Integrand size = 20, antiderivative size = 155 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {(A b-a B) e (b d-a e)^3 x}{b^5}+\frac {(A b-a B) (b d-a e)^2 (d+e x)^2}{2 b^4}+\frac {(A b-a B) (b d-a e) (d+e x)^3}{3 b^3}+\frac {(A b-a B) (d+e x)^4}{4 b^2}+\frac {B (d+e x)^5}{5 b e}+\frac {(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6} \] Output:
(A*b-B*a)*e*(-a*e+b*d)^3*x/b^5+1/2*(A*b-B*a)*(-a*e+b*d)^2*(e*x+d)^2/b^4+1/ 3*(A*b-B*a)*(-a*e+b*d)*(e*x+d)^3/b^3+1/4*(A*b-B*a)*(e*x+d)^4/b^2+1/5*B*(e* x+d)^5/b/e+(A*b-B*a)*(-a*e+b*d)^4*ln(b*x+a)/b^6
Time = 0.08 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {b x \left (60 a^4 B e^4-30 a^3 b e^3 (8 B d+2 A e+B e x)+10 a^2 b^2 e^2 \left (3 A e (8 d+e x)+2 B \left (18 d^2+6 d e x+e^2 x^2\right )\right )-5 a b^3 e \left (4 A e \left (18 d^2+6 d e x+e^2 x^2\right )+B \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+b^4 \left (5 A e \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )+12 B \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )\right )\right )+60 (A b-a B) (b d-a e)^4 \log (a+b x)}{60 b^6} \] Input:
Integrate[((A + B*x)*(d + e*x)^4)/(a + b*x),x]
Output:
(b*x*(60*a^4*B*e^4 - 30*a^3*b*e^3*(8*B*d + 2*A*e + B*e*x) + 10*a^2*b^2*e^2 *(3*A*e*(8*d + e*x) + 2*B*(18*d^2 + 6*d*e*x + e^2*x^2)) - 5*a*b^3*e*(4*A*e *(18*d^2 + 6*d*e*x + e^2*x^2) + B*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3* e^3*x^3)) + b^4*(5*A*e*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) + 12*B*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4))) + 60* (A*b - a*B)*(b*d - a*e)^4*Log[a + b*x])/(60*b^6)
Time = 0.31 (sec) , antiderivative size = 155, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)}+\frac {e (A b-a B) (b d-a e)^3}{b^5}+\frac {e (d+e x) (A b-a B) (b d-a e)^2}{b^4}+\frac {e (d+e x)^2 (A b-a B) (b d-a e)}{b^3}+\frac {e (d+e x)^3 (A b-a B)}{b^2}+\frac {B (d+e x)^4}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6}+\frac {e x (A b-a B) (b d-a e)^3}{b^5}+\frac {(d+e x)^2 (A b-a B) (b d-a e)^2}{2 b^4}+\frac {(d+e x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac {(d+e x)^4 (A b-a B)}{4 b^2}+\frac {B (d+e x)^5}{5 b e}\) |
Input:
Int[((A + B*x)*(d + e*x)^4)/(a + b*x),x]
Output:
((A*b - a*B)*e*(b*d - a*e)^3*x)/b^5 + ((A*b - a*B)*(b*d - a*e)^2*(d + e*x) ^2)/(2*b^4) + ((A*b - a*B)*(b*d - a*e)*(d + e*x)^3)/(3*b^3) + ((A*b - a*B) *(d + e*x)^4)/(4*b^2) + (B*(d + e*x)^5)/(5*b*e) + ((A*b - a*B)*(b*d - a*e) ^4*Log[a + b*x])/b^6
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]))))
Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(147)=294\).
