Integrand size = 20, antiderivative size = 141 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {e^2 (3 b B d+A b e-3 a B e) x}{b^4}+\frac {B e^3 x^2}{2 b^3}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}-\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^5 (a+b x)}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) \log (a+b x)}{b^5} \] Output:
e^2*(A*b*e-3*B*a*e+3*B*b*d)*x/b^4+1/2*B*e^3*x^2/b^3-1/2*(A*b-B*a)*(-a*e+b* d)^3/b^5/(b*x+a)^2-(-a*e+b*d)^2*(3*A*b*e-4*B*a*e+B*b*d)/b^5/(b*x+a)+3*e*(- a*e+b*d)*(A*b*e-2*B*a*e+B*b*d)*ln(b*x+a)/b^5
Time = 0.08 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.74 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {-A b \left (5 a^3 e^3+a^2 b e^2 (-9 d+4 e x)+a b^2 e \left (3 d^2-12 d e x-4 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x-2 e^3 x^3\right )\right )+B \left (7 a^4 e^3+a^3 b e^2 (-15 d+2 e x)+a^2 b^2 e \left (9 d^2-12 d e x-11 e^2 x^2\right )+b^4 x \left (-2 d^3+6 d e^2 x^2+e^3 x^3\right )-a b^3 \left (d^3-12 d^2 e x-12 d e^2 x^2+4 e^3 x^3\right )\right )+6 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^2 \log (a+b x)}{2 b^5 (a+b x)^2} \] Input:
Integrate[((A + B*x)*(d + e*x)^3)/(a + b*x)^3,x]
Output:
(-(A*b*(5*a^3*e^3 + a^2*b*e^2*(-9*d + 4*e*x) + a*b^2*e*(3*d^2 - 12*d*e*x - 4*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x - 2*e^3*x^3))) + B*(7*a^4*e^3 + a^3*b*e ^2*(-15*d + 2*e*x) + a^2*b^2*e*(9*d^2 - 12*d*e*x - 11*e^2*x^2) + b^4*x*(-2 *d^3 + 6*d*e^2*x^2 + e^3*x^3) - a*b^3*(d^3 - 12*d^2*e*x - 12*d*e^2*x^2 + 4 *e^3*x^3)) + 6*e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^2*Log[a + b*x])/(2*b^5*(a + b*x)^2)
Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^2 (-3 a B e+A b e+3 b B d)}{b^4}+\frac {3 e (b d-a e) (-2 a B e+A b e+b B d)}{b^4 (a+b x)}+\frac {(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^4 (a+b x)^2}+\frac {(A b-a B) (b d-a e)^3}{b^4 (a+b x)^3}+\frac {B e^3 x}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 (a+b x)}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}+\frac {3 e (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5}+\frac {e^2 x (-3 a B e+A b e+3 b B d)}{b^4}+\frac {B e^3 x^2}{2 b^3}\) |
Input:
Int[((A + B*x)*(d + e*x)^3)/(a + b*x)^3,x]
Output:
(e^2*(3*b*B*d + A*b*e - 3*a*B*e)*x)/b^4 + (B*e^3*x^2)/(2*b^3) - ((A*b - a* B)*(b*d - a*e)^3)/(2*b^5*(a + b*x)^2) - ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e))/(b^5*(a + b*x)) + (3*e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*Log [a + b*x])/b^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 = 0.21 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.90
| method | result | size |
| default | \(\frac {e^{2} \left (\frac {1}{2} B b e \,x^{2}+A b e x -3 B a e x +3 B b d x \right )}{b^{4}}-\frac {-A \,a^{3} b \,e^{3}+3 A \,a^{2} b^{2} d \,e^{2}-3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}+B \,a^{4} e^{3}-3 B \,a^{3} b d \,e^{2}+3 B \,a^{2} b^{2} d^{2} e -B a \,b^{3} d^{3}}{2 b^{5} \left (b x +a \right )^{2}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -4 B \,a^{3} e^{3}+9 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +b^{3} B \,d^{3}}{b^{5} \left (b x +a \right )}-\frac {3 e \left (A a b \,e^{2}-A \,b^{2} d e -2 B \,a^{2} e^{2}+3 B a b d e -b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(268\) |
