3} d^{2} e^{2}+21 A x \,a^{2} b \,e^{4}+28 A x a \,b^{2} d \,e^{3}+56 A x \,b^{3} d^{2} e^{2}+7 B x \,a^{3} e^{4}+28 B x \,a^{2} b d \,e^{3}+168 B x a \,b^{2} d^{2} e^{2}-448 B x \,b^{3} d^{3} e +5 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+8 A a \,b^{2} d^{2} e^{2}+16 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+8 B \,a^{2} b \,d^{2} e^{2}+48 B a \,b^{2} d^{3} e -128 b^{3} B \,d^{4}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{5}}\) | \(301\) |
| orering | \(-\frac {2 \left (-35 B \,x^{4} b^{3} e^{4}+35 A \,x^{3} b^{3} e^{4}+105 B \,x^{3} a \,b^{2} e^{4}-280 B \,x^{3} b^{3} d \,e^{3}+35 A \,x^{2} a \,b^{2} e^{4}+70 A \,x^{2} b^{3} d \,e^{3}+35 B \,x^{2} a^{2} b \,e^{4}+210 B \,x^{2} a \,b^{2} d \,e^{3}-560 B \,x^{2} b^{3} d^{2} e^{2}+21 A x \,a^{2} b \,e^{4}+28 A x a \,b^{2} d \,e^{3}+56 A x \,b^{3} d^{2} e^{2}+7 B x \,a^{3} e^{4}+28 B x \,a^{2} b d \,e^{3}+168 B x a \,b^{2} d^{2} e^{2}-448 B x \,b^{3} d^{3} e +5 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+8 A a \,b^{2} d^{2} e^{2}+16 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+8 B \,a^{2} b \,d^{2} e^{2}+48 B a \,b^{2} d^{3} e -128 b^{3} B \,d^{4}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{5}}\) | \(301\) |
| risch | \(\frac {2 b^{3} B \sqrt {e x +d}}{e^{5}}-\frac {2 \left (35 A \,x^{3} b^{3} e^{4}+105 B \,x^{3} a \,b^{2} e^{4}-140 B \,x^{3} b^{3} d \,e^{3}+35 A \,x^{2} a \,b^{2} e^{4}+70 A \,x^{2} b^{3} d \,e^{3}+35 B \,x^{2} a^{2} b \,e^{4}+210 B \,x^{2} a \,b^{2} d \,e^{3}-350 B \,x^{2} b^{3} d^{2} e^{2}+21 A x \,a^{2} b \,e^{4}+28 A x a \,b^{2} d \,e^{3}+56 A x \,b^{3} d^{2} e^{2}+7 B x \,a^{3} e^{4}+28 B x \,a^{2} b d \,e^{3}+168 B x a \,b^{2} d^{2} e^{2}-308 B x \,b^{3} d^{3} e +5 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+8 A a \,b^{2} d^{2} e^{2}+16 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+8 B \,a^{2} b \,d^{2} e^{2}+48 B a \,b^{2} d^{3} e -93 b^{3} B \,d^{4}\right )}{35 e^{5} \sqrt {e x +d}\, \left (e^{3} x^{3}+3 e^{2} d \,x^{2}+3 d^{2} x e +d^{3}\right )}\) | \(335\) |
Input:
int((b*x+a)^3*(B*x+A)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/35*((-70*x^3*(-B*x+A)*b^3-70*a*x^2*(3*B*x+A)*b^2-42*a^2*x*(5/3*B*x+A)*b- 10*a^3*(7/5*B*x+A))*e^4-12*(35/3*x^2*(-4*B*x+A)*b^3+14/3*a*x*(15/2*B*x+A)* b^2+a^2*(14/3*B*x+A)*b+1/3*a^3*B)*d*e^3-16*b*((-70*B*x^2+7*A*x)*b^2+a*(21* B*x+A)*b+a^2*B)*d^2*e^2-32*((-28*B*x+A)*b+3*B*a)*b^2*d^3*e+256*b^3*B*d^4)/ (e*x+d)^(7/2)/e^5
Time = 0.10 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (35 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} - 5 \, A a^{3} e^{4} - 16 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 8 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \, {\left (8 \, B b^{3} d e^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 35 \, {\left (16 \, B b^{3} d^{2} e^{2} - 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} - {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \, {\left (64 \, B b^{3} d^{3} e - 8 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \] Input:
integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(9/2),x, algorithm="fricas")
Output:
2/35*(35*B*b^3*e^4*x^4 + 128*B*b^3*d^4 - 5*A*a^3*e^4 - 16*(3*B*a*b^2 + A*b ^3)*d^3*e - 8*(B*a^2*b + A*a*b^2)*d^2*e^2 - 2*(B*a^3 + 3*A*a^2*b)*d*e^3 + 35*(8*B*b^3*d*e^3 - (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 35*(16*B*b^3*d^2*e^2 - 2*(3*B*a*b^2 + A*b^3)*d*e^3 - (B*a^2*b + A*a*b^2)*e^4)*x^2 + 7*(64*B*b^3*d ^3*e - 8*(3*B*a*b^2 + A*b^3)*d^2*e^2 - 4*(B*a^2*b + A*a*b^2)*d*e^3 - (B*a^ 3 + 3*A*a^2*b)*e^4)*x)*sqrt(e*x + d)/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^ 2 + 4*d^3*e^6*x + d^4*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 2144 vs. \(2 (165) = 330\).
