\(\int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx\) [1123]

Optimal result

Integrand size = 20, antiderivative size = 187 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx=\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) x}{b^5}-\frac {(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac {e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^2}{b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^3}{3 b^6}+\frac {B e^4 (a+b x)^4}{4 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) \log (a+b x)}{b^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 187, 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} \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx=\frac {e^3 (a+b x)^3 (-5 a B e+A b e+4 b B d)}{3 b^6}+\frac {e^2 (a+b x)^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac {(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac {(b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{b^6}+\frac {2 e x (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac {B e^4 (a+b x)^4}{4 b^6} \]

[In]

Int[((A + B*x)*(d + e*x)^4)/(a + b*x)^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.95 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx=\frac {B \left (12 a^5 e^4-48 a^4 b e^3 (d+e x)+6 a^3 b^2 e^2 \left (12 d^2+24 d e x-5 e^2 x^2\right )+b^5 e x^2 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )+2 a^2 b^3 e \left (-24 d^3-72 d^2 e x+48 d e^2 x^2+5 e^3 x^3\right )+a b^4 \left (12 d^4+48 d^3 e x-108 d^2 e^2 x^2-32 d e^3 x^3-5 e^4 x^4\right )\right )-4 A b \left (3 a^4 e^4-3 a^3 b e^3 (4 d+3 e x)+6 a^2 b^2 e^2 \left (3 d^2+4 d e x-e^2 x^2\right )+2 a b^3 e \left (-6 d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+b^4 \left (3 d^4-18 d^2 e^2 x^2-6 d e^3 x^3-e^4 x^4\right )\right )+12 (b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x) \log (a+b x)}{12 b^6 (a+b x)} \]

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(a + b*x)^2,x]

[Out]

(B*(12*a^5*e^4 - 48*a^4*b*e^3*(d + e*x) + 6*a^3*b^2*e^2*(12*d^2 + 24*d*e*x - 5*e^2*x^2) + b^5*e*x^2*(48*d^3 +
36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) + 2*a^2*b^3*e*(-24*d^3 - 72*d^2*e*x + 48*d*e^2*x^2 + 5*e^3*x^3) + a*b^4
*(12*d^4 + 48*d^3*e*x - 108*d^2*e^2*x^2 - 32*d*e^3*x^3 - 5*e^4*x^4)) - 4*A*b*(3*a^4*e^4 - 3*a^3*b*e^3*(4*d + 3
*e*x) + 6*a^2*b^2*e^2*(3*d^2 + 4*d*e*x - e^2*x^2) + 2*a*b^3*e*(-6*d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3) + b
^4*(3*d^4 - 18*d^2*e^2*x^2 - 6*d*e^3*x^3 - e^4*x^4)) + 12*(b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e)*(a + b*x)*
Log[a + b*x])/(12*b^6*(a + b*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(420\) vs. \(2(183)=366\).