| norman | \(\frac {\frac {e^{2} \left (A b e -2 B a e +3 B b d \right ) x^{3}}{b^{2}}-\frac {9 A \,a^{3} b \,e^{3}-9 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}-18 B \,a^{4} e^{3}+27 B \,a^{3} b d \,e^{2}-9 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}}{2 b^{5}}-\frac {\left (6 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -12 B \,a^{3} e^{3}+18 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +b^{3} B \,d^{3}\right ) x}{b^{4}}+\frac {B \,e^{3} x^{4}}{2 b}}{\left (b x +a \right )^{2}}-\frac {3 e \left (A a b \,e^{2}-A \,b^{2} d e -2 B \,a^{2} e^{2}+3 B a b d e -b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(268\) |
| risch | \(\frac {B \,e^{3} x^{2}}{2 b^{3}}+\frac {e^{3} A x}{b^{3}}-\frac {3 e^{3} B a x}{b^{4}}+\frac {3 e^{2} B d x}{b^{3}}+\frac {\left (-3 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}-3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-9 B \,a^{2} b d \,e^{2}+6 B a \,b^{2} d^{2} e -b^{3} B \,d^{3}\right ) x -\frac {5 A \,a^{3} b \,e^{3}-9 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}-7 B \,a^{4} e^{3}+15 B \,a^{3} b d \,e^{2}-9 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}}{2 b}}{b^{4} \left (b x +a \right )^{2}}-\frac {3 e^{3} \ln \left (b x +a \right ) A a}{b^{4}}+\frac {3 e^{2} \ln \left (b x +a \right ) A d}{b^{3}}+\frac {6 e^{3} \ln \left (b x +a \right ) B \,a^{2}}{b^{5}}-\frac {9 e^{2} \ln \left (b x +a \right ) B a d}{b^{4}}+\frac {3 e \ln \left (b x +a \right ) B \,d^{2}}{b^{3}}\) | \(304\) |
| parallelrisch | \(-\frac {-6 A \ln \left (b x +a \right ) a^{2} b^{2} d \,e^{2}-9 B \,a^{2} b^{2} d^{2} e +18 B \ln \left (b x +a \right ) x^{2} a \,b^{3} d \,e^{2}+9 A \,a^{3} b \,e^{3}+27 B \,a^{3} b d \,e^{2}-9 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e +B a \,b^{3} d^{3}-12 A x a \,b^{3} d \,e^{2}+36 B x \,a^{2} b^{2} d \,e^{2}-12 B x a \,b^{3} d^{2} e +18 B \ln \left (b x +a \right ) a^{3} b d \,e^{2}-6 B \ln \left (b x +a \right ) a^{2} b^{2} d^{2} e +A \,b^{4} d^{3}-18 B \,a^{4} e^{3}+6 A x \,b^{4} d^{2} e -24 B x \,a^{3} b \,e^{3}+6 A \ln \left (b x +a \right ) a^{3} b \,e^{3}+6 A \ln \left (b x +a \right ) x^{2} a \,b^{3} e^{3}-6 A \ln \left (b x +a \right ) x^{2} b^{4} d \,e^{2}-12 B \ln \left (b x +a \right ) x^{2} a^{2} b^{2} e^{3}-6 B \ln \left (b x +a \right ) x^{2} b^{4} d^{2} e -2 A \,x^{3} b^{4} e^{3}+2 B x \,b^{4} d^{3}-12 B \ln \left (b x +a \right ) a^{4} e^{3}-B \,x^{4} e^{3} b^{4}+12 A \ln \left (b x +a \right ) x \,a^{2} b^{2} e^{3}-24 B \ln \left (b x +a \right ) x \,a^{3} b \,e^{3}-12 A \ln \left (b x +a \right ) x a \,b^{3} d \,e^{2}+36 B \ln \left (b x +a \right ) x \,a^{2} b^{2} d \,e^{2}-12 B \ln \left (b x +a \right ) x a \,b^{3} d^{2} e +4 B \,x^{3} a \,b^{3} e^{3}-6 B \,x^{3} b^{4} d \,e^{2}+12 A x \,a^{2} b^{2} e^{3}}{2 b^{5} \left (b x +a \right )^{2}}\) | \(507\) |
Input:
int((B*x+A)*(e*x+d)^3/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
e^2/b^4*(1/2*B*b*e*x^2+A*b*e*x-3*B*a*e*x+3*B*b*d*x)-1/2/b^5*(-A*a^3*b*e^3+ 3*A*a^2*b^2*d*e^2-3*A*a*b^3*d^2*e+A*b^4*d^3+B*a^4*e^3-3*B*a^3*b*d*e^2+3*B* a^2*b^2*d^2*e-B*a*b^3*d^3)/(b*x+a)^2-1/b^5*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+ 3*A*b^3*d^2*e-4*B*a^3*e^3+9*B*a^2*b*d*e^2-6*B*a*b^2*d^2*e+B*b^3*d^3)/(b*x+ a)-3/b^5*e*(A*a*b*e^2-A*b^2*d*e-2*B*a^2*e^2+3*B*a*b*d*e-B*b^2*d^2)*ln(b*x+ a)
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (137) = 274\).
