Integrand size = 22, antiderivative size = 169 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{3 e^5 (d+e x)^{3/2}}-\frac {6 b (b d-a e) (2 b B d-A b e-a B e)}{e^5 \sqrt {d+e x}}-\frac {2 b^2 (4 b B d-A b e-3 a B e) \sqrt {d+e x}}{e^5}+\frac {2 b^3 B (d+e x)^{3/2}}{3 e^5} \] Output:
-2/5*(-a*e+b*d)^3*(-A*e+B*d)/e^5/(e*x+d)^(5/2)+2/3*(-a*e+b*d)^2*(-3*A*b*e- B*a*e+4*B*b*d)/e^5/(e*x+d)^(3/2)-6*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)/e^5 /(e*x+d)^(1/2)-2*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(1/2)/e^5+2/3*b^3*B* (e*x+d)^(3/2)/e^5
Time = 0.19 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (a^3 e^3 (2 B d+3 A e+5 B e x)+3 a^2 b e^2 \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )-3 a b^2 e \left (-A e \left (8 d^2+20 d e x+15 e^2 x^2\right )+3 B \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )+b^3 \left (-3 A e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )\right )}{15 e^5 (d+e x)^{5/2}} \] Input:
Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]
Output:
(-2*(a^3*e^3*(2*B*d + 3*A*e + 5*B*e*x) + 3*a^2*b*e^2*(A*e*(2*d + 5*e*x) + B*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) - 3*a*b^2*e*(-(A*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) + 3*B*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)) + b^3 *(-3*A*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 + 3 20*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4))))/(15*e^5*(d + e *x)^(5/2))
Time = 0.30 (sec) , antiderivative size = 169, 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.
| \(\displaystyle \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^2 (3 a B e+A b e-4 b B d)}{e^4 \sqrt {d+e x}}-\frac {3 b (b d-a e) (a B e+A b e-2 b B d)}{e^4 (d+e x)^{3/2}}+\frac {(a e-b d)^2 (a B e+3 A b e-4 b B d)}{e^4 (d+e x)^{5/2}}+\frac {(a e-b d)^3 (A e-B d)}{e^4 (d+e x)^{7/2}}+\frac {b^3 B \sqrt {d+e x}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^2 \sqrt {d+e x} (-3 a B e-A b e+4 b B d)}{e^5}-\frac {6 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 \sqrt {d+e x}}+\frac {2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (d+e x)^{3/2}}-\frac {2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac {2 b^3 B (d+e x)^{3/2}}{3 e^5}\) |
Input:
Int[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]
Output:
(-2*(b*d - a*e)^3*(B*d - A*e))/(5*e^5*(d + e*x)^(5/2)) + (2*(b*d - a*e)^2* (4*b*B*d - 3*A*b*e - a*B*e))/(3*e^5*(d + e*x)^(3/2)) - (6*b*(b*d - a*e)*(2 *b*B*d - A*b*e - a*B*e))/(e^5*Sqrt[d + e*x]) - (2*b^2*(4*b*B*d - A*b*e - 3 *a*B*e)*Sqrt[d + e*x])/e^5 + (2*b^3*B*(d + e*x)^(3/2))/(3*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 = 1.14
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (10 B \,x^{4}+30 A \,x^{3}\right ) b^{3}-90 a \,x^{2} \left (-B x +A \right ) b^{2}-30 a^{2} x \left (3 B x +A \right ) b -6 a^{3} \left (\frac {5 B x}{3}+A \right )\right ) e^{4}-12 \left (-15 \left (-\frac {4 B x}{9}+A \right ) x^{2} b^{3}+10 a \left (-\frac {9 B x}{2}+A \right ) x \,b^{2}+a^{2} \left (10 B x +A \right ) b +\frac {a^{3} B}{3}\right ) d \,e^{3}-48 \left (-5 x \left (-2 B x +A \right ) b^{2}+a \left (-15 B x +A \right ) b +a^{2} B \right ) b \,d^{2} e^{2}+96 b^{2} \left (\left (-\frac {20 B x}{3}+A \right ) b +3 B a \right ) d^{3} e -256 b^{3} B \,d^{4}}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(193\) |
