\(\int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx\) [1133]

Optimal result

Integrand size = 20, antiderivative size = 230 \[ \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx=\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) x}{b^6}-\frac {(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}-\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^7 (a+b x)}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^2}{2 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^3}{3 b^7}+\frac {B e^5 (a+b x)^4}{4 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) \log (a+b x)}{b^7} \]

[Out]

10*e^2*(-a*e+b*d)^2*(A*b*e-2*B*a*e+B*b*d)*x/b^6-1/2*(A*b-B*a)*(-a*e+b*d)^5/b^7/(b*x+a)^2-(-a*e+b*d)^4*(5*A*b*e
-6*B*a*e+B*b*d)/b^7/(b*x+a)+5/2*e^3*(-a*e+b*d)*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^2/b^7+1/3*e^4*(A*b*e-6*B*a*e+5*
B*b*d)*(b*x+a)^3/b^7+1/4*B*e^5*(b*x+a)^4/b^7+5*e*(-a*e+b*d)^3*(2*A*b*e-3*B*a*e+B*b*d)*ln(b*x+a)/b^7

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 230, 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} \[ \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx=\frac {e^4 (a+b x)^3 (-6 a B e+A b e+5 b B d)}{3 b^7}+\frac {5 e^3 (a+b x)^2 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}-\frac {(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{b^7 (a+b x)}-\frac {(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}+\frac {5 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^7}+\frac {10 e^2 x (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^6}+\frac {B e^5 (a+b x)^4}{4 b^7} \]

[In]

Int[((A + B*x)*(d + e*x)^5)/(a + b*x)^3,x]

[Out]

(10*e^2*(b*d - a*e)^2*(b*B*d + A*b*e - 2*a*B*e)*x)/b^6 - ((A*b - a*B)*(b*d - a*e)^5)/(2*b^7*(a + b*x)^2) - ((b
*d - a*e)^4*(b*B*d + 5*A*b*e - 6*a*B*e))/(b^7*(a + b*x)) + (5*e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*B*e)*(a +
b*x)^2)/(2*b^7) + (e^4*(5*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^3)/(3*b^7) + (B*e^5*(a + b*x)^4)/(4*b^7) + (5*e*
(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*a*B*e)*Log[a + b*x])/b^7

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 {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e)}{b^6}+\frac {(A b-a B) (b d-a e)^5}{b^6 (a+b x)^3}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^6 (a+b x)^2}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e)}{b^6 (a+b x)}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)}{b^6}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^2}{b^6}+\frac {B e^5 (a+b x)^3}{b^6}\right ) \, dx \\ & = \frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) x}{b^6}-\frac {(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}-\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^7 (a+b x)}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^2}{2 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^3}{3 b^7}+\frac {B e^5 (a+b x)^4}{4 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) \log (a+b x)}{b^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx=\frac {12 b e^2 \left (-10 a^3 B e^3+10 b^3 d^2 (B d+A e)-15 a b^2 d e (2 B d+A e)+6 a^2 b e^2 (5 B d+A e)\right ) x+6 b^2 e^3 \left (6 a^2 B e^2+5 b^2 d (2 B d+A e)-3 a b e (5 B d+A e)\right ) x^2+4 b^3 e^4 (5 b B d+A b e-3 a B e) x^3+3 b^4 B e^5 x^4-\frac {6 (A b-a B) (b d-a e)^5}{(a+b x)^2}-\frac {12 (b d-a e)^4 (b B d+5 A b e-6 a B e)}{a+b x}+60 e (b d-a e)^3 (b B d+2 A b e-3 a B e) \log (a+b x)}{12 b^7} \]

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x)^3,x]

[Out]

(12*b*e^2*(-10*a^3*B*e^3 + 10*b^3*d^2*(B*d + A*e) - 15*a*b^2*d*e*(2*B*d + A*e) + 6*a^2*b*e^2*(5*B*d + A*e))*x
+ 6*b^2*e^3*(6*a^2*B*e^2 + 5*b^2*d*(2*B*d + A*e) - 3*a*b*e*(5*B*d + A*e))*x^2 + 4*b^3*e^4*(5*b*B*d + A*b*e - 3
*a*B*e)*x^3 + 3*b^4*B*e^5*x^4 - (6*(A*b - a*B)*(b*d - a*e)^5)/(a + b*x)^2 - (12*(b*d - a*e)^4*(b*B*d + 5*A*b*e
- 6*a*B*e))/(a + b*x) + 60*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*a*B*e)*Log[a + b*x])/(12*b^7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(222)=444\).