Time = 0.10 (sec) , antiderivative size = 442, normalized size of antiderivative = 3.13 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {B b^{4} e^{3} x^{4} - {\left (B a b^{3} + A b^{4}\right )} d^{3} + 3 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e - 3 \, {\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} + {\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \, {\left (3 \, B b^{4} d e^{2} - {\left (2 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + {\left (12 \, B a b^{3} d e^{2} - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \, {\left (B b^{4} d^{3} - 3 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{2} - {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \, {\left (B a^{2} b^{2} d^{2} e - {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} + {\left (2 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (B b^{4} d^{2} e - {\left (3 \, B a b^{3} - A b^{4}\right )} d e^{2} + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (B a b^{3} d^{2} e - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d e^{2} + {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \] Input:
integrate((B*x+A)*(e*x+d)^3/(b*x+a)^3,x, algorithm="fricas")
Output:
1/2*(B*b^4*e^3*x^4 - (B*a*b^3 + A*b^4)*d^3 + 3*(3*B*a^2*b^2 - A*a*b^3)*d^2 *e - 3*(5*B*a^3*b - 3*A*a^2*b^2)*d*e^2 + (7*B*a^4 - 5*A*a^3*b)*e^3 + 2*(3* B*b^4*d*e^2 - (2*B*a*b^3 - A*b^4)*e^3)*x^3 + (12*B*a*b^3*d*e^2 - (11*B*a^2 *b^2 - 4*A*a*b^3)*e^3)*x^2 - 2*(B*b^4*d^3 - 3*(2*B*a*b^3 - A*b^4)*d^2*e + 6*(B*a^2*b^2 - A*a*b^3)*d*e^2 - (B*a^3*b - 2*A*a^2*b^2)*e^3)*x + 6*(B*a^2* b^2*d^2*e - (3*B*a^3*b - A*a^2*b^2)*d*e^2 + (2*B*a^4 - A*a^3*b)*e^3 + (B*b ^4*d^2*e - (3*B*a*b^3 - A*b^4)*d*e^2 + (2*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 2*(B*a*b^3*d^2*e - (3*B*a^2*b^2 - A*a*b^3)*d*e^2 + (2*B*a^3*b - A*a^2*b^2) *e^3)*x)*log(b*x + a))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (141) = 282\).
Time = 2.04 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {B e^{3} x^{2}}{2 b^{3}} + x \left (\frac {A e^{3}}{b^{3}} - \frac {3 B a e^{3}}{b^{4}} + \frac {3 B d e^{2}}{b^{3}}\right ) + \frac {- 5 A a^{3} b e^{3} + 9 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e - A b^{4} d^{3} + 7 B a^{4} e^{3} - 15 B a^{3} b d e^{2} + 9 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3} + x \left (- 6 A a^{2} b^{2} e^{3} + 12 A a b^{3} d e^{2} - 6 A b^{4} d^{2} e + 8 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 12 B a b^{3} d^{2} e - 2 B b^{4} d^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {3 e \left (a e - b d\right ) \left (- A b e + 2 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{5}} \] Input:
integrate((B*x+A)*(e*x+d)**3/(b*x+a)**3,x)
Output:
B*e**3*x**2/(2*b**3) + x*(A*e**3/b**3 - 3*B*a*e**3/b**4 + 3*B*d*e**2/b**3) + (-5*A*a**3*b*e**3 + 9*A*a**2*b**2*d*e**2 - 3*A*a*b**3*d**2*e - A*b**4*d **3 + 7*B*a**4*e**3 - 15*B*a**3*b*d*e**2 + 9*B*a**2*b**2*d**2*e - B*a*b**3 *d**3 + x*(-6*A*a**2*b**2*e**3 + 12*A*a*b**3*d*e**2 - 6*A*b**4*d**2*e + 8* B*a**3*b*e**3 - 18*B*a**2*b**2*d*e**2 + 12*B*a*b**3*d**2*e - 2*B*b**4*d**3 ))/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 3*e*(a*e - b*d)*(-A*b*e + 2* B*a*e - B*b*d)*log(a + b*x)/b**5
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (137) = 274\).