Time = 0.71 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.25

method	result	size
norman	\(\frac {\frac {\left (4 A \,a^{4} b \,e^{4}-12 A \,a^{3} b^{2} d \,e^{3}+12 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-5 B \,a^{5} e^{4}+16 B \,a^{4} b d \,e^{3}-18 B \,a^{3} b^{2} d^{2} e^{2}+8 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) x}{b^{5} a}+\frac {B \,e^{4} x^{5}}{4 b}+\frac {e \left (4 A \,a^{2} b \,e^{3}-12 A a \,b^{2} d \,e^{2}+12 A \,b^{3} d^{2} e -5 B \,a^{3} e^{3}+16 B \,a^{2} b d \,e^{2}-18 B a \,b^{2} d^{2} e +8 b^{3} B \,d^{3}\right ) x^{2}}{2 b^{4}}-\frac {e^{2} \left (4 A a b \,e^{2}-12 A \,b^{2} d e -5 B \,a^{2} e^{2}+16 B a b d e -18 b^{2} B \,d^{2}\right ) x^{3}}{6 b^{3}}+\frac {e^{3} \left (4 A b e -5 B a e +16 B b d \right ) x^{4}}{12 b^{2}}}{b x +a}-\frac {\left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -5 B \,a^{4} e^{4}+16 B \,a^{3} b d \,e^{3}-18 B \,a^{2} b^{2} d^{2} e^{2}+8 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(421\)
default	\(\frac {e \left (\frac {1}{4} b^{3} B \,x^{4} e^{3}+\frac {1}{3} A \,b^{3} e^{3} x^{3}-\frac {2}{3} B a \,b^{2} e^{3} x^{3}+\frac {4}{3} B \,b^{3} d \,e^{2} x^{3}-A a \,b^{2} e^{3} x^{2}+2 A \,b^{3} d \,e^{2} x^{2}+\frac {3}{2} B \,a^{2} b \,e^{3} x^{2}-4 B a \,b^{2} d \,e^{2} x^{2}+3 B \,b^{3} d^{2} e \,x^{2}+3 A \,a^{2} b \,e^{3} x -8 A a \,b^{2} d \,e^{2} x +6 A \,b^{3} d^{2} e x -4 B \,a^{3} e^{3} x +12 B \,a^{2} b d \,e^{2} x -12 B a \,b^{2} d^{2} e x +4 b^{3} B \,d^{3} x \right )}{b^{5}}+\frac {\left (-4 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}-12 A a \,b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e +5 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-8 B a \,b^{3} d^{3} e +B \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}-\frac {A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}}{b^{6} \left (b x +a \right )}\)	\(440\)
risch	\(\frac {12 \ln \left (b x +a \right ) A \,a^{2} d \,e^{3}}{b^{4}}-\frac {12 \ln \left (b x +a \right ) A a \,d^{2} e^{2}}{b^{3}}-\frac {16 \ln \left (b x +a \right ) B \,a^{3} d \,e^{3}}{b^{5}}+\frac {18 \ln \left (b x +a \right ) B \,a^{2} d^{2} e^{2}}{b^{4}}-\frac {8 \ln \left (b x +a \right ) B a \,d^{3} e}{b^{3}}-\frac {e^{4} A a \,x^{2}}{b^{3}}+\frac {2 e^{3} A d \,x^{2}}{b^{2}}+\frac {5 \ln \left (b x +a \right ) B \,a^{4} e^{4}}{b^{6}}+\frac {B \,a^{5} e^{4}}{b^{6} \left (b x +a \right )}+\frac {B a \,d^{4}}{b^{2} \left (b x +a \right )}-\frac {4 \ln \left (b x +a \right ) A \,a^{3} e^{4}}{b^{5}}+\frac {4 \ln \left (b x +a \right ) A \,d^{3} e}{b^{2}}-\frac {2 e^{4} B a \,x^{3}}{3 b^{3}}+\frac {4 e^{3} B d \,x^{3}}{3 b^{2}}-\frac {4 e^{3} B a d \,x^{2}}{b^{3}}-\frac {8 e^{3} A a d x}{b^{3}}+\frac {12 e^{3} B \,a^{2} d x}{b^{4}}-\frac {12 e^{2} B a \,d^{2} x}{b^{3}}+\frac {3 e^{4} B \,a^{2} x^{2}}{2 b^{4}}+\frac {3 e^{2} B \,d^{2} x^{2}}{b^{2}}+\frac {3 e^{4} A \,a^{2} x}{b^{4}}+\frac {6 e^{2} A \,d^{2} x}{b^{2}}-\frac {4 e^{4} B \,a^{3} x}{b^{5}}+\frac {4 e B \,d^{3} x}{b^{2}}-\frac {A \,a^{4} e^{4}}{b^{5} \left (b x +a \right )}+\frac {e^{4} B \,x^{4}}{4 b^{2}}+\frac {e^{4} A \,x^{3}}{3 b^{2}}-\frac {A \,d^{4}}{b \left (b x +a \right )}+\frac {\ln \left (b x +a \right ) B \,d^{4}}{b^{2}}+\frac {4 A \,a^{3} d \,e^{3}}{b^{4} \left (b x +a \right )}-\frac {6 A \,a^{2} d^{2} e^{2}}{b^{3} \left (b x +a \right )}+\frac {4 A a \,d^{3} e}{b^{2} \left (b x +a \right )}-\frac {4 B \,a^{4} d \,e^{3}}{b^{5} \left (b x +a \right )}+\frac {6 B \,a^{3} d^{2} e^{2}}{b^{4} \left (b x +a \right )}-\frac {4 B \,a^{2} d^{3} e}{b^{3} \left (b x +a \right )}\)	\(564\)
parallelrisch	\(-\frac {48 A \ln \left (b x +a \right ) x \,a^{3} b^{2} e^{4}-48 A \ln \left (b x +a \right ) x \,b^{5} d^{3} e -60 B \ln \left (b x +a \right ) x \,a^{4} b \,e^{4}+48 A \ln \left (b x +a \right ) a^{4} b \,e^{4}-12 B \ln \left (b x +a \right ) a \,b^{4} d^{4}+5 B \,x^{4} a \,b^{4} e^{4}-16 B \,x^{4} b^{5} d \,e^{3}+8 A \,x^{3} a \,b^{4} e^{4}-24 A \,x^{3} b^{5} d \,e^{3}-10 B \,x^{3} a^{2} b^{3} e^{4}-36 B \,x^{3} b^{5} d^{2} e^{2}-24 A \,x^{2} a^{2} b^{3} e^{4}-72 A \,x^{2} b^{5} d^{2} e^{2}+30 B \,x^{2} a^{3} b^{2} e^{4}-48 B \,x^{2} b^{5} d^{3} e -144 A \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{3}+144 A \ln \left (b x +a \right ) a^{2} b^{3} d^{2} e^{2}-48 A \ln \left (b x +a \right ) a \,b^{4} d^{3} e +192 B \ln \left (b x +a \right ) a^{4} b d \,e^{3}-216 B \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{2}-4 A \,x^{4} b^{5} e^{4}-60 B \ln \left (b x +a \right ) a^{5} e^{4}-48 A a \,b^{4} d^{3} e +192 B \,a^{4} b d \,e^{3}-216 B \,a^{3} b^{2} d^{2} e^{2}+96 B \,a^{2} b^{3} d^{3} e -144 A \,a^{3} b^{2} d \,e^{3}+144 A \,a^{2} b^{3} d^{2} e^{2}-12 B \ln \left (b x +a \right ) x \,b^{5} d^{4}+12 A \,b^{5} d^{4}+96 B \ln \left (b x +a \right ) a^{2} b^{3} d^{3} e -144 A \ln \left (b x +a \right ) x \,a^{2} b^{3} d \,e^{3}+144 A \ln \left (b x +a \right ) x a \,b^{4} d^{2} e^{2}+192 B \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{3}-216 B \ln \left (b x +a \right ) x \,a^{2} b^{3} d^{2} e^{2}+96 B \ln \left (b x +a \right ) x a \,b^{4} d^{3} e +32 B \,x^{3} a \,b^{4} d \,e^{3}+72 A \,x^{2} a \,b^{4} d \,e^{3}-96 B \,x^{2} a^{2} b^{3} d \,e^{3}+108 B \,x^{2} a \,b^{4} d^{2} e^{2}-60 B \,a^{5} e^{4}-12 B a \,b^{4} d^{4}-3 B \,x^{5} e^{4} b^{5}+48 A \,a^{4} b \,e^{4}}{12 b^{6} \left (b x +a \right )}\)	\(684\)