Time = 0.71 (sec) , antiderivative size = 587, normalized size of antiderivative = 2.55

method	result	size
norman	\(\frac {-\frac {30 A \,a^{5} b \,e^{5}-90 A \,a^{4} b^{2} d \,e^{4}+90 A \,a^{3} b^{3} d^{2} e^{3}-30 A \,a^{2} b^{4} d^{3} e^{2}+5 A a \,b^{5} d^{4} e +A \,b^{6} d^{5}-45 B \,a^{6} e^{5}+150 B \,a^{5} b d \,e^{4}-180 B \,a^{4} b^{2} d^{2} e^{3}+90 B \,a^{3} b^{3} d^{3} e^{2}-15 B \,a^{2} b^{4} d^{4} e +B a \,b^{5} d^{5}}{2 b^{7}}-\frac {\left (20 A \,a^{4} b \,e^{5}-60 A \,a^{3} b^{2} d \,e^{4}+60 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e -30 B \,a^{5} e^{5}+100 B \,a^{4} b d \,e^{4}-120 B \,a^{3} b^{2} d^{2} e^{3}+60 B \,a^{2} b^{3} d^{3} e^{2}-10 B a \,b^{4} d^{4} e +B \,b^{5} d^{5}\right ) x}{b^{6}}+\frac {B \,e^{5} x^{6}}{4 b}+\frac {5 e^{2} \left (2 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e -3 B \,a^{3} e^{3}+10 B \,a^{2} b d \,e^{2}-12 B a \,b^{2} d^{2} e +6 b^{3} B \,d^{3}\right ) x^{3}}{3 b^{4}}-\frac {5 e^{3} \left (2 A a b \,e^{2}-6 A \,b^{2} d e -3 B \,a^{2} e^{2}+10 B a b d e -12 b^{2} B \,d^{2}\right ) x^{4}}{12 b^{3}}+\frac {e^{4} \left (2 A b e -3 B a e +10 B b d \right ) x^{5}}{6 b^{2}}}{\left (b x +a \right )^{2}}-\frac {5 e \left (2 A \,a^{3} b \,e^{4}-6 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}-2 A \,b^{4} d^{3} e -3 B \,a^{4} e^{4}+10 B \,a^{3} b d \,e^{3}-12 B \,a^{2} b^{2} d^{2} e^{2}+6 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{7}}\)	\(587\)
default	\(\frac {e^{2} \left (\frac {1}{4} b^{3} B \,x^{4} e^{3}+\frac {1}{3} A \,b^{3} e^{3} x^{3}-B a \,b^{2} e^{3} x^{3}+\frac {5}{3} B \,b^{3} d \,e^{2} x^{3}-\frac {3}{2} A a \,b^{2} e^{3} x^{2}+\frac {5}{2} A \,b^{3} d \,e^{2} x^{2}+3 B \,a^{2} b \,e^{3} x^{2}-\frac {15}{2} B a \,b^{2} d \,e^{2} x^{2}+5 B \,b^{3} d^{2} e \,x^{2}+6 A \,a^{2} b \,e^{3} x -15 A a \,b^{2} d \,e^{2} x +10 A \,b^{3} d^{2} e x -10 B \,a^{3} e^{3} x +30 B \,a^{2} b d \,e^{2} x -30 B a \,b^{2} d^{2} e x +10 b^{3} B \,d^{3} x \right )}{b^{6}}-\frac {5 e \left (2 A \,a^{3} b \,e^{4}-6 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}-2 A \,b^{4} d^{3} e -3 B \,a^{4} e^{4}+10 B \,a^{3} b d \,e^{3}-12 B \,a^{2} b^{2} d^{2} e^{2}+6 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{7}}-\frac {-A \,a^{5} b \,e^{5}+5 A \,a^{4} b^{2} d \,e^{4}-10 A \,a^{3} b^{3} d^{2} e^{3}+10 A \,a^{2} b^{4} d^{3} e^{2}-5 A a \,b^{5} d^{4} e +A \,b^{6} d^{5}+B \,a^{6} e^{5}-5 B \,a^{5} b d \,e^{4}+10 B \,a^{4} b^{2} d^{2} e^{3}-10 B \,a^{3} b^{3} d^{3} e^{2}+5 B \,a^{2} b^{4} d^{4} e -B a \,b^{5} d^{5}}{2 b^{7} \left (b x +a \right )^{2}}-\frac {5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e -6 B \,a^{5} e^{5}+25 B \,a^{4} b d \,e^{4}-40 B \,a^{3} b^{2} d^{2} e^{3}+30 B \,a^{2} b^{3} d^{3} e^{2}-10 B a \,b^{4} d^{4} e +B \,b^{5} d^{5}}{b^{7} \left (b x +a \right )}\)	\(618\)
risch	\(\frac {5 e \ln \left (b x +a \right ) B \,d^{4}}{b^{3}}-\frac {15 e^{4} B a d \,x^{2}}{2 b^{4}}-\frac {15 e^{4} A a d x}{b^{4}}+\frac {30 e^{4} B \,a^{2} d x}{b^{5}}+\frac {e^{5} B \,x^{4}}{4 b^{3}}+\frac {e^{5} A \,x^{3}}{3 b^{3}}-\frac {30 e^{3} B a \,d^{2} x}{b^{4}}-\frac {e^{5} B a \,x^{3}}{b^{4}}+\frac {5 e^{4} B d \,x^{3}}{3 b^{3}}-\frac {10 e^{5} \ln \left (b x +a \right ) A \,a^{3}}{b^{6}}+\frac {10 e^{2} \ln \left (b x +a \right ) A \,d^{3}}{b^{3}}+\frac {15 e^{5} \ln \left (b x +a \right ) B \,a^{4}}{b^{7}}-\frac {50 e^{4} \ln \left (b x +a \right ) B \,a^{3} d}{b^{6}}+\frac {60 e^{3} \ln \left (b x +a \right ) B \,a^{2} d^{2}}{b^{5}}-\frac {30 e^{2} \ln \left (b x +a \right ) B a \,d^{3}}{b^{4}}+\frac {\left (-5 A \,a^{4} b \,e^{5}+20 A \,a^{3} b^{2} d \,e^{4}-30 A \,a^{2} b^{3} d^{2} e^{3}+20 A a \,b^{4} d^{3} e^{2}-5 A \,b^{5} d^{4} e +6 B \,a^{5} e^{5}-25 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-30 B \,a^{2} b^{3} d^{3} e^{2}+10 B a \,b^{4} d^{4} e -B \,b^{5} d^{5}\right ) x -\frac {9 A \,a^{5} b \,e^{5}-35 A \,a^{4} b^{2} d \,e^{4}+50 A \,a^{3} b^{3} d^{2} e^{3}-30 A \,a^{2} b^{4} d^{3} e^{2}+5 A a \,b^{5} d^{4} e +A \,b^{6} d^{5}-11 B \,a^{6} e^{5}+45 B \,a^{5} b d \,e^{4}-70 B \,a^{4} b^{2} d^{2} e^{3}+50 B \,a^{3} b^{3} d^{3} e^{2}-15 B \,a^{2} b^{4} d^{4} e +B a \,b^{5} d^{5}}{2 b}}{b^{6} \left (b x +a \right )^{2}}-\frac {3 e^{5} A a \,x^{2}}{2 b^{4}}+\frac {5 e^{4} A d \,x^{2}}{2 b^{3}}+\frac {3 e^{5} B \,a^{2} x^{2}}{b^{5}}+\frac {5 e^{3} B \,d^{2} x^{2}}{b^{3}}+\frac {6 e^{5} A \,a^{2} x}{b^{5}}+\frac {10 e^{3} A \,d^{2} x}{b^{3}}-\frac {10 e^{5} B \,a^{3} x}{b^{6}}+\frac {10 e^{2} B \,d^{3} x}{b^{3}}+\frac {30 e^{4} \ln \left (b x +a \right ) A \,a^{2} d}{b^{5}}-\frac {30 e^{3} \ln \left (b x +a \right ) A a \,d^{2}}{b^{4}}\)	\(677\)
parallelrisch	\(\text {Expression too large to display}\)	\(1077\)