Time = 0.04 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.00 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=-\frac {{\left (B a b^{3} + A b^{4}\right )} d^{3} - 3 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \, {\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} - {\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \, {\left (B b^{4} d^{3} - 3 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} - {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {B b e^{3} x^{2} + 2 \, {\left (3 \, B b d e^{2} - {\left (3 \, B a - A b\right )} e^{3}\right )} x}{2 \, b^{4}} + \frac {3 \, {\left (B b^{2} d^{2} e - {\left (3 \, B a b - A b^{2}\right )} d e^{2} + {\left (2 \, B a^{2} - A a b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \] Input:
integrate((B*x+A)*(e*x+d)^3/(b*x+a)^3,x, algorithm="maxima")
Output:
-1/2*((B*a*b^3 + A*b^4)*d^3 - 3*(3*B*a^2*b^2 - A*a*b^3)*d^2*e + 3*(5*B*a^3 *b - 3*A*a^2*b^2)*d*e^2 - (7*B*a^4 - 5*A*a^3*b)*e^3 + 2*(B*b^4*d^3 - 3*(2* B*a*b^3 - A*b^4)*d^2*e + 3*(3*B*a^2*b^2 - 2*A*a*b^3)*d*e^2 - (4*B*a^3*b - 3*A*a^2*b^2)*e^3)*x)/(b^7*x^2 + 2*a*b^6*x + a^2*b^5) + 1/2*(B*b*e^3*x^2 + 2*(3*B*b*d*e^2 - (3*B*a - A*b)*e^3)*x)/b^4 + 3*(B*b^2*d^2*e - (3*B*a*b - A *b^2)*d*e^2 + (2*B*a^2 - A*a*b)*e^3)*log(b*x + a)/b^5
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (137) = 274\).
Time = 0.12 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.01 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {3 \, {\left (B b^{2} d^{2} e - 3 \, B a b d e^{2} + A b^{2} d e^{2} + 2 \, B a^{2} e^{3} - A a b e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {B b^{3} e^{3} x^{2} + 6 \, B b^{3} d e^{2} x - 6 \, B a b^{2} e^{3} x + 2 \, A b^{3} e^{3} x}{2 \, b^{6}} - \frac {B a b^{3} d^{3} + A b^{4} d^{3} - 9 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 15 \, B a^{3} b d e^{2} - 9 \, A a^{2} b^{2} d e^{2} - 7 \, B a^{4} e^{3} + 5 \, A a^{3} b e^{3} + 2 \, {\left (B b^{4} d^{3} - 6 \, B a b^{3} d^{2} e + 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} - 6 \, A a b^{3} d e^{2} - 4 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \] Input:
integrate((B*x+A)*(e*x+d)^3/(b*x+a)^3,x, algorithm="giac")
Output:
3*(B*b^2*d^2*e - 3*B*a*b*d*e^2 + A*b^2*d*e^2 + 2*B*a^2*e^3 - A*a*b*e^3)*lo g(abs(b*x + a))/b^5 + 1/2*(B*b^3*e^3*x^2 + 6*B*b^3*d*e^2*x - 6*B*a*b^2*e^3 *x + 2*A*b^3*e^3*x)/b^6 - 1/2*(B*a*b^3*d^3 + A*b^4*d^3 - 9*B*a^2*b^2*d^2*e + 3*A*a*b^3*d^2*e + 15*B*a^3*b*d*e^2 - 9*A*a^2*b^2*d*e^2 - 7*B*a^4*e^3 + 5*A*a^3*b*e^3 + 2*(B*b^4*d^3 - 6*B*a*b^3*d^2*e + 3*A*b^4*d^2*e + 9*B*a^2*b ^2*d*e^2 - 6*A*a*b^3*d*e^2 - 4*B*a^3*b*e^3 + 3*A*a^2*b^2*e^3)*x)/((b*x + a )^2*b^5)