Time = 0.22 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.61
| method | result | size |
| norman | \(-\frac {\left (A \,a^{3} b \,e^{4}-4 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -B \,a^{4} e^{4}+4 B \,a^{3} b d \,e^{3}-6 B \,a^{2} b^{2} d^{2} e^{2}+4 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) x}{b^{5}}+\frac {B \,e^{4} x^{5}}{5 b}+\frac {e \left (A \,a^{2} b \,e^{3}-4 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e -B \,a^{3} e^{3}+4 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x^{2}}{2 b^{4}}-\frac {e^{2} \left (A a b \,e^{2}-4 A \,b^{2} d e -B \,a^{2} e^{2}+4 B a b d e -6 b^{2} B \,d^{2}\right ) x^{3}}{3 b^{3}}+\frac {e^{3} \left (A b e -B a e +4 B b d \right ) x^{4}}{4 b^{2}}+\frac {\left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(404\) |
| default | \(-\frac {-B \,b^{4} d \,e^{3} x^{4}+\frac {1}{3} A a \,b^{3} e^{4} x^{3}-\frac {4}{3} A \,b^{4} d \,e^{3} x^{3}-\frac {1}{3} B \,a^{2} b^{2} e^{4} x^{3}-2 B \,b^{4} d^{2} e^{2} x^{3}-3 A \,b^{4} d^{2} e^{2} x^{2}+\frac {1}{2} B \,a^{3} b \,e^{4} x^{2}-2 B \,b^{4} d^{3} e \,x^{2}+A \,a^{3} b \,e^{4} x -4 A \,b^{4} d^{3} e x -4 A \,a^{2} b^{2} d \,e^{3} x +6 A a \,b^{3} d^{2} e^{2} x +4 B \,a^{3} b d \,e^{3} x -6 B \,a^{2} b^{2} d^{2} e^{2} x +4 B a \,b^{3} d^{3} e x -2 B \,a^{2} b^{2} d \,e^{3} x^{2}+3 B a \,b^{3} d^{2} e^{2} x^{2}+\frac {1}{4} B a \,b^{3} e^{4} x^{4}+\frac {4}{3} B a \,b^{3} d \,e^{3} x^{3}+2 A a \,b^{3} d \,e^{3} x^{2}-\frac {1}{5} b^{4} B \,x^{5} e^{4}-\frac {1}{4} A \,b^{4} e^{4} x^{4}-B \,a^{4} e^{4} x -B \,b^{4} d^{4} x -\frac {1}{2} A \,a^{2} b^{2} e^{4} x^{2}}{b^{5}}+\frac {\left (A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(464\) |
| risch | \(-\frac {4 \ln \left (b x +a \right ) A \,a^{3} d \,e^{3}}{b^{4}}+\frac {6 \ln \left (b x +a \right ) A \,a^{2} d^{2} e^{2}}{b^{3}}-\frac {4 \ln \left (b x +a \right ) A a \,d^{3} e}{b^{2}}+\frac {4 \ln \left (b x +a \right ) B \,a^{4} d \,e^{3}}{b^{5}}-\frac {6 \ln \left (b x +a \right ) B \,a^{3} d^{2} e^{2}}{b^{4}}+\frac {4 \ln \left (b x +a \right ) B \,a^{2} d^{3} e}{b^{3}}+\frac {A \,e^{4} x^{4}}{4 b}+\frac {B \,d^{4} x}{b}+\frac {\ln \left (b x +a \right ) A \,d^{4}}{b}+\frac {4 A \,a^{2} d \,e^{3} x}{b^{3}}+\frac {A \,a^{2} e^{4} x^{2}}{2 b^{3}}+\frac {\ln \left (b x +a \right ) A \,a^{4} e^{4}}{b^{5}}-\frac {\ln \left (b x +a \right ) B \,a^{5} e^{4}}{b^{6}}-\frac {\ln \left (b x +a \right ) B a \,d^{4}}{b^{2}}+\frac {6 B \,a^{2} d^{2} e^{2} x}{b^{3}}-\frac {4 B a \,d^{3} e x}{b^{2}}+\frac {2 B \,a^{2} d \,e^{3} x^{2}}{b^{3}}-\frac {3 B a \,d^{2} e^{2} x^{2}}{b^{2}}-\frac {4 B a d \,e^{3} x^{3}}{3 b^{2}}-\frac {2 A a d \,e^{3} x^{2}}{b^{2}}-\frac {6 A a \,d^{2} e^{2} x}{b^{2}}-\frac {4 B \,a^{3} d \,e^{3} x}{b^{4}}+\frac {B \,e^{4} x^{5}}{5 b}+\frac {B d \,e^{3} x^{4}}{b}-\frac {A a \,e^{4} x^{3}}{3 b^{2}}+\frac {4 A d \,e^{3} x^{3}}{3 b}+\frac {B \,a^{2} e^{4} x^{3}}{3 b^{3}}+\frac {2 B \,d^{2} e^{2} x^{3}}{b}+\frac {3 A \,d^{2} e^{2} x^{2}}{b}-\frac {B \,a^{3} e^{4} x^{2}}{2 b^{4}}+\frac {2 B \,d^{3} e \,x^{2}}{b}-\frac {A \,a^{3} e^{4} x}{b^{4}}+\frac {4 A \,d^{3} e x}{b}-\frac {B a \,e^{4} x^{4}}{4 b^{2}}+\frac {B \,a^{4} e^{4} x}{b^{5}}\) | \(521\) |
| parallelrisch | \(\frac {240 A x \,a^{2} b^{3} d \,e^{3}-360 A x a \,b^{4} d^{2} e^{2}-240 B x \,a^{3} b^{2} d \,e^{3}+360 B x \,a^{2} b^{3} d^{2} e^{2}-240 A \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{3}+360 A \ln \left (b x +a \right ) a^{2} b^{3} d^{2} e^{2}-240 B x a \,b^{4} d^{3} e -240 A \ln \left (b x +a \right ) a \,b^{4} d^{3} e +240 B \ln \left (b x +a \right ) a^{4} b d \,e^{3}-80 B \,x^{3} a \,b^{4} d \,e^{3}-120 A \,x^{2} a \,b^{4} d \,e^{3}+120 B \,x^{2} a^{2} b^{3} d \,e^{3}-180 B \,x^{2} a \,b^{4} d^{2} e^{2}+120 B \,x^{2} b^{5} d^{3} e -60 A x \,a^{3} b^{2} e^{4}+240 A x \,b^{5} d^{3} e +60 B x \,a^{4} b \,e^{4}+60 A \ln \left (b x +a \right ) a^{4} b \,e^{4}-60 B \ln \left (b x +a \right ) a \,b^{4} d^{4}-15 B \,x^{4} a \,b^{4} e^{4}+60 B \,x^{4} b^{5} d \,e^{3}-20 A \,x^{3} a \,b^{4} e^{4}+80 A \,x^{3} b^{5} d \,e^{3}+20 B \,x^{3} a^{2} b^{3} e^{4}+120 B \,x^{3} b^{5} d^{2} e^{2}+30 A \,x^{2} a^{2} b^{3} e^{4}+180 A \,x^{2} b^{5} d^{2} e^{2}-30 B \,x^{2} a^{3} b^{2} e^{4}-360 B \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{2}+240 B \ln \left (b x +a \right ) a^{2} b^{3} d^{3} e +15 A \,x^{4} b^{5} e^{4}+60 B x \,b^{5} d^{4}+60 A \ln \left (b x +a \right ) b^{5} d^{4}-60 B \ln \left (b x +a \right ) a^{5} e^{4}+12 B \,x^{5} e^{4} b^{5}}{60 b^{6}}\) | \(522\) |
Input:
int((B*x+A)*(e*x+d)^4/(b*x+a),x,method=_RETURNVERBOSE)
Output:
-(A*a^3*b*e^4-4*A*a^2*b^2*d*e^3+6*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e-B*a^4*e^4+ 4*B*a^3*b*d*e^3-6*B*a^2*b^2*d^2*e^2+4*B*a*b^3*d^3*e-B*b^4*d^4)/b^5*x+1/5*B /b*e^4*x^5+1/2/b^4*e*(A*a^2*b*e^3-4*A*a*b^2*d*e^2+6*A*b^3*d^2*e-B*a^3*e^3+ 4*B*a^2*b*d*e^2-6*B*a*b^2*d^2*e+4*B*b^3*d^3)*x^2-1/3/b^3*e^2*(A*a*b*e^2-4* A*b^2*d*e-B*a^2*e^2+4*B*a*b*d*e-6*B*b^2*d^2)*x^3+1/4/b^2*e^3*(A*b*e-B*a*e+ 4*B*b*d)*x^4+(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*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (149) = 298\).
