\(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx\) [1044]

Optimal result

Integrand size = 20, antiderivative size = 149 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx=\frac {b^3 B x}{e^4}-\frac {(b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^3}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{2 e^5 (d+e x)^2}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{e^5 (d+e x)}-\frac {b^2 (4 b B d-A b e-3 a B e) \log (d+e x)}{e^5} \]

[Out]

b^3*B*x/e^4-1/3*(-a*e+b*d)^3*(-A*e+B*d)/e^5/(e*x+d)^3+1/2*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)/e^5/(e*x+d)^2-
3*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)/e^5/(e*x+d)-b^2*(-A*b*e-3*B*a*e+4*B*b*d)*ln(e*x+d)/e^5

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx=-\frac {b^2 \log (d+e x) (-3 a B e-A b e+4 b B d)}{e^5}-\frac {3 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (d+e x)}+\frac {(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{2 e^5 (d+e x)^2}-\frac {(b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^3}+\frac {b^3 B x}{e^4} \]

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^4,x]

[Out]

(b^3*B*x)/e^4 - ((b*d - a*e)^3*(B*d - A*e))/(3*e^5*(d + e*x)^3) + ((b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e))/
(2*e^5*(d + e*x)^2) - (3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))/(e^5*(d + e*x)) - (b^2*(4*b*B*d - A*b*e - 3*
a*B*e)*Log[d + e*x])/e^5

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^3 B}{e^4}+\frac {(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^4}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^3}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^2}+\frac {b^2 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)}\right ) \, dx \\ & = \frac {b^3 B x}{e^4}-\frac {(b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^3}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{2 e^5 (d+e x)^2}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{e^5 (d+e x)}-\frac {b^2 (4 b B d-A b e-3 a B e) \log (d+e x)}{e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx=\frac {-a^3 e^3 (2 A e+B (d+3 e x))-3 a^2 b e^2 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+3 a b^2 e \left (-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )+B d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )+b^3 \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )-6 b^2 (4 b B d-A b e-3 a B e) (d+e x)^3 \log (d+e x)}{6 e^5 (d+e x)^3} \]

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^4,x]

[Out]

(-(a^3*e^3*(2*A*e + B*(d + 3*e*x))) - 3*a^2*b*e^2*(A*e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2)) + 3*a*b^
2*e*(-2*A*e*(d^2 + 3*d*e*x + 3*e^2*x^2) + B*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) + b^3*(A*d*e*(11*d^2 + 27*d*e*
x + 18*e^2*x^2) - 2*B*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4)) - 6*b^2*(4*b*B*d - A*b*
e - 3*a*B*e)*(d + e*x)^3*Log[d + e*x])/(6*e^5*(d + e*x)^3)