[In]

int((B*x+A)*(e*x+d)^4/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

((4*A*a^4*b*e^4-12*A*a^3*b^2*d*e^3+12*A*a^2*b^3*d^2*e^2-4*A*a*b^4*d^3*e+A*b^5*d^4-5*B*a^5*e^4+16*B*a^4*b*d*e^3
-18*B*a^3*b^2*d^2*e^2+8*B*a^2*b^3*d^3*e-B*a*b^4*d^4)/b^5/a*x+1/4*B/b*e^4*x^5+1/2*e*(4*A*a^2*b*e^3-12*A*a*b^2*d
*e^2+12*A*b^3*d^2*e-5*B*a^3*e^3+16*B*a^2*b*d*e^2-18*B*a*b^2*d^2*e+8*B*b^3*d^3)/b^4*x^2-1/6*e^2*(4*A*a*b*e^2-12
*A*b^2*d*e-5*B*a^2*e^2+16*B*a*b*d*e-18*B*b^2*d^2)/b^3*x^3+1/12*e^3*(4*A*b*e-5*B*a*e+16*B*b*d)/b^2*x^4)/(b*x+a)
-(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e-5*B*a^4*e^4+16*B*a^3*b*d*e^3-18*B*a^2*b^2*
d^2*e^2+8*B*a*b^3*d^3*e-B*b^4*d^4)/b^6*ln(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (182) = 364\).

Time = 0.23 (sec) , antiderivative size = 610, normalized size of antiderivative = 3.26 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx=\frac {3 \, B b^{5} e^{4} x^{5} + 12 \, {\left (B a b^{4} - A b^{5}\right )} d^{4} - 48 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 72 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 48 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + 12 \, {\left (B a^{5} - A a^{4} b\right )} e^{4} + {\left (16 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} e^{4}\right )} x^{4} + 2 \, {\left (18 \, B b^{5} d^{2} e^{2} - 4 \, {\left (4 \, B a b^{4} - 3 \, A b^{5}\right )} d e^{3} + {\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 6 \, {\left (8 \, B b^{5} d^{3} e - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} d^{2} e^{2} + 4 \, {\left (4 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} d e^{3} - {\left (5 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 12 \, {\left (4 \, B a b^{4} d^{3} e - 6 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} - {\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (B a b^{4} d^{4} - 4 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} d e^{3} + {\left (5 \, B a^{5} - 4 \, A a^{4} b\right )} e^{4} + {\left (B b^{5} d^{4} - 4 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} + {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \]

[In]

integrate((B*x+A)*(e*x+d)^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/12*(3*B*b^5*e^4*x^5 + 12*(B*a*b^4 - A*b^5)*d^4 - 48*(B*a^2*b^3 - A*a*b^4)*d^3*e + 72*(B*a^3*b^2 - A*a^2*b^3)
*d^2*e^2 - 48*(B*a^4*b - A*a^3*b^2)*d*e^3 + 12*(B*a^5 - A*a^4*b)*e^4 + (16*B*b^5*d*e^3 - (5*B*a*b^4 - 4*A*b^5)
*e^4)*x^4 + 2*(18*B*b^5*d^2*e^2 - 4*(4*B*a*b^4 - 3*A*b^5)*d*e^3 + (5*B*a^2*b^3 - 4*A*a*b^4)*e^4)*x^3 + 6*(8*B*
b^5*d^3*e - 6*(3*B*a*b^4 - 2*A*b^5)*d^2*e^2 + 4*(4*B*a^2*b^3 - 3*A*a*b^4)*d*e^3 - (5*B*a^3*b^2 - 4*A*a^2*b^3)*
e^4)*x^2 + 12*(4*B*a*b^4*d^3*e - 6*(2*B*a^2*b^3 - A*a*b^4)*d^2*e^2 + 4*(3*B*a^3*b^2 - 2*A*a^2*b^3)*d*e^3 - (4*
B*a^4*b - 3*A*a^3*b^2)*e^4)*x + 12*(B*a*b^4*d^4 - 4*(2*B*a^2*b^3 - A*a*b^4)*d^3*e + 6*(3*B*a^3*b^2 - 2*A*a^2*b
^3)*d^2*e^2 - 4*(4*B*a^4*b - 3*A*a^3*b^2)*d*e^3 + (5*B*a^5 - 4*A*a^4*b)*e^4 + (B*b^5*d^4 - 4*(2*B*a*b^4 - A*b^
5)*d^3*e + 6*(3*B*a^2*b^3 - 2*A*a*b^4)*d^2*e^2 - 4*(4*B*a^3*b^2 - 3*A*a^2*b^3)*d*e^3 + (5*B*a^4*b - 4*A*a^3*b^
2)*e^4)*x)*log(b*x + a))/(b^7*x + a*b^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (189) = 378\).