Time = 0.09 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.60 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {12 \, B b^{5} e^{4} x^{5} + 15 \, {\left (4 \, B b^{5} d e^{3} - {\left (B a b^{4} - A b^{5}\right )} e^{4}\right )} x^{4} + 20 \, {\left (6 \, B b^{5} d^{2} e^{2} - 4 \, {\left (B a b^{4} - A b^{5}\right )} d e^{3} + {\left (B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} + 30 \, {\left (4 \, B b^{5} d^{3} e - 6 \, {\left (B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{3} - {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 60 \, {\left (B b^{5} d^{4} - 4 \, {\left (B a b^{4} - A b^{5}\right )} d^{3} e + 6 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} + {\left (B a^{4} b - A a^{3} b^{2}\right )} e^{4}\right )} x - 60 \, {\left ({\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} - A a^{4} b\right )} e^{4}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \] Input:
integrate((B*x+A)*(e*x+d)^4/(b*x+a),x, algorithm="fricas")
Output:
1/60*(12*B*b^5*e^4*x^5 + 15*(4*B*b^5*d*e^3 - (B*a*b^4 - A*b^5)*e^4)*x^4 + 20*(6*B*b^5*d^2*e^2 - 4*(B*a*b^4 - A*b^5)*d*e^3 + (B*a^2*b^3 - A*a*b^4)*e^ 4)*x^3 + 30*(4*B*b^5*d^3*e - 6*(B*a*b^4 - A*b^5)*d^2*e^2 + 4*(B*a^2*b^3 - A*a*b^4)*d*e^3 - (B*a^3*b^2 - A*a^2*b^3)*e^4)*x^2 + 60*(B*b^5*d^4 - 4*(B*a *b^4 - A*b^5)*d^3*e + 6*(B*a^2*b^3 - A*a*b^4)*d^2*e^2 - 4*(B*a^3*b^2 - A*a ^2*b^3)*d*e^3 + (B*a^4*b - A*a^3*b^2)*e^4)*x - 60*((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)*log(b*x + a))/b^6
Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (136) = 272\).
Time = 0.55 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.27 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {B e^{4} x^{5}}{5 b} + x^{4} \left (\frac {A e^{4}}{4 b} - \frac {B a e^{4}}{4 b^{2}} + \frac {B d e^{3}}{b}\right ) + x^{3} \left (- \frac {A a e^{4}}{3 b^{2}} + \frac {4 A d e^{3}}{3 b} + \frac {B a^{2} e^{4}}{3 b^{3}} - \frac {4 B a d e^{3}}{3 b^{2}} + \frac {2 B d^{2} e^{2}}{b}\right ) + x^{2} \left (\frac {A a^{2} e^{4}}{2 b^{3}} - \frac {2 A a d e^{3}}{b^{2}} + \frac {3 A d^{2} e^{2}}{b} - \frac {B a^{3} e^{4}}{2 b^{4}} + \frac {2 B a^{2} d e^{3}}{b^{3}} - \frac {3 B a d^{2} e^{2}}{b^{2}} + \frac {2 B d^{3} e}{b}\right ) + x \left (- \frac {A a^{3} e^{4}}{b^{4}} + \frac {4 A a^{2} d e^{3}}{b^{3}} - \frac {6 A a d^{2} e^{2}}{b^{2}} + \frac {4 A d^{3} e}{b} + \frac {B a^{4} e^{4}}{b^{5}} - \frac {4 B a^{3} d e^{3}}{b^{4}} + \frac {6 B a^{2} d^{2} e^{2}}{b^{3}} - \frac {4 B a d^{3} e}{b^{2}} + \frac {B d^{4}}{b}\right ) - \frac {\left (- A b + B a\right ) \left (a e - b d\right )^{4} \log {\left (a + b x \right )}}{b^{6}} \] Input:
integrate((B*x+A)*(e*x+d)**4/(b*x+a),x)
Output:
B*e**4*x**5/(5*b) + x**4*(A*e**4/(4*b) - B*a*e**4/(4*b**2) + B*d*e**3/b) + x**3*(-A*a*e**4/(3*b**2) + 4*A*d*e**3/(3*b) + B*a**2*e**4/(3*b**3) - 4*B* a*d*e**3/(3*b**2) + 2*B*d**2*e**2/b) + x**2*(A*a**2*e**4/(2*b**3) - 2*A*a* d*e**3/b**2 + 3*A*d**2*e**2/b - B*a**3*e**4/(2*b**4) + 2*B*a**2*d*e**3/b** 3 - 3*B*a*d**2*e**2/b**2 + 2*B*d**3*e/b) + x*(-A*a**3*e**4/b**4 + 4*A*a**2 *d*e**3/b**3 - 6*A*a*d**2*e**2/b**2 + 4*A*d**3*e/b + B*a**4*e**4/b**5 - 4* B*a**3*d*e**3/b**4 + 6*B*a**2*d**2*e**2/b**3 - 4*B*a*d**3*e/b**2 + B*d**4/ b) - (-A*b + B*a)*(a*e - b*d)**4*log(a + b*x)/b**6
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (149) = 298\).