[In]

int((B*x+A)*(e*x+d)^5/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

(-1/2*(30*A*a^5*b*e^5-90*A*a^4*b^2*d*e^4+90*A*a^3*b^3*d^2*e^3-30*A*a^2*b^4*d^3*e^2+5*A*a*b^5*d^4*e+A*b^6*d^5-4
5*B*a^6*e^5+150*B*a^5*b*d*e^4-180*B*a^4*b^2*d^2*e^3+90*B*a^3*b^3*d^3*e^2-15*B*a^2*b^4*d^4*e+B*a*b^5*d^5)/b^7-(
20*A*a^4*b*e^5-60*A*a^3*b^2*d*e^4+60*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e-30*B*a^5*e^5+100*B*a^4
*b*d*e^4-120*B*a^3*b^2*d^2*e^3+60*B*a^2*b^3*d^3*e^2-10*B*a*b^4*d^4*e+B*b^5*d^5)/b^6*x+1/4*B/b*e^5*x^6+5/3*e^2*
(2*A*a^2*b*e^3-6*A*a*b^2*d*e^2+6*A*b^3*d^2*e-3*B*a^3*e^3+10*B*a^2*b*d*e^2-12*B*a*b^2*d^2*e+6*B*b^3*d^3)/b^4*x^
3-5/12*e^3*(2*A*a*b*e^2-6*A*b^2*d*e-3*B*a^2*e^2+10*B*a*b*d*e-12*B*b^2*d^2)/b^3*x^4+1/6*e^4*(2*A*b*e-3*B*a*e+10
*B*b*d)/b^2*x^5)/(b*x+a)^2-5/b^7*e*(2*A*a^3*b*e^4-6*A*a^2*b^2*d*e^3+6*A*a*b^3*d^2*e^2-2*A*b^4*d^3*e-3*B*a^4*e^
4+10*B*a^3*b*d*e^3-12*B*a^2*b^2*d^2*e^2+6*B*a*b^3*d^3*e-B*b^4*d^4)*ln(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (222) = 444\).