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.82

method	result	size
norman	\(\frac {\frac {b^{3} B \,x^{4}}{e}-\frac {2 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+6 A a \,b^{2} d^{2} e^{2}-11 A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+6 B \,a^{2} b \,d^{2} e^{2}-33 B a \,b^{2} d^{3} e +44 b^{3} B \,d^{4}}{6 e^{5}}-\frac {3 \left (A a \,b^{2} e^{2}-A \,b^{3} d e +B \,a^{2} b \,e^{2}-3 B a \,b^{2} d e +4 b^{3} B \,d^{2}\right ) x^{2}}{e^{3}}-\frac {\left (3 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}-9 A \,b^{3} d^{2} e +B \,a^{3} e^{3}+6 B \,a^{2} b d \,e^{2}-27 B a \,b^{2} d^{2} e +36 b^{3} B \,d^{3}\right ) x}{2 e^{4}}}{\left (e x +d \right )^{3}}+\frac {b^{2} \left (A b e +3 B a e -4 B b d \right ) \ln \left (e x +d \right )}{e^{5}}\)	\(271\)
default	\(\frac {b^{3} B x}{e^{4}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{2 e^{5} \left (e x +d \right )^{2}}+\frac {b^{2} \left (A b e +3 B a e -4 B b d \right ) \ln \left (e x +d \right )}{e^{5}}\)	\(272\)
risch	\(\frac {b^{3} B x}{e^{4}}+\frac {\left (-3 A a \,b^{2} e^{3}+3 A \,b^{3} d \,e^{2}-3 B \,a^{2} b \,e^{3}+9 B a \,b^{2} d \,e^{2}-6 b^{3} B \,d^{2} e \right ) x^{2}+\left (-\frac {3}{2} A \,a^{2} b \,e^{3}-3 A a \,b^{2} d \,e^{2}+\frac {9}{2} A \,b^{3} d^{2} e -\frac {1}{2} B \,a^{3} e^{3}-3 B \,a^{2} b d \,e^{2}+\frac {27}{2} B a \,b^{2} d^{2} e -10 b^{3} B \,d^{3}\right ) x -\frac {2 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+6 A a \,b^{2} d^{2} e^{2}-11 A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+6 B \,a^{2} b \,d^{2} e^{2}-33 B a \,b^{2} d^{3} e +26 b^{3} B \,d^{4}}{6 e}}{e^{4} \left (e x +d \right )^{3}}+\frac {b^{3} \ln \left (e x +d \right ) A}{e^{4}}+\frac {3 b^{2} \ln \left (e x +d \right ) B a}{e^{4}}-\frac {4 b^{3} \ln \left (e x +d \right ) B d}{e^{5}}\)	\(290\)
parallelrisch	\(\frac {-6 A a \,b^{2} d^{2} e^{2}-24 B \ln \left (e x +d \right ) x^{3} b^{3} d \,e^{3}+54 B \,x^{2} a \,b^{2} d \,e^{3}-18 A x a \,b^{2} d \,e^{3}-18 B x \,a^{2} b d \,e^{3}+81 B x a \,b^{2} d^{2} e^{2}+54 B \ln \left (e x +d \right ) x a \,b^{2} d^{2} e^{2}-44 b^{3} B \,d^{4}-2 a^{3} A \,e^{4}-B \,a^{3} d \,e^{3}+11 A \,b^{3} d^{3} e -3 A \,a^{2} b d \,e^{3}+54 B \ln \left (e x +d \right ) x^{2} a \,b^{2} d \,e^{3}+6 A \ln \left (e x +d \right ) x^{3} b^{3} e^{4}+18 B \ln \left (e x +d \right ) x^{3} a \,b^{2} e^{4}-6 B \,a^{2} b \,d^{2} e^{2}+33 B a \,b^{2} d^{3} e -18 A \,x^{2} a \,b^{2} e^{4}+18 A \,x^{2} b^{3} d \,e^{3}-18 B \,x^{2} a^{2} b \,e^{4}-72 B \,x^{2} b^{3} d^{2} e^{2}-9 A x \,a^{2} b \,e^{4}+27 A x \,b^{3} d^{2} e^{2}-108 B x \,b^{3} d^{3} e +6 A \ln \left (e x +d \right ) b^{3} d^{3} e +6 B \,x^{4} b^{3} e^{4}+18 B \ln \left (e x +d \right ) a \,b^{2} d^{3} e -3 B x \,a^{3} e^{4}-24 B \ln \left (e x +d \right ) b^{3} d^{4}+18 A \ln \left (e x +d \right ) x^{2} b^{3} d \,e^{3}-72 B \ln \left (e x +d \right ) x^{2} b^{3} d^{2} e^{2}+18 A \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{2}-72 B \ln \left (e x +d \right ) x \,b^{3} d^{3} e}{6 e^{5} \left (e x +d \right )^{3}}\)	\(483\)

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

(b^3*B/e*x^4-1/6*(2*A*a^3*e^4+3*A*a^2*b*d*e^3+6*A*a*b^2*d^2*e^2-11*A*b^3*d^3*e+B*a^3*d*e^3+6*B*a^2*b*d^2*e^2-3
3*B*a*b^2*d^3*e+44*B*b^3*d^4)/e^5-3*(A*a*b^2*e^2-A*b^3*d*e+B*a^2*b*e^2-3*B*a*b^2*d*e+4*B*b^3*d^2)/e^3*x^2-1/2*
(3*A*a^2*b*e^3+6*A*a*b^2*d*e^2-9*A*b^3*d^2*e+B*a^3*e^3+6*B*a^2*b*d*e^2-27*B*a*b^2*d^2*e+36*B*b^3*d^3)/e^4*x)/(
e*x+d)^3+b^2/e^5*(A*b*e+3*B*a*e-4*B*b*d)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (145) = 290\).