Time = 0.79 (sec) , antiderivative size = 2144, normalized size of antiderivative = 12.84 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(9/2),x)
Output:
Piecewise((-10*A*a**3*e**4/(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*s qrt(d + e*x) + 105*d*e**7*x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 12*A*a**2*b*d*e**3/(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e*x) + 105*d*e**7*x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 42 *A*a**2*b*e**4*x/(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e* x) + 105*d*e**7*x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 16*A*a* b**2*d**2*e**2/(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e*x) + 105*d*e**7*x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 56*A*a*b* *2*d*e**3*x/(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e*x) + 105*d*e**7*x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 70*A*a*b**2* e**4*x**2/(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e*x) + 10 5*d*e**7*x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 32*A*b**3*d**3 *e/(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e*x) + 105*d*e** 7*x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 112*A*b**3*d**2*e**2* x/(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e*x) + 105*d*e**7 *x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 140*A*b**3*d*e**3*x**2 /(35*d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e*x) + 105*d*e**7* x**2*sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 70*A*b**3*e**4*x**3/(35 *d**3*e**5*sqrt(d + e*x) + 105*d**2*e**6*x*sqrt(d + e*x) + 105*d*e**7*x**2 *sqrt(d + e*x) + 35*e**8*x**3*sqrt(d + e*x)) - 4*B*a**3*d*e**3/(35*d**3...
Time = 0.04 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (\frac {35 \, \sqrt {e x + d} B b^{3}}{e^{4}} - \frac {5 \, B b^{3} d^{4} + 5 \, A a^{3} e^{4} - 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 15 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 35 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{3} + 35 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{2} - 7 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {7}{2}} e^{4}}\right )}}{35 \, e} \] Input:
integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(9/2),x, algorithm="maxima")
Output:
2/35*(35*sqrt(e*x + d)*B*b^3/e^4 - (5*B*b^3*d^4 + 5*A*a^3*e^4 - 5*(3*B*a*b ^2 + A*b^3)*d^3*e + 15*(B*a^2*b + A*a*b^2)*d^2*e^2 - 5*(B*a^3 + 3*A*a^2*b) *d*e^3 - 35*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)^3 + 35*(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)^2 - 7*( 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))/((e*x + d)^(7/2)*e^4))/e
Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (153) = 306\).
Time = 0.13 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {2 \, \sqrt {e x + d} B b^{3}}{e^{5}} + \frac {2 \, {\left (140 \, {\left (e x + d\right )}^{3} B b^{3} d - 70 \, {\left (e x + d\right )}^{2} B b^{3} d^{2} + 28 \, {\left (e x + d\right )} B b^{3} d^{3} - 5 \, B b^{3} d^{4} - 105 \, {\left (e x + d\right )}^{3} B a b^{2} e - 35 \, {\left (e x + d\right )}^{3} A b^{3} e + 105 \, {\left (e x + d\right )}^{2} B a b^{2} d e + 35 \, {\left (e x + d\right )}^{2} A b^{3} d e - 63 \, {\left (e x + d\right )} B a b^{2} d^{2} e - 21 \, {\left (e x + d\right )} A b^{3} d^{2} e + 15 \, B a b^{2} d^{3} e + 5 \, A b^{3} d^{3} e - 35 \, {\left (e x + d\right )}^{2} B a^{2} b e^{2} - 35 \, {\left (e x + d\right )}^{2} A a b^{2} e^{2} + 42 \, {\left (e x + d\right )} B a^{2} b d e^{2} + 42 \, {\left (e x + d\right )} A a b^{2} d e^{2} - 15 \, B a^{2} b d^{2} e^{2} - 15 \, A a b^{2} d^{2} e^{2} - 7 \, {\left (e x + d\right )} B a^{3} e^{3} - 21 \, {\left (e x + d\right )} A a^{2} b e^{3} + 5 \, B a^{3} d e^{3} + 15 \, A a^{2} b d e^{3} - 5 \, A a^{3} e^{4}\right )}}{35 \, {\left (e x + d\right )}^{\frac {7}{2}} e^{5}} \] Input:
integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(9/2),x, algorithm="giac")
Output:
2*sqrt(e*x + d)*B*b^3/e^5 + 2/35*(140*(e*x + d)^3*B*b^3*d - 70*(e*x + d)^2 *B*b^3*d^2 + 28*(e*x + d)*B*b^3*d^3 - 5*B*b^3*d^4 - 105*(e*x + d)^3*B*a*b^ 2*e - 35*(e*x + d)^3*A*b^3*e + 105*(e*x + d)^2*B*a*b^2*d*e + 35*(e*x + d)^ 2*A*b^3*d*e - 63*(e*x + d)*B*a*b^2*d^2*e - 21*(e*x + d)*A*b^3*d^2*e + 15*B *a*b^2*d^3*e + 5*A*b^3*d^3*e - 35*(e*x + d)^2*B*a^2*b*e^2 - 35*(e*x + d)^2 *A*a*b^2*e^2 + 42*(e*x + d)*B*a^2*b*d*e^2 + 42*(e*x + d)*A*a*b^2*d*e^2 - 1 5*B*a^2*b*d^2*e^2 - 15*A*a*b^2*d^2*e^2 - 7*(e*x + d)*B*a^3*e^3 - 21*(e*x + d)*A*a^2*b*e^3 + 5*B*a^3*d*e^3 + 15*A*a^2*b*d*e^3 - 5*A*a^3*e^4)/((e*x + d)^(7/2)*e^5)
Time = 1.05 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2\,\left (2\,B\,a^3\,d\,e^3+7\,B\,a^3\,e^4\,x+5\,A\,a^3\,e^4+8\,B\,a^2\,b\,d^2\,e^2+28\,B\,a^2\,b\,d\,e^3\,x+6\,A\,a^2\,b\,d\,e^3+35\,B\,a^2\,b\,e^4\,x^2+21\,A\,a^2\,b\,e^4\,x+48\,B\,a\,b^2\,d^3\,e+168\,B\,a\,b^2\,d^2\,e^2\,x+8\,A\,a\,b^2\,d^2\,e^2+210\,B\,a\,b^2\,d\,e^3\,x^2+28\,A\,a\,b^2\,d\,e^3\,x+105\,B\,a\,b^2\,e^4\,x^3+35\,A\,a\,b^2\,e^4\,x^2-128\,B\,b^3\,d^4-448\,B\,b^3\,d^3\,e\,x+16\,A\,b^3\,d^3\,e-560\,B\,b^3\,d^2\,e^2\,x^2+56\,A\,b^3\,d^2\,e^2\,x-280\,B\,b^3\,d\,e^3\,x^3+70\,A\,b^3\,d\,e^3\,x^2-35\,B\,b^3\,e^4\,x^4+35\,A\,b^3\,e^4\,x^3\right )}{35\,e^5\,{\left (d+e\,x\right )}^{7/2}} \] Input:
int(((A + B*x)*(a + b*x)^3)/(d + e*x)^(9/2),x)
Output:
-(2*(5*A*a^3*e^4 - 128*B*b^3*d^4 + 16*A*b^3*d^3*e + 2*B*a^3*d*e^3 + 7*B*a^ 3*e^4*x + 35*A*b^3*e^4*x^3 - 35*B*b^3*e^4*x^4 - 448*B*b^3*d^3*e*x + 8*A*a* b^2*d^2*e^2 + 8*B*a^2*b*d^2*e^2 + 35*A*a*b^2*e^4*x^2 + 35*B*a^2*b*e^4*x^2 + 105*B*a*b^2*e^4*x^3 + 56*A*b^3*d^2*e^2*x + 70*A*b^3*d*e^3*x^2 - 280*B*b^ 3*d*e^3*x^3 - 560*B*b^3*d^2*e^2*x^2 + 6*A*a^2*b*d*e^3 + 48*B*a*b^2*d^3*e + 21*A*a^2*b*e^4*x + 28*A*a*b^2*d*e^3*x + 28*B*a^2*b*d*e^3*x + 168*B*a*b^2* d^2*e^2*x + 210*B*a*b^2*d*e^3*x^2))/(35*e^5*(d + e*x)^(7/2))
Time = 0.15 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {2 b^{4} e^{4} x^{4}-8 a \,b^{3} e^{4} x^{3}+16 b^{4} d \,e^{3} x^{3}-4 a^{2} b^{2} e^{4} x^{2}-16 a \,b^{3} d \,e^{3} x^{2}+32 b^{4} d^{2} e^{2} x^{2}-\frac {8}{5} a^{3} b \,e^{4} x -\frac {16}{5} a^{2} b^{2} d \,e^{3} x -\frac {64}{5} a \,b^{3} d^{2} e^{2} x +\frac {128}{5} b^{4} d^{3} e x -\frac {2}{7} a^{4} e^{4}-\frac {16}{35} a^{3} b d \,e^{3}-\frac {32}{35} a^{2} b^{2} d^{2} e^{2}-\frac {128}{35} a \,b^{3} d^{3} e +\frac {256}{35} b^{4} d^{4}}{\sqrt {e x +d}\, e^{5} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:
int((b*x+a)^3*(B*x+A)/(e*x+d)^(9/2),x)
Output:
(2*( - 5*a**4*e**4 - 8*a**3*b*d*e**3 - 28*a**3*b*e**4*x - 16*a**2*b**2*d** 2*e**2 - 56*a**2*b**2*d*e**3*x - 70*a**2*b**2*e**4*x**2 - 64*a*b**3*d**3*e - 224*a*b**3*d**2*e**2*x - 280*a*b**3*d*e**3*x**2 - 140*a*b**3*e**4*x**3 + 128*b**4*d**4 + 448*b**4*d**3*e*x + 560*b**4*d**2*e**2*x**2 + 280*b**4*d *e**3*x**3 + 35*b**4*e**4*x**4))/(35*sqrt(d + e*x)*e**5*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))