Time = 0.24 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.97 \[ \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx=\frac {3 \, B b^{6} e^{5} x^{6} - 6 \, {\left (B a b^{5} + A b^{6}\right )} d^{5} + 30 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} d^{4} e - 60 \, {\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{3} e^{2} + 60 \, {\left (7 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} d^{2} e^{3} - 30 \, {\left (9 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} d e^{4} + 6 \, {\left (11 \, B a^{6} - 9 \, A a^{5} b\right )} e^{5} + 2 \, {\left (10 \, B b^{6} d e^{4} - {\left (3 \, B a b^{5} - 2 \, A b^{6}\right )} e^{5}\right )} x^{5} + 5 \, {\left (12 \, B b^{6} d^{2} e^{3} - 2 \, {\left (5 \, B a b^{5} - 3 \, A b^{6}\right )} d e^{4} + {\left (3 \, B a^{2} b^{4} - 2 \, A a b^{5}\right )} e^{5}\right )} x^{4} + 20 \, {\left (6 \, B b^{6} d^{3} e^{2} - 6 \, {\left (2 \, B a b^{5} - A b^{6}\right )} d^{2} e^{3} + 2 \, {\left (5 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} d e^{4} - {\left (3 \, B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 6 \, {\left (40 \, B a b^{5} d^{3} e^{2} - 10 \, {\left (11 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )} d^{2} e^{3} + 5 \, {\left (21 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} d e^{4} - {\left (34 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} - 12 \, {\left (B b^{6} d^{5} - 5 \, {\left (2 \, B a b^{5} - A b^{6}\right )} d^{4} e + 20 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} d^{2} e^{3} - 5 \, {\left (B a^{4} b^{2} + A a^{3} b^{3}\right )} d e^{4} + {\left (4 \, B a^{5} b - A a^{4} b^{2}\right )} e^{5}\right )} x + 60 \, {\left (B a^{2} b^{4} d^{4} e - 2 \, {\left (3 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} + 6 \, {\left (2 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} - 2 \, {\left (5 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} d e^{4} + {\left (3 \, B a^{6} - 2 \, A a^{5} b\right )} e^{5} + {\left (B b^{6} d^{4} e - 2 \, {\left (3 \, B a b^{5} - A b^{6}\right )} d^{3} e^{2} + 6 \, {\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} d^{2} e^{3} - 2 \, {\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d e^{4} + {\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} + 2 \, {\left (B a b^{5} d^{4} e - 2 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} d^{3} e^{2} + 6 \, {\left (2 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{2} e^{3} - 2 \, {\left (5 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} d e^{4} + {\left (3 \, B a^{5} b - 2 \, A a^{4} b^{2}\right )} e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \]

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/12*(3*B*b^6*e^5*x^6 - 6*(B*a*b^5 + A*b^6)*d^5 + 30*(3*B*a^2*b^4 - A*a*b^5)*d^4*e - 60*(5*B*a^3*b^3 - 3*A*a^2
*b^4)*d^3*e^2 + 60*(7*B*a^4*b^2 - 5*A*a^3*b^3)*d^2*e^3 - 30*(9*B*a^5*b - 7*A*a^4*b^2)*d*e^4 + 6*(11*B*a^6 - 9*
A*a^5*b)*e^5 + 2*(10*B*b^6*d*e^4 - (3*B*a*b^5 - 2*A*b^6)*e^5)*x^5 + 5*(12*B*b^6*d^2*e^3 - 2*(5*B*a*b^5 - 3*A*b
^6)*d*e^4 + (3*B*a^2*b^4 - 2*A*a*b^5)*e^5)*x^4 + 20*(6*B*b^6*d^3*e^2 - 6*(2*B*a*b^5 - A*b^6)*d^2*e^3 + 2*(5*B*
a^2*b^4 - 3*A*a*b^5)*d*e^4 - (3*B*a^3*b^3 - 2*A*a^2*b^4)*e^5)*x^3 + 6*(40*B*a*b^5*d^3*e^2 - 10*(11*B*a^2*b^4 -
4*A*a*b^5)*d^2*e^3 + 5*(21*B*a^3*b^3 - 11*A*a^2*b^4)*d*e^4 - (34*B*a^4*b^2 - 21*A*a^3*b^3)*e^5)*x^2 - 12*(B*b
^6*d^5 - 5*(2*B*a*b^5 - A*b^6)*d^4*e + 20*(B*a^2*b^4 - A*a*b^5)*d^3*e^2 - 10*(B*a^3*b^3 - 2*A*a^2*b^4)*d^2*e^3
- 5*(B*a^4*b^2 + A*a^3*b^3)*d*e^4 + (4*B*a^5*b - A*a^4*b^2)*e^5)*x + 60*(B*a^2*b^4*d^4*e - 2*(3*B*a^3*b^3 - A
*a^2*b^4)*d^3*e^2 + 6*(2*B*a^4*b^2 - A*a^3*b^3)*d^2*e^3 - 2*(5*B*a^5*b - 3*A*a^4*b^2)*d*e^4 + (3*B*a^6 - 2*A*a
^5*b)*e^5 + (B*b^6*d^4*e - 2*(3*B*a*b^5 - A*b^6)*d^3*e^2 + 6*(2*B*a^2*b^4 - A*a*b^5)*d^2*e^3 - 2*(5*B*a^3*b^3
- 3*A*a^2*b^4)*d*e^4 + (3*B*a^4*b^2 - 2*A*a^3*b^3)*e^5)*x^2 + 2*(B*a*b^5*d^4*e - 2*(3*B*a^2*b^4 - A*a*b^5)*d^3
*e^2 + 6*(2*B*a^3*b^3 - A*a^2*b^4)*d^2*e^3 - 2*(5*B*a^4*b^2 - 3*A*a^3*b^3)*d*e^4 + (3*B*a^5*b - 2*A*a^4*b^2)*e
^5)*x)*log(b*x + a))/(b^9*x^2 + 2*a*b^8*x + a^2*b^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (233) = 466\).