Time = 0.22 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.72 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx=\frac {6 \, B b^{3} e^{4} x^{4} + 18 \, B b^{3} d e^{3} x^{3} - 26 \, B b^{3} d^{4} - 2 \, A a^{3} e^{4} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 6 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 18 \, {\left (B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 3 \, {\left (18 \, B b^{3} d^{3} e - 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x - 6 \, {\left (4 \, B b^{3} d^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + {\left (4 \, B b^{3} d e^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (4 \, B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3}\right )} x^{2} + 3 \, {\left (4 \, B b^{3} d^{3} e - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(6*B*b^3*e^4*x^4 + 18*B*b^3*d*e^3*x^3 - 26*B*b^3*d^4 - 2*A*a^3*e^4 + 11*(3*B*a*b^2 + A*b^3)*d^3*e - 6*(B*a
^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3 - 18*(B*b^3*d^2*e^2 - (3*B*a*b^2 + A*b^3)*d*e^3 + (B*a^2*b
+ A*a*b^2)*e^4)*x^2 - 3*(18*B*b^3*d^3*e - 9*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 6*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^
3 + 3*A*a^2*b)*e^4)*x - 6*(4*B*b^3*d^4 - (3*B*a*b^2 + A*b^3)*d^3*e + (4*B*b^3*d*e^3 - (3*B*a*b^2 + A*b^3)*e^4)
*x^3 + 3*(4*B*b^3*d^2*e^2 - (3*B*a*b^2 + A*b^3)*d*e^3)*x^2 + 3*(4*B*b^3*d^3*e - (3*B*a*b^2 + A*b^3)*d^2*e^2)*x
)*log(e*x + d))/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (143) = 286\).

Time = 5.90 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.26 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx=\frac {B b^{3} x}{e^{4}} + \frac {b^{2} \left (A b e + 3 B a e - 4 B b d\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- 2 A a^{3} e^{4} - 3 A a^{2} b d e^{3} - 6 A a b^{2} d^{2} e^{2} + 11 A b^{3} d^{3} e - B a^{3} d e^{3} - 6 B a^{2} b d^{2} e^{2} + 33 B a b^{2} d^{3} e - 26 B b^{3} d^{4} + x^{2} \left (- 18 A a b^{2} e^{4} + 18 A b^{3} d e^{3} - 18 B a^{2} b e^{4} + 54 B a b^{2} d e^{3} - 36 B b^{3} d^{2} e^{2}\right ) + x \left (- 9 A a^{2} b e^{4} - 18 A a b^{2} d e^{3} + 27 A b^{3} d^{2} e^{2} - 3 B a^{3} e^{4} - 18 B a^{2} b d e^{3} + 81 B a b^{2} d^{2} e^{2} - 60 B b^{3} d^{3} e\right )}{6 d^{3} e^{5} + 18 d^{2} e^{6} x + 18 d e^{7} x^{2} + 6 e^{8} x^{3}} \]

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**4,x)

[Out]

B*b**3*x/e**4 + b**2*(A*b*e + 3*B*a*e - 4*B*b*d)*log(d + e*x)/e**5 + (-2*A*a**3*e**4 - 3*A*a**2*b*d*e**3 - 6*A
*a*b**2*d**2*e**2 + 11*A*b**3*d**3*e - B*a**3*d*e**3 - 6*B*a**2*b*d**2*e**2 + 33*B*a*b**2*d**3*e - 26*B*b**3*d
**4 + x**2*(-18*A*a*b**2*e**4 + 18*A*b**3*d*e**3 - 18*B*a**2*b*e**4 + 54*B*a*b**2*d*e**3 - 36*B*b**3*d**2*e**2
) + x*(-9*A*a**2*b*e**4 - 18*A*a*b**2*d*e**3 + 27*A*b**3*d**2*e**2 - 3*B*a**3*e**4 - 18*B*a**2*b*d*e**3 + 81*B
*a*b**2*d**2*e**2 - 60*B*b**3*d**3*e))/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx=\frac {B b^{3} x}{e^{4}} - \frac {26 \, B b^{3} d^{4} + 2 \, A a^{3} e^{4} - 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 18 \, {\left (2 \, B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \, {\left (20 \, B b^{3} d^{3} e - 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{6 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} - \frac {{\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} \log \left (e x + d\right )}{e^{5}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^4,x, algorithm="maxima")

[Out]

B*b^3*x/e^4 - 1/6*(26*B*b^3*d^4 + 2*A*a^3*e^4 - 11*(3*B*a*b^2 + A*b^3)*d^3*e + 6*(B*a^2*b + A*a*b^2)*d^2*e^2 +
(B*a^3 + 3*A*a^2*b)*d*e^3 + 18*(2*B*b^3*d^2*e^2 - (3*B*a*b^2 + A*b^3)*d*e^3 + (B*a^2*b + A*a*b^2)*e^4)*x^2 +
3*(20*B*b^3*d^3*e - 9*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 6*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x)/
(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) - (4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*log(e*x + d)/e^5