Time = 1.12 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.06 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=x\,\left (\frac {A\,e^3+3\,B\,d\,e^2}{b^3}-\frac {3\,B\,a\,e^3}{b^4}\right )-\frac {\frac {-7\,B\,a^4\,e^3+15\,B\,a^3\,b\,d\,e^2+5\,A\,a^3\,b\,e^3-9\,B\,a^2\,b^2\,d^2\,e-9\,A\,a^2\,b^2\,d\,e^2+B\,a\,b^3\,d^3+3\,A\,a\,b^3\,d^2\,e+A\,b^4\,d^3}{2\,b}+x\,\left (-4\,B\,a^3\,e^3+9\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3-6\,B\,a\,b^2\,d^2\,e-6\,A\,a\,b^2\,d\,e^2+B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}+\frac {\ln \left (a+b\,x\right )\,\left (6\,B\,a^2\,e^3-9\,B\,a\,b\,d\,e^2-3\,A\,a\,b\,e^3+3\,B\,b^2\,d^2\,e+3\,A\,b^2\,d\,e^2\right )}{b^5}+\frac {B\,e^3\,x^2}{2\,b^3} \] Input:
int(((A + B*x)*(d + e*x)^3)/(a + b*x)^3,x)
Output:
x*((A*e^3 + 3*B*d*e^2)/b^3 - (3*B*a*e^3)/b^4) - ((A*b^4*d^3 - 7*B*a^4*e^3 + 5*A*a^3*b*e^3 + B*a*b^3*d^3 - 9*A*a^2*b^2*d*e^2 - 9*B*a^2*b^2*d^2*e + 3* A*a*b^3*d^2*e + 15*B*a^3*b*d*e^2)/(2*b) + x*(B*b^3*d^3 - 4*B*a^3*e^3 + 3*A *a^2*b*e^3 + 3*A*b^3*d^2*e - 6*A*a*b^2*d*e^2 - 6*B*a*b^2*d^2*e + 9*B*a^2*b *d*e^2))/(a^2*b^4 + b^6*x^2 + 2*a*b^5*x) + (log(a + b*x)*(6*B*a^2*e^3 - 3* A*a*b*e^3 + 3*A*b^2*d*e^2 + 3*B*b^2*d^2*e - 9*B*a*b*d*e^2))/b^5 + (B*e^3*x ^2)/(2*b^3)
Time = 0.16 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {6 \,\mathrm {log}\left (b x +a \right ) a^{4} e^{3}-12 \,\mathrm {log}\left (b x +a \right ) a^{3} b d \,e^{2}+6 \,\mathrm {log}\left (b x +a \right ) a^{3} b \,e^{3} x +6 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} d^{2} e -12 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} d \,e^{2} x +6 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} d^{2} e x -6 a^{3} b \,e^{3} x +12 a^{2} b^{2} d \,e^{2} x -3 a^{2} b^{2} e^{3} x^{2}-6 a \,b^{3} d^{2} e x +6 a \,b^{3} d \,e^{2} x^{2}+a \,b^{3} e^{3} x^{3}+2 b^{4} d^{3} x}{2 a \,b^{4} \left (b x +a \right )} \] Input:
int((B*x+A)*(e*x+d)^3/(b*x+a)^3,x)
Output:
(6*log(a + b*x)*a**4*e**3 - 12*log(a + b*x)*a**3*b*d*e**2 + 6*log(a + b*x) *a**3*b*e**3*x + 6*log(a + b*x)*a**2*b**2*d**2*e - 12*log(a + b*x)*a**2*b* *2*d*e**2*x + 6*log(a + b*x)*a*b**3*d**2*e*x - 6*a**3*b*e**3*x + 12*a**2*b **2*d*e**2*x - 3*a**2*b**2*e**3*x**2 - 6*a*b**3*d**2*e*x + 6*a*b**3*d*e**2 *x**2 + a*b**3*e**3*x**3 + 2*b**4*d**3*x)/(2*a*b**4*(a + b*x))