Time = 4.75 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.67 \[ \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx=\frac {B e^{5} x^{4}}{4 b^{3}} + x^{3} \left (\frac {A e^{5}}{3 b^{3}} - \frac {B a e^{5}}{b^{4}} + \frac {5 B d e^{4}}{3 b^{3}}\right ) + x^{2} \left (- \frac {3 A a e^{5}}{2 b^{4}} + \frac {5 A d e^{4}}{2 b^{3}} + \frac {3 B a^{2} e^{5}}{b^{5}} - \frac {15 B a d e^{4}}{2 b^{4}} + \frac {5 B d^{2} e^{3}}{b^{3}}\right ) + x \left (\frac {6 A a^{2} e^{5}}{b^{5}} - \frac {15 A a d e^{4}}{b^{4}} + \frac {10 A d^{2} e^{3}}{b^{3}} - \frac {10 B a^{3} e^{5}}{b^{6}} + \frac {30 B a^{2} d e^{4}}{b^{5}} - \frac {30 B a d^{2} e^{3}}{b^{4}} + \frac {10 B d^{3} e^{2}}{b^{3}}\right ) + \frac {- 9 A a^{5} b e^{5} + 35 A a^{4} b^{2} d e^{4} - 50 A a^{3} b^{3} d^{2} e^{3} + 30 A a^{2} b^{4} d^{3} e^{2} - 5 A a b^{5} d^{4} e - A b^{6} d^{5} + 11 B a^{6} e^{5} - 45 B a^{5} b d e^{4} + 70 B a^{4} b^{2} d^{2} e^{3} - 50 B a^{3} b^{3} d^{3} e^{2} + 15 B a^{2} b^{4} d^{4} e - B a b^{5} d^{5} + x \left (- 10 A a^{4} b^{2} e^{5} + 40 A a^{3} b^{3} d e^{4} - 60 A a^{2} b^{4} d^{2} e^{3} + 40 A a b^{5} d^{3} e^{2} - 10 A b^{6} d^{4} e + 12 B a^{5} b e^{5} - 50 B a^{4} b^{2} d e^{4} + 80 B a^{3} b^{3} d^{2} e^{3} - 60 B a^{2} b^{4} d^{3} e^{2} + 20 B a b^{5} d^{4} e - 2 B b^{6} d^{5}\right )}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac {5 e \left (a e - b d\right )^{3} \left (- 2 A b e + 3 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{7}} \]

[In]

integrate((B*x+A)*(e*x+d)**5/(b*x+a)**3,x)

[Out]

B*e**5*x**4/(4*b**3) + x**3*(A*e**5/(3*b**3) - B*a*e**5/b**4 + 5*B*d*e**4/(3*b**3)) + x**2*(-3*A*a*e**5/(2*b**
4) + 5*A*d*e**4/(2*b**3) + 3*B*a**2*e**5/b**5 - 15*B*a*d*e**4/(2*b**4) + 5*B*d**2*e**3/b**3) + x*(6*A*a**2*e**
5/b**5 - 15*A*a*d*e**4/b**4 + 10*A*d**2*e**3/b**3 - 10*B*a**3*e**5/b**6 + 30*B*a**2*d*e**4/b**5 - 30*B*a*d**2*
e**3/b**4 + 10*B*d**3*e**2/b**3) + (-9*A*a**5*b*e**5 + 35*A*a**4*b**2*d*e**4 - 50*A*a**3*b**3*d**2*e**3 + 30*A
*a**2*b**4*d**3*e**2 - 5*A*a*b**5*d**4*e - A*b**6*d**5 + 11*B*a**6*e**5 - 45*B*a**5*b*d*e**4 + 70*B*a**4*b**2*
d**2*e**3 - 50*B*a**3*b**3*d**3*e**2 + 15*B*a**2*b**4*d**4*e - B*a*b**5*d**5 + x*(-10*A*a**4*b**2*e**5 + 40*A*
a**3*b**3*d*e**4 - 60*A*a**2*b**4*d**2*e**3 + 40*A*a*b**5*d**3*e**2 - 10*A*b**6*d**4*e + 12*B*a**5*b*e**5 - 50
*B*a**4*b**2*d*e**4 + 80*B*a**3*b**3*d**2*e**3 - 60*B*a**2*b**4*d**3*e**2 + 20*B*a*b**5*d**4*e - 2*B*b**6*d**5
))/(2*a**2*b**7 + 4*a*b**8*x + 2*b**9*x**2) + 5*e*(a*e - b*d)**3*(-2*A*b*e + 3*B*a*e - B*b*d)*log(a + b*x)/b**
7