Time = 1.08 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx=\frac {B e^{4} x^{4}}{4 b^{2}} + x^{3} \left (\frac {A e^{4}}{3 b^{2}} - \frac {2 B a e^{4}}{3 b^{3}} + \frac {4 B d e^{3}}{3 b^{2}}\right ) + x^{2} \left (- \frac {A a e^{4}}{b^{3}} + \frac {2 A d e^{3}}{b^{2}} + \frac {3 B a^{2} e^{4}}{2 b^{4}} - \frac {4 B a d e^{3}}{b^{3}} + \frac {3 B d^{2} e^{2}}{b^{2}}\right ) + x \left (\frac {3 A a^{2} e^{4}}{b^{4}} - \frac {8 A a d e^{3}}{b^{3}} + \frac {6 A d^{2} e^{2}}{b^{2}} - \frac {4 B a^{3} e^{4}}{b^{5}} + \frac {12 B a^{2} d e^{3}}{b^{4}} - \frac {12 B a d^{2} e^{2}}{b^{3}} + \frac {4 B d^{3} e}{b^{2}}\right ) + \frac {- A a^{4} b e^{4} + 4 A a^{3} b^{2} d e^{3} - 6 A a^{2} b^{3} d^{2} e^{2} + 4 A a b^{4} d^{3} e - A b^{5} d^{4} + B a^{5} e^{4} - 4 B a^{4} b d e^{3} + 6 B a^{3} b^{2} d^{2} e^{2} - 4 B a^{2} b^{3} d^{3} e + B a b^{4} d^{4}}{a b^{6} + b^{7} x} + \frac {\left (a e - b d\right )^{3} \left (- 4 A b e + 5 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{6}} \]

[In]

integrate((B*x+A)*(e*x+d)**4/(b*x+a)**2,x)

[Out]

B*e**4*x**4/(4*b**2) + x**3*(A*e**4/(3*b**2) - 2*B*a*e**4/(3*b**3) + 4*B*d*e**3/(3*b**2)) + x**2*(-A*a*e**4/b*
*3 + 2*A*d*e**3/b**2 + 3*B*a**2*e**4/(2*b**4) - 4*B*a*d*e**3/b**3 + 3*B*d**2*e**2/b**2) + x*(3*A*a**2*e**4/b**
4 - 8*A*a*d*e**3/b**3 + 6*A*d**2*e**2/b**2 - 4*B*a**3*e**4/b**5 + 12*B*a**2*d*e**3/b**4 - 12*B*a*d**2*e**2/b**
3 + 4*B*d**3*e/b**2) + (-A*a**4*b*e**4 + 4*A*a**3*b**2*d*e**3 - 6*A*a**2*b**3*d**2*e**2 + 4*A*a*b**4*d**3*e -
A*b**5*d**4 + B*a**5*e**4 - 4*B*a**4*b*d*e**3 + 6*B*a**3*b**2*d**2*e**2 - 4*B*a**2*b**3*d**3*e + B*a*b**4*d**4
)/(a*b**6 + b**7*x) + (a*e - b*d)**3*(-4*A*b*e + 5*B*a*e - B*b*d)*log(a + b*x)/b**6

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (182) = 364\).

Time = 0.21 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.20 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx=\frac {{\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} - A a^{4} b\right )} e^{4}}{b^{7} x + a b^{6}} + \frac {3 \, B b^{3} e^{4} x^{4} + 4 \, {\left (4 \, B b^{3} d e^{3} - {\left (2 \, B a b^{2} - A b^{3}\right )} e^{4}\right )} x^{3} + 6 \, {\left (6 \, B b^{3} d^{2} e^{2} - 4 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d e^{3} + {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{4}\right )} x^{2} + 12 \, {\left (4 \, B b^{3} d^{3} e - 6 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e^{3} - {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} e^{4}\right )} x}{12 \, b^{5}} + \frac {{\left (B b^{4} d^{4} - 4 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]