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.87 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx=\frac {B b^{3} x}{e^{4}} - \frac {{\left (4 \, B b^{3} d - 3 \, B a b^{2} e - A b^{3} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} - \frac {26 \, B b^{3} d^{4} - 33 \, B a b^{2} d^{3} e - 11 \, A b^{3} d^{3} e + 6 \, B a^{2} b d^{2} e^{2} + 6 \, A a b^{2} d^{2} e^{2} + B a^{3} d e^{3} + 3 \, A a^{2} b d e^{3} + 2 \, A a^{3} e^{4} + 18 \, {\left (2 \, B b^{3} d^{2} e^{2} - 3 \, B a b^{2} d e^{3} - A b^{3} d e^{3} + B a^{2} b e^{4} + A a b^{2} e^{4}\right )} x^{2} + 3 \, {\left (20 \, B b^{3} d^{3} e - 27 \, B a b^{2} d^{2} e^{2} - 9 \, A b^{3} d^{2} e^{2} + 6 \, B a^{2} b d e^{3} + 6 \, A a b^{2} d e^{3} + B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{5}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^4,x, algorithm="giac")

[Out]

B*b^3*x/e^4 - (4*B*b^3*d - 3*B*a*b^2*e - A*b^3*e)*log(abs(e*x + d))/e^5 - 1/6*(26*B*b^3*d^4 - 33*B*a*b^2*d^3*e
- 11*A*b^3*d^3*e + 6*B*a^2*b*d^2*e^2 + 6*A*a*b^2*d^2*e^2 + B*a^3*d*e^3 + 3*A*a^2*b*d*e^3 + 2*A*a^3*e^4 + 18*(
2*B*b^3*d^2*e^2 - 3*B*a*b^2*d*e^3 - A*b^3*d*e^3 + B*a^2*b*e^4 + A*a*b^2*e^4)*x^2 + 3*(20*B*b^3*d^3*e - 27*B*a*
b^2*d^2*e^2 - 9*A*b^3*d^2*e^2 + 6*B*a^2*b*d*e^3 + 6*A*a*b^2*d*e^3 + B*a^3*e^4 + 3*A*a^2*b*e^4)*x)/((e*x + d)^3
*e^5)

Mupad [B] (verification not implemented)

Time = 1.33 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (A\,b^3\,e-4\,B\,b^3\,d+3\,B\,a\,b^2\,e\right )}{e^5}-\frac {\frac {B\,a^3\,d\,e^3+2\,A\,a^3\,e^4+6\,B\,a^2\,b\,d^2\,e^2+3\,A\,a^2\,b\,d\,e^3-33\,B\,a\,b^2\,d^3\,e+6\,A\,a\,b^2\,d^2\,e^2+26\,B\,b^3\,d^4-11\,A\,b^3\,d^3\,e}{6\,e}+x\,\left (\frac {B\,a^3\,e^3}{2}+3\,B\,a^2\,b\,d\,e^2+\frac {3\,A\,a^2\,b\,e^3}{2}-\frac {27\,B\,a\,b^2\,d^2\,e}{2}+3\,A\,a\,b^2\,d\,e^2+10\,B\,b^3\,d^3-\frac {9\,A\,b^3\,d^2\,e}{2}\right )+x^2\,\left (3\,B\,a^2\,b\,e^3-9\,B\,a\,b^2\,d\,e^2+3\,A\,a\,b^2\,e^3+6\,B\,b^3\,d^2\,e-3\,A\,b^3\,d\,e^2\right )}{d^3\,e^4+3\,d^2\,e^5\,x+3\,d\,e^6\,x^2+e^7\,x^3}+\frac {B\,b^3\,x}{e^4} \]

[In]

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^4,x)

[Out]

(log(d + e*x)*(A*b^3*e - 4*B*b^3*d + 3*B*a*b^2*e))/e^5 - ((2*A*a^3*e^4 + 26*B*b^3*d^4 - 11*A*b^3*d^3*e + B*a^3
*d*e^3 + 6*A*a*b^2*d^2*e^2 + 6*B*a^2*b*d^2*e^2 + 3*A*a^2*b*d*e^3 - 33*B*a*b^2*d^3*e)/(6*e) + x*((B*a^3*e^3)/2
+ 10*B*b^3*d^3 + (3*A*a^2*b*e^3)/2 - (9*A*b^3*d^2*e)/2 + 3*A*a*b^2*d*e^2 - (27*B*a*b^2*d^2*e)/2 + 3*B*a^2*b*d*
e^2) + x^2*(3*A*a*b^2*e^3 + 3*B*a^2*b*e^3 - 3*A*b^3*d*e^2 + 6*B*b^3*d^2*e - 9*B*a*b^2*d*e^2))/(d^3*e^4 + e^7*x
^3 + 3*d^2*e^5*x + 3*d*e^6*x^2) + (B*b^3*x)/e^4