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (222) = 444\).

Time = 0.20 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.57 \[ \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx=-\frac {{\left (B a b^{5} + A b^{6}\right )} d^{5} - 5 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \, {\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \, {\left (7 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (9 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} d e^{4} - {\left (11 \, B a^{6} - 9 \, A a^{5} b\right )} e^{5} + 2 \, {\left (B b^{6} d^{5} - 5 \, {\left (2 \, B a b^{5} - A b^{6}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{4} - 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (5 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (6 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} e^{5}\right )} x}{2 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac {3 \, B b^{3} e^{5} x^{4} + 4 \, {\left (5 \, B b^{3} d e^{4} - {\left (3 \, B a b^{2} - A b^{3}\right )} e^{5}\right )} x^{3} + 6 \, {\left (10 \, B b^{3} d^{2} e^{3} - 5 \, {\left (3 \, B a b^{2} - A b^{3}\right )} d e^{4} + 3 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} e^{5}\right )} x^{2} + 12 \, {\left (10 \, B b^{3} d^{3} e^{2} - 10 \, {\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} e^{3} + 15 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} d e^{4} - 2 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{5}\right )} x}{12 \, b^{6}} + \frac {5 \, {\left (B b^{4} d^{4} e - 2 \, {\left (3 \, B a b^{3} - A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{3} - 2 \, {\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \]

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*((B*a*b^5 + A*b^6)*d^5 - 5*(3*B*a^2*b^4 - A*a*b^5)*d^4*e + 10*(5*B*a^3*b^3 - 3*A*a^2*b^4)*d^3*e^2 - 10*(7
*B*a^4*b^2 - 5*A*a^3*b^3)*d^2*e^3 + 5*(9*B*a^5*b - 7*A*a^4*b^2)*d*e^4 - (11*B*a^6 - 9*A*a^5*b)*e^5 + 2*(B*b^6*
d^5 - 5*(2*B*a*b^5 - A*b^6)*d^4*e + 10*(3*B*a^2*b^4 - 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 - 3*A*a^2*b^4)*d^2*
e^3 + 5*(5*B*a^4*b^2 - 4*A*a^3*b^3)*d*e^4 - (6*B*a^5*b - 5*A*a^4*b^2)*e^5)*x)/(b^9*x^2 + 2*a*b^8*x + a^2*b^7)
+ 1/12*(3*B*b^3*e^5*x^4 + 4*(5*B*b^3*d*e^4 - (3*B*a*b^2 - A*b^3)*e^5)*x^3 + 6*(10*B*b^3*d^2*e^3 - 5*(3*B*a*b^2
- A*b^3)*d*e^4 + 3*(2*B*a^2*b - A*a*b^2)*e^5)*x^2 + 12*(10*B*b^3*d^3*e^2 - 10*(3*B*a*b^2 - A*b^3)*d^2*e^3 + 1
5*(2*B*a^2*b - A*a*b^2)*d*e^4 - 2*(5*B*a^3 - 3*A*a^2*b)*e^5)*x)/b^6 + 5*(B*b^4*d^4*e - 2*(3*B*a*b^3 - A*b^4)*d
^3*e^2 + 6*(2*B*a^2*b^2 - A*a*b^3)*d^2*e^3 - 2*(5*B*a^3*b - 3*A*a^2*b^2)*d*e^4 + (3*B*a^4 - 2*A*a^3*b)*e^5)*lo
g(b*x + a)/b^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (222) = 444\).