| risch | \(\frac {2 b^{2} \left (e b B x +3 A b e +9 B a e -11 B b d \right ) \sqrt {e x +d}}{3 e^{5}}-\frac {2 \left (45 A \,x^{2} b^{2} e^{3}+45 B \,x^{2} a b \,e^{3}-90 B \,x^{2} b^{2} d \,e^{2}+15 A x a b \,e^{3}+75 A x \,b^{2} d \,e^{2}+5 B x \,a^{2} e^{3}+65 B x a b d \,e^{2}-160 B x \,b^{2} d^{2} e +3 a^{2} A \,e^{3}+9 A a b d \,e^{2}+33 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}+26 B a b \,d^{2} e -73 b^{2} B \,d^{3}\right ) \left (a e -d b \right )}{15 e^{5} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d e x +d^{2}\right )}\) | \(220\) |
| derivativedivides | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{3} e \sqrt {e x +d}+6 B a \,b^{2} e \sqrt {e x +d}-8 B \,b^{3} d \sqrt {e x +d}-\frac {6 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{5}}\) | \(286\) |
| default | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{3} e \sqrt {e x +d}+6 B a \,b^{2} e \sqrt {e x +d}-8 B \,b^{3} d \sqrt {e x +d}-\frac {6 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{5}}\) | \(286\) |
| gosper | \(-\frac {2 \left (-5 B \,x^{4} b^{3} e^{4}-15 A \,x^{3} b^{3} e^{4}-45 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+45 A \,x^{2} a \,b^{2} e^{4}-90 A \,x^{2} b^{3} d \,e^{3}+45 B \,x^{2} a^{2} b \,e^{4}-270 B \,x^{2} a \,b^{2} d \,e^{3}+240 B \,x^{2} b^{3} d^{2} e^{2}+15 A x \,a^{2} b \,e^{4}+60 A x a \,b^{2} d \,e^{3}-120 A x \,b^{3} d^{2} e^{2}+5 B x \,a^{3} e^{4}+60 B x \,a^{2} b d \,e^{3}-360 B x a \,b^{2} d^{2} e^{2}+320 B x \,b^{3} d^{3} e +3 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+24 A a \,b^{2} d^{2} e^{2}-48 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+24 B \,a^{2} b \,d^{2} e^{2}-144 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(301\) |
| trager | \(-\frac {2 \left (-5 B \,x^{4} b^{3} e^{4}-15 A \,x^{3} b^{3} e^{4}-45 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+45 A \,x^{2} a \,b^{2} e^{4}-90 A \,x^{2} b^{3} d \,e^{3}+45 B \,x^{2} a^{2} b \,e^{4}-270 B \,x^{2} a \,b^{2} d \,e^{3}+240 B \,x^{2} b^{3} d^{2} e^{2}+15 A x \,a^{2} b \,e^{4}+60 A x a \,b^{2} d \,e^{3}-120 A x \,b^{3} d^{2} e^{2}+5 B x \,a^{3} e^{4}+60 B x \,a^{2} b d \,e^{3}-360 B x a \,b^{2} d^{2} e^{2}+320 B x \,b^{3} d^{3} e +3 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+24 A a \,b^{2} d^{2} e^{2}-48 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+24 B \,a^{2} b \,d^{2} e^{2}-144 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(301\) |
| orering | \(-\frac {2 \left (-5 B \,x^{4} b^{3} e^{4}-15 A \,x^{3} b^{3} e^{4}-45 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+45 A \,x^{2} a \,b^{2} e^{4}-90 A \,x^{2} b^{3} d \,e^{3}+45 B \,x^{2} a^{2} b \,e^{4}-270 B \,x^{2} a \,b^{2} d \,e^{3}+240 B \,x^{2} b^{3} d^{2} e^{2}+15 A x \,a^{2} b \,e^{4}+60 A x a \,b^{2} d \,e^{3}-120 A x \,b^{3} d^{2} e^{2}+5 B x \,a^{3} e^{4}+60 B x \,a^{2} b d \,e^{3}-360 B x a \,b^{2} d^{2} e^{2}+320 B x \,b^{3} d^{3} e +3 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+24 A a \,b^{2} d^{2} e^{2}-48 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+24 B \,a^{2} b \,d^{2} e^{2}-144 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\) | \(301\) |
Input:
int((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/15*(((10*B*x^4+30*A*x^3)*b^3-90*a*x^2*(-B*x+A)*b^2-30*a^2*x*(3*B*x+A)*b- 6*a^3*(5/3*B*x+A))*e^4-12*(-15*(-4/9*B*x+A)*x^2*b^3+10*a*(-9/2*B*x+A)*x*b^ 2+a^2*(10*B*x+A)*b+1/3*a^3*B)*d*e^3-48*(-5*x*(-2*B*x+A)*b^2+a*(-15*B*x+A)* b+a^2*B)*b*d^2*e^2+96*b^2*((-20/3*B*x+A)*b+3*B*a)*d^3*e-256*b^3*B*d^4)/(e* x+d)^(5/2)/e^5