[In]

integrate((B*x+A)*(e*x+d)^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

((B*a*b^4 - A*b^5)*d^4 - 4*(B*a^2*b^3 - A*a*b^4)*d^3*e + 6*(B*a^3*b^2 - A*a^2*b^3)*d^2*e^2 - 4*(B*a^4*b - A*a^
3*b^2)*d*e^3 + (B*a^5 - A*a^4*b)*e^4)/(b^7*x + a*b^6) + 1/12*(3*B*b^3*e^4*x^4 + 4*(4*B*b^3*d*e^3 - (2*B*a*b^2
- A*b^3)*e^4)*x^3 + 6*(6*B*b^3*d^2*e^2 - 4*(2*B*a*b^2 - A*b^3)*d*e^3 + (3*B*a^2*b - 2*A*a*b^2)*e^4)*x^2 + 12*(
4*B*b^3*d^3*e - 6*(2*B*a*b^2 - A*b^3)*d^2*e^2 + 4*(3*B*a^2*b - 2*A*a*b^2)*d*e^3 - (4*B*a^3 - 3*A*a^2*b)*e^4)*x
)/b^5 + (B*b^4*d^4 - 4*(2*B*a*b^3 - A*b^4)*d^3*e + 6*(3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e^2 - 4*(4*B*a^3*b - 3*A*a^
2*b^2)*d*e^3 + (5*B*a^4 - 4*A*a^3*b)*e^4)*log(b*x + a)/b^6

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (182) = 364\).

Time = 0.29 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.91 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx=\frac {{\left (3 \, B e^{4} + \frac {4 \, {\left (4 \, B b^{2} d e^{3} - 5 \, B a b e^{4} + A b^{2} e^{4}\right )}}{{\left (b x + a\right )} b} + \frac {12 \, {\left (3 \, B b^{4} d^{2} e^{2} - 8 \, B a b^{3} d e^{3} + 2 \, A b^{4} d e^{3} + 5 \, B a^{2} b^{2} e^{4} - 2 \, A a b^{3} e^{4}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {24 \, {\left (2 \, B b^{6} d^{3} e - 9 \, B a b^{5} d^{2} e^{2} + 3 \, A b^{6} d^{2} e^{2} + 12 \, B a^{2} b^{4} d e^{3} - 6 \, A a b^{5} d e^{3} - 5 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{4}}{12 \, b^{6}} - \frac {{\left (B b^{4} d^{4} - 8 \, B a b^{3} d^{3} e + 4 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} - 12 \, A a b^{3} d^{2} e^{2} - 16 \, B a^{3} b d e^{3} + 12 \, A a^{2} b^{2} d e^{3} + 5 \, B a^{4} e^{4} - 4 \, A a^{3} b e^{4}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{6}} + \frac {\frac {B a b^{8} d^{4}}{b x + a} - \frac {A b^{9} d^{4}}{b x + a} - \frac {4 \, B a^{2} b^{7} d^{3} e}{b x + a} + \frac {4 \, A a b^{8} d^{3} e}{b x + a} + \frac {6 \, B a^{3} b^{6} d^{2} e^{2}}{b x + a} - \frac {6 \, A a^{2} b^{7} d^{2} e^{2}}{b x + a} - \frac {4 \, B a^{4} b^{5} d e^{3}}{b x + a} + \frac {4 \, A a^{3} b^{6} d e^{3}}{b x + a} + \frac {B a^{5} b^{4} e^{4}}{b x + a} - \frac {A a^{4} b^{5} e^{4}}{b x + a}}{b^{10}} \]

[In]

integrate((B*x+A)*(e*x+d)^4/(b*x+a)^2,x, algorithm="giac")

[Out]