Time = 0.29 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.75 \[ \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx=\frac {5 \, {\left (B b^{4} d^{4} e - 6 \, B a b^{3} d^{3} e^{2} + 2 \, A b^{4} d^{3} e^{2} + 12 \, B a^{2} b^{2} d^{2} e^{3} - 6 \, A a b^{3} d^{2} e^{3} - 10 \, B a^{3} b d e^{4} + 6 \, A a^{2} b^{2} d e^{4} + 3 \, B a^{4} e^{5} - 2 \, A a^{3} b e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {B a b^{5} d^{5} + A b^{6} d^{5} - 15 \, B a^{2} b^{4} d^{4} e + 5 \, A a b^{5} d^{4} e + 50 \, B a^{3} b^{3} d^{3} e^{2} - 30 \, A a^{2} b^{4} d^{3} e^{2} - 70 \, B a^{4} b^{2} d^{2} e^{3} + 50 \, A a^{3} b^{3} d^{2} e^{3} + 45 \, B a^{5} b d e^{4} - 35 \, A a^{4} b^{2} d e^{4} - 11 \, B a^{6} e^{5} + 9 \, A a^{5} b e^{5} + 2 \, {\left (B b^{6} d^{5} - 10 \, B a b^{5} d^{4} e + 5 \, A b^{6} d^{4} e + 30 \, B a^{2} b^{4} d^{3} e^{2} - 20 \, A a b^{5} d^{3} e^{2} - 40 \, B a^{3} b^{3} d^{2} e^{3} + 30 \, A a^{2} b^{4} d^{2} e^{3} + 25 \, B a^{4} b^{2} d e^{4} - 20 \, A a^{3} b^{3} d e^{4} - 6 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{7}} + \frac {3 \, B b^{9} e^{5} x^{4} + 20 \, B b^{9} d e^{4} x^{3} - 12 \, B a b^{8} e^{5} x^{3} + 4 \, A b^{9} e^{5} x^{3} + 60 \, B b^{9} d^{2} e^{3} x^{2} - 90 \, B a b^{8} d e^{4} x^{2} + 30 \, A b^{9} d e^{4} x^{2} + 36 \, B a^{2} b^{7} e^{5} x^{2} - 18 \, A a b^{8} e^{5} x^{2} + 120 \, B b^{9} d^{3} e^{2} x - 360 \, B a b^{8} d^{2} e^{3} x + 120 \, A b^{9} d^{2} e^{3} x + 360 \, B a^{2} b^{7} d e^{4} x - 180 \, A a b^{8} d e^{4} x - 120 \, B a^{3} b^{6} e^{5} x + 72 \, A a^{2} b^{7} e^{5} x}{12 \, b^{12}} \]

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^3,x, algorithm="giac")

[Out]

5*(B*b^4*d^4*e - 6*B*a*b^3*d^3*e^2 + 2*A*b^4*d^3*e^2 + 12*B*a^2*b^2*d^2*e^3 - 6*A*a*b^3*d^2*e^3 - 10*B*a^3*b*d
*e^4 + 6*A*a^2*b^2*d*e^4 + 3*B*a^4*e^5 - 2*A*a^3*b*e^5)*log(abs(b*x + a))/b^7 - 1/2*(B*a*b^5*d^5 + A*b^6*d^5 -
15*B*a^2*b^4*d^4*e + 5*A*a*b^5*d^4*e + 50*B*a^3*b^3*d^3*e^2 - 30*A*a^2*b^4*d^3*e^2 - 70*B*a^4*b^2*d^2*e^3 + 5
0*A*a^3*b^3*d^2*e^3 + 45*B*a^5*b*d*e^4 - 35*A*a^4*b^2*d*e^4 - 11*B*a^6*e^5 + 9*A*a^5*b*e^5 + 2*(B*b^6*d^5 - 10
*B*a*b^5*d^4*e + 5*A*b^6*d^4*e + 30*B*a^2*b^4*d^3*e^2 - 20*A*a*b^5*d^3*e^2 - 40*B*a^3*b^3*d^2*e^3 + 30*A*a^2*b
^4*d^2*e^3 + 25*B*a^4*b^2*d*e^4 - 20*A*a^3*b^3*d*e^4 - 6*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*x)/((b*x + a)^2*b^7) +
1/12*(3*B*b^9*e^5*x^4 + 20*B*b^9*d*e^4*x^3 - 12*B*a*b^8*e^5*x^3 + 4*A*b^9*e^5*x^3 + 60*B*b^9*d^2*e^3*x^2 - 90
*B*a*b^8*d*e^4*x^2 + 30*A*b^9*d*e^4*x^2 + 36*B*a^2*b^7*e^5*x^2 - 18*A*a*b^8*e^5*x^2 + 120*B*b^9*d^3*e^2*x - 36
0*B*a*b^8*d^2*e^3*x + 120*A*b^9*d^2*e^3*x + 360*B*a^2*b^7*d*e^4*x - 180*A*a*b^8*d*e^4*x - 120*B*a^3*b^6*e^5*x
+ 72*A*a^2*b^7*e^5*x)/b^12