Time = 0.08 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} + 48 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 24 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \, {\left (8 \, B b^{3} d e^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} - 15 \, {\left (16 \, B b^{3} d^{2} e^{2} - 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 5 \, {\left (64 \, B b^{3} d^{3} e - 24 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \] Input:
integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="fricas")
Output:
2/15*(5*B*b^3*e^4*x^4 - 128*B*b^3*d^4 - 3*A*a^3*e^4 + 48*(3*B*a*b^2 + A*b^ 3)*d^3*e - 24*(B*a^2*b + A*a*b^2)*d^2*e^2 - 2*(B*a^3 + 3*A*a^2*b)*d*e^3 - 5*(8*B*b^3*d*e^3 - 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 - 15*(16*B*b^3*d^2*e^2 - 6*(3*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 - 5*(64*B*b^ 3*d^3*e - 24*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 12*(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^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e ^6*x + d^3*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 1654 vs. \(2 (167) = 334\).
Time = 0.57 (sec) , antiderivative size = 1654, normalized size of antiderivative = 9.79 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(7/2),x)
Output:
Piecewise((-6*A*a**3*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 12*A*a**2*b*d*e**3/(15*d**2*e**5*s qrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 3 0*A*a**2*b*e**4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 48*A*a*b**2*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 120*A* a*b**2*d*e**3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 90*A*a*b**2*e**4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 96*A*b** 3*d**3*e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7 *x**2*sqrt(d + e*x)) + 240*A*b**3*d**2*e**2*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 180*A*b**3*d*e **3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7 *x**2*sqrt(d + e*x)) + 30*A*b**3*e**4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 3 0*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 4*B*a**3*d*e**3/( 15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt (d + e*x)) - 10*B*a**3*e**4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sq rt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 48*B*a**2*b*d**2*e**2/(15*d**2 *e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e* x)) - 120*B*a**2*b*d*e**3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*s...
Time = 0.05 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{3} - 3 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} \sqrt {e x + d}\right )}}{e^{4}} - \frac {3 \, B b^{3} d^{4} + 3 \, A a^{3} e^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 45 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{4}}\right )}}{15 \, e} \] Input:
integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="maxima")
Output:
2/15*(5*((e*x + d)^(3/2)*B*b^3 - 3*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*sqr t(e*x + d))/e^4 - (3*B*b^3*d^4 + 3*A*a^3*e^4 - 3*(3*B*a*b^2 + A*b^3)*d^3*e + 9*(B*a^2*b + A*a*b^2)*d^2*e^2 - 3*(B*a^3 + 3*A*a^2*b)*d*e^3 + 45*(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 - 5*(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)^(5/2)*e^4))/e
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (153) = 306\).