1/12*(3*B*e^4 + 4*(4*B*b^2*d*e^3 - 5*B*a*b*e^4 + A*b^2*e^4)/((b*x + a)*b) + 12*(3*B*b^4*d^2*e^2 - 8*B*a*b^3*d*
e^3 + 2*A*b^4*d*e^3 + 5*B*a^2*b^2*e^4 - 2*A*a*b^3*e^4)/((b*x + a)^2*b^2) + 24*(2*B*b^6*d^3*e - 9*B*a*b^5*d^2*e
^2 + 3*A*b^6*d^2*e^2 + 12*B*a^2*b^4*d*e^3 - 6*A*a*b^5*d*e^3 - 5*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)/((b*x + a)^3*
b^3))*(b*x + a)^4/b^6 - (B*b^4*d^4 - 8*B*a*b^3*d^3*e + 4*A*b^4*d^3*e + 18*B*a^2*b^2*d^2*e^2 - 12*A*a*b^3*d^2*e
^2 - 16*B*a^3*b*d*e^3 + 12*A*a^2*b^2*d*e^3 + 5*B*a^4*e^4 - 4*A*a^3*b*e^4)*log(abs(b*x + a)/((b*x + a)^2*abs(b)
))/b^6 + (B*a*b^8*d^4/(b*x + a) - A*b^9*d^4/(b*x + a) - 4*B*a^2*b^7*d^3*e/(b*x + a) + 4*A*a*b^8*d^3*e/(b*x + a
) + 6*B*a^3*b^6*d^2*e^2/(b*x + a) - 6*A*a^2*b^7*d^2*e^2/(b*x + a) - 4*B*a^4*b^5*d*e^3/(b*x + a) + 4*A*a^3*b^6*
d*e^3/(b*x + a) + B*a^5*b^4*e^4/(b*x + a) - A*a^4*b^5*e^4/(b*x + a))/b^10

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.60 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx=x^3\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{3\,b^2}-\frac {2\,B\,a\,e^4}{3\,b^3}\right )-x^2\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^2}-\frac {2\,B\,a\,e^4}{b^3}\right )}{b}-\frac {d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b^2}+\frac {B\,a^2\,e^4}{2\,b^4}\right )+x\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^2}-\frac {2\,B\,a\,e^4}{b^3}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b^2}+\frac {B\,a^2\,e^4}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^2}-\frac {2\,B\,a\,e^4}{b^3}\right )}{b^2}+\frac {2\,d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (5\,B\,a^4\,e^4-16\,B\,a^3\,b\,d\,e^3-4\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3-8\,B\,a\,b^3\,d^3\,e-12\,A\,a\,b^3\,d^2\,e^2+B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{b^6}-\frac {-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {B\,e^4\,x^4}{4\,b^2} \]

[In]

int(((A + B*x)*(d + e*x)^4)/(a + b*x)^2,x)

[Out]

x^3*((A*e^4 + 4*B*d*e^3)/(3*b^2) - (2*B*a*e^4)/(3*b^3)) - x^2*((a*((A*e^4 + 4*B*d*e^3)/b^2 - (2*B*a*e^4)/b^3))
/b - (d*e^2*(2*A*e + 3*B*d))/b^2 + (B*a^2*e^4)/(2*b^4)) + x*((2*a*((2*a*((A*e^4 + 4*B*d*e^3)/b^2 - (2*B*a*e^4)
/b^3))/b - (2*d*e^2*(2*A*e + 3*B*d))/b^2 + (B*a^2*e^4)/b^4))/b - (a^2*((A*e^4 + 4*B*d*e^3)/b^2 - (2*B*a*e^4)/b
^3))/b^2 + (2*d^2*e*(3*A*e + 2*B*d))/b^2) + (log(a + b*x)*(5*B*a^4*e^4 + B*b^4*d^4 - 4*A*a^3*b*e^4 + 4*A*b^4*d
^3*e - 12*A*a*b^3*d^2*e^2 + 12*A*a^2*b^2*d*e^3 + 18*B*a^2*b^2*d^2*e^2 - 8*B*a*b^3*d^3*e - 16*B*a^3*b*d*e^3))/b
^6 - (A*b^5*d^4 - B*a^5*e^4 + A*a^4*b*e^4 - B*a*b^4*d^4 - 4*A*a^3*b^2*d*e^3 + 4*B*a^2*b^3*d^3*e + 6*A*a^2*b^3*
d^2*e^2 - 6*B*a^3*b^2*d^2*e^2 - 4*A*a*b^4*d^3*e + 4*B*a^4*b*d*e^3)/(b*(a*b^5 + b^6*x)) + (B*e^4*x^4)/(4*b^2)