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.96 \[ \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx=x^3\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{3\,b^3}-\frac {B\,a\,e^5}{b^4}\right )-\frac {x\,\left (-6\,B\,a^5\,e^5+25\,B\,a^4\,b\,d\,e^4+5\,A\,a^4\,b\,e^5-40\,B\,a^3\,b^2\,d^2\,e^3-20\,A\,a^3\,b^2\,d\,e^4+30\,B\,a^2\,b^3\,d^3\,e^2+30\,A\,a^2\,b^3\,d^2\,e^3-10\,B\,a\,b^4\,d^4\,e-20\,A\,a\,b^4\,d^3\,e^2+B\,b^5\,d^5+5\,A\,b^5\,d^4\,e\right )+\frac {-11\,B\,a^6\,e^5+45\,B\,a^5\,b\,d\,e^4+9\,A\,a^5\,b\,e^5-70\,B\,a^4\,b^2\,d^2\,e^3-35\,A\,a^4\,b^2\,d\,e^4+50\,B\,a^3\,b^3\,d^3\,e^2+50\,A\,a^3\,b^3\,d^2\,e^3-15\,B\,a^2\,b^4\,d^4\,e-30\,A\,a^2\,b^4\,d^3\,e^2+B\,a\,b^5\,d^5+5\,A\,a\,b^5\,d^4\,e+A\,b^6\,d^5}{2\,b}}{a^2\,b^6+2\,a\,b^7\,x+b^8\,x^2}-x^2\,\left (\frac {3\,a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^3}-\frac {3\,B\,a\,e^5}{b^4}\right )}{2\,b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{2\,b^3}+\frac {3\,B\,a^2\,e^5}{2\,b^5}\right )-x\,\left (\frac {3\,a^2\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^3}-\frac {3\,B\,a\,e^5}{b^4}\right )}{b^2}-\frac {3\,a\,\left (\frac {3\,a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^3}-\frac {3\,B\,a\,e^5}{b^4}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b^3}+\frac {3\,B\,a^2\,e^5}{b^5}\right )}{b}-\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{b^3}+\frac {B\,a^3\,e^5}{b^6}\right )+\frac {\ln \left (a+b\,x\right )\,\left (15\,B\,a^4\,e^5-50\,B\,a^3\,b\,d\,e^4-10\,A\,a^3\,b\,e^5+60\,B\,a^2\,b^2\,d^2\,e^3+30\,A\,a^2\,b^2\,d\,e^4-30\,B\,a\,b^3\,d^3\,e^2-30\,A\,a\,b^3\,d^2\,e^3+5\,B\,b^4\,d^4\,e+10\,A\,b^4\,d^3\,e^2\right )}{b^7}+\frac {B\,e^5\,x^4}{4\,b^3} \]

[In]

int(((A + B*x)*(d + e*x)^5)/(a + b*x)^3,x)

[Out]

x^3*((A*e^5 + 5*B*d*e^4)/(3*b^3) - (B*a*e^5)/b^4) - (x*(B*b^5*d^5 - 6*B*a^5*e^5 + 5*A*a^4*b*e^5 + 5*A*b^5*d^4*
e - 20*A*a*b^4*d^3*e^2 - 20*A*a^3*b^2*d*e^4 + 30*A*a^2*b^3*d^2*e^3 + 30*B*a^2*b^3*d^3*e^2 - 40*B*a^3*b^2*d^2*e
^3 - 10*B*a*b^4*d^4*e + 25*B*a^4*b*d*e^4) + (A*b^6*d^5 - 11*B*a^6*e^5 + 9*A*a^5*b*e^5 + B*a*b^5*d^5 - 35*A*a^4
*b^2*d*e^4 - 15*B*a^2*b^4*d^4*e - 30*A*a^2*b^4*d^3*e^2 + 50*A*a^3*b^3*d^2*e^3 + 50*B*a^3*b^3*d^3*e^2 - 70*B*a^
4*b^2*d^2*e^3 + 5*A*a*b^5*d^4*e + 45*B*a^5*b*d*e^4)/(2*b))/(a^2*b^6 + b^8*x^2 + 2*a*b^7*x) - x^2*((3*a*((A*e^5
+ 5*B*d*e^4)/b^3 - (3*B*a*e^5)/b^4))/(2*b) - (5*d*e^3*(A*e + 2*B*d))/(2*b^3) + (3*B*a^2*e^5)/(2*b^5)) - x*((3
*a^2*((A*e^5 + 5*B*d*e^4)/b^3 - (3*B*a*e^5)/b^4))/b^2 - (3*a*((3*a*((A*e^5 + 5*B*d*e^4)/b^3 - (3*B*a*e^5)/b^4)
)/b - (5*d*e^3*(A*e + 2*B*d))/b^3 + (3*B*a^2*e^5)/b^5))/b - (10*d^2*e^2*(A*e + B*d))/b^3 + (B*a^3*e^5)/b^6) +
(log(a + b*x)*(15*B*a^4*e^5 - 10*A*a^3*b*e^5 + 5*B*b^4*d^4*e + 10*A*b^4*d^3*e^2 - 30*A*a*b^3*d^2*e^3 + 30*A*a^
2*b^2*d*e^4 - 30*B*a*b^3*d^3*e^2 + 60*B*a^2*b^2*d^2*e^3 - 50*B*a^3*b*d*e^4))/b^7 + (B*e^5*x^4)/(4*b^3)