Time = 0.13 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (90 \, {\left (e x + d\right )}^{2} B b^{3} d^{2} - 20 \, {\left (e x + d\right )} B b^{3} d^{3} + 3 \, B b^{3} d^{4} - 135 \, {\left (e x + d\right )}^{2} B a b^{2} d e - 45 \, {\left (e x + d\right )}^{2} A b^{3} d e + 45 \, {\left (e x + d\right )} B a b^{2} d^{2} e + 15 \, {\left (e x + d\right )} A b^{3} d^{2} e - 9 \, B a b^{2} d^{3} e - 3 \, A b^{3} d^{3} e + 45 \, {\left (e x + d\right )}^{2} B a^{2} b e^{2} + 45 \, {\left (e x + d\right )}^{2} A a b^{2} e^{2} - 30 \, {\left (e x + d\right )} B a^{2} b d e^{2} - 30 \, {\left (e x + d\right )} A a b^{2} d e^{2} + 9 \, B a^{2} b d^{2} e^{2} + 9 \, A a b^{2} d^{2} e^{2} + 5 \, {\left (e x + d\right )} B a^{3} e^{3} + 15 \, {\left (e x + d\right )} A a^{2} b e^{3} - 3 \, B a^{3} d e^{3} - 9 \, A a^{2} b d e^{3} + 3 \, A a^{3} e^{4}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{5}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{3} e^{10} - 12 \, \sqrt {e x + d} B b^{3} d e^{10} + 9 \, \sqrt {e x + d} B a b^{2} e^{11} + 3 \, \sqrt {e x + d} A b^{3} e^{11}\right )}}{3 \, e^{15}} \] Input:
integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="giac")
Output:
-2/15*(90*(e*x + d)^2*B*b^3*d^2 - 20*(e*x + d)*B*b^3*d^3 + 3*B*b^3*d^4 - 1 35*(e*x + d)^2*B*a*b^2*d*e - 45*(e*x + d)^2*A*b^3*d*e + 45*(e*x + d)*B*a*b ^2*d^2*e + 15*(e*x + d)*A*b^3*d^2*e - 9*B*a*b^2*d^3*e - 3*A*b^3*d^3*e + 45 *(e*x + d)^2*B*a^2*b*e^2 + 45*(e*x + d)^2*A*a*b^2*e^2 - 30*(e*x + d)*B*a^2 *b*d*e^2 - 30*(e*x + d)*A*a*b^2*d*e^2 + 9*B*a^2*b*d^2*e^2 + 9*A*a*b^2*d^2* e^2 + 5*(e*x + d)*B*a^3*e^3 + 15*(e*x + d)*A*a^2*b*e^3 - 3*B*a^3*d*e^3 - 9 *A*a^2*b*d*e^3 + 3*A*a^3*e^4)/((e*x + d)^(5/2)*e^5) + 2/3*((e*x + d)^(3/2) *B*b^3*e^10 - 12*sqrt(e*x + d)*B*b^3*d*e^10 + 9*sqrt(e*x + d)*B*a*b^2*e^11 + 3*sqrt(e*x + d)*A*b^3*e^11)/e^15
Time = 1.01 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx=-\frac {2\,\left (2\,B\,a^3\,d\,e^3+5\,B\,a^3\,e^4\,x+3\,A\,a^3\,e^4+24\,B\,a^2\,b\,d^2\,e^2+60\,B\,a^2\,b\,d\,e^3\,x+6\,A\,a^2\,b\,d\,e^3+45\,B\,a^2\,b\,e^4\,x^2+15\,A\,a^2\,b\,e^4\,x-144\,B\,a\,b^2\,d^3\,e-360\,B\,a\,b^2\,d^2\,e^2\,x+24\,A\,a\,b^2\,d^2\,e^2-270\,B\,a\,b^2\,d\,e^3\,x^2+60\,A\,a\,b^2\,d\,e^3\,x-45\,B\,a\,b^2\,e^4\,x^3+45\,A\,a\,b^2\,e^4\,x^2+128\,B\,b^3\,d^4+320\,B\,b^3\,d^3\,e\,x-48\,A\,b^3\,d^3\,e+240\,B\,b^3\,d^2\,e^2\,x^2-120\,A\,b^3\,d^2\,e^2\,x+40\,B\,b^3\,d\,e^3\,x^3-90\,A\,b^3\,d\,e^3\,x^2-5\,B\,b^3\,e^4\,x^4-15\,A\,b^3\,e^4\,x^3\right )}{15\,e^5\,{\left (d+e\,x\right )}^{5/2}} \] Input:
int(((A + B*x)*(a + b*x)^3)/(d + e*x)^(7/2),x)
Output:
-(2*(3*A*a^3*e^4 + 128*B*b^3*d^4 - 48*A*b^3*d^3*e + 2*B*a^3*d*e^3 + 5*B*a^ 3*e^4*x - 15*A*b^3*e^4*x^3 - 5*B*b^3*e^4*x^4 + 320*B*b^3*d^3*e*x + 24*A*a* b^2*d^2*e^2 + 24*B*a^2*b*d^2*e^2 + 45*A*a*b^2*e^4*x^2 + 45*B*a^2*b*e^4*x^2 - 45*B*a*b^2*e^4*x^3 - 120*A*b^3*d^2*e^2*x - 90*A*b^3*d*e^3*x^2 + 40*B*b^ 3*d*e^3*x^3 + 240*B*b^3*d^2*e^2*x^2 + 6*A*a^2*b*d*e^3 - 144*B*a*b^2*d^3*e + 15*A*a^2*b*e^4*x + 60*A*a*b^2*d*e^3*x + 60*B*a^2*b*d*e^3*x - 360*B*a*b^2 *d^2*e^2*x - 270*B*a*b^2*d*e^3*x^2))/(15*e^5*(d + e*x)^(5/2))
Time = 0.16 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx=\frac {\frac {2}{3} b^{4} e^{4} x^{4}+8 a \,b^{3} e^{4} x^{3}-\frac {16}{3} b^{4} d \,e^{3} x^{3}-12 a^{2} b^{2} e^{4} x^{2}+48 a \,b^{3} d \,e^{3} x^{2}-32 b^{4} d^{2} e^{2} x^{2}-\frac {8}{3} a^{3} b \,e^{4} x -16 a^{2} b^{2} d \,e^{3} x +64 a \,b^{3} d^{2} e^{2} x -\frac {128}{3} b^{4} d^{3} e x -\frac {2}{5} a^{4} e^{4}-\frac {16}{15} a^{3} b d \,e^{3}-\frac {32}{5} a^{2} b^{2} d^{2} e^{2}+\frac {128}{5} a \,b^{3} d^{3} e -\frac {256}{15} b^{4} d^{4}}{\sqrt {e x +d}\, e^{5} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:
int((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x)
Output:
(2*( - 3*a**4*e**4 - 8*a**3*b*d*e**3 - 20*a**3*b*e**4*x - 48*a**2*b**2*d** 2*e**2 - 120*a**2*b**2*d*e**3*x - 90*a**2*b**2*e**4*x**2 + 192*a*b**3*d**3 *e + 480*a*b**3*d**2*e**2*x + 360*a*b**3*d*e**3*x**2 + 60*a*b**3*e**4*x**3 - 128*b**4*d**4 - 320*b**4*d**3*e*x - 240*b**4*d**2*e**2*x**2 - 40*b**4*d *e**3*x**3 + 5*b**4*e**4*x**4))/(15*sqrt(d + e*x)*e**5*(d**2 + 2*d*e*x + e **2*x**2))