Time = 0.04 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.58 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {12 \, B b^{4} e^{4} x^{5} + 15 \, {\left (4 \, B b^{4} d e^{3} - {\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{4} + 20 \, {\left (6 \, B b^{4} d^{2} e^{2} - 4 \, {\left (B a b^{3} - A b^{4}\right )} d e^{3} + {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{3} + 30 \, {\left (4 \, B b^{4} d^{3} e - 6 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} - {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x^{2} + 60 \, {\left (B b^{4} d^{4} - 4 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} - A a^{3} b\right )} e^{4}\right )} x}{60 \, b^{5}} - \frac {{\left ({\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} - A a^{4} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \] Input:
integrate((B*x+A)*(e*x+d)^4/(b*x+a),x, algorithm="maxima")
Output:
1/60*(12*B*b^4*e^4*x^5 + 15*(4*B*b^4*d*e^3 - (B*a*b^3 - A*b^4)*e^4)*x^4 + 20*(6*B*b^4*d^2*e^2 - 4*(B*a*b^3 - A*b^4)*d*e^3 + (B*a^2*b^2 - A*a*b^3)*e^ 4)*x^3 + 30*(4*B*b^4*d^3*e - 6*(B*a*b^3 - A*b^4)*d^2*e^2 + 4*(B*a^2*b^2 - A*a*b^3)*d*e^3 - (B*a^3*b - A*a^2*b^2)*e^4)*x^2 + 60*(B*b^4*d^4 - 4*(B*a*b ^3 - A*b^4)*d^3*e + 6*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 - 4*(B*a^3*b - A*a^2*b ^2)*d*e^3 + (B*a^4 - A*a^3*b)*e^4)*x)/b^5 - ((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)*log(b*x + a)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (149) = 298\).
Time = 0.12 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.01 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {12 \, B b^{4} e^{4} x^{5} + 60 \, B b^{4} d e^{3} x^{4} - 15 \, B a b^{3} e^{4} x^{4} + 15 \, A b^{4} e^{4} x^{4} + 120 \, B b^{4} d^{2} e^{2} x^{3} - 80 \, B a b^{3} d e^{3} x^{3} + 80 \, A b^{4} d e^{3} x^{3} + 20 \, B a^{2} b^{2} e^{4} x^{3} - 20 \, A a b^{3} e^{4} x^{3} + 120 \, B b^{4} d^{3} e x^{2} - 180 \, B a b^{3} d^{2} e^{2} x^{2} + 180 \, A b^{4} d^{2} e^{2} x^{2} + 120 \, B a^{2} b^{2} d e^{3} x^{2} - 120 \, A a b^{3} d e^{3} x^{2} - 30 \, B a^{3} b e^{4} x^{2} + 30 \, A a^{2} b^{2} e^{4} x^{2} + 60 \, B b^{4} d^{4} x - 240 \, B a b^{3} d^{3} e x + 240 \, A b^{4} d^{3} e x + 360 \, B a^{2} b^{2} d^{2} e^{2} x - 360 \, A a b^{3} d^{2} e^{2} x - 240 \, B a^{3} b d e^{3} x + 240 \, A a^{2} b^{2} d e^{3} x + 60 \, B a^{4} e^{4} x - 60 \, A a^{3} b e^{4} x}{60 \, b^{5}} - \frac {{\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \] Input:
integrate((B*x+A)*(e*x+d)^4/(b*x+a),x, algorithm="giac")
Output:
1/60*(12*B*b^4*e^4*x^5 + 60*B*b^4*d*e^3*x^4 - 15*B*a*b^3*e^4*x^4 + 15*A*b^ 4*e^4*x^4 + 120*B*b^4*d^2*e^2*x^3 - 80*B*a*b^3*d*e^3*x^3 + 80*A*b^4*d*e^3* x^3 + 20*B*a^2*b^2*e^4*x^3 - 20*A*a*b^3*e^4*x^3 + 120*B*b^4*d^3*e*x^2 - 18 0*B*a*b^3*d^2*e^2*x^2 + 180*A*b^4*d^2*e^2*x^2 + 120*B*a^2*b^2*d*e^3*x^2 - 120*A*a*b^3*d*e^3*x^2 - 30*B*a^3*b*e^4*x^2 + 30*A*a^2*b^2*e^4*x^2 + 60*B*b ^4*d^4*x - 240*B*a*b^3*d^3*e*x + 240*A*b^4*d^3*e*x + 360*B*a^2*b^2*d^2*e^2 *x - 360*A*a*b^3*d^2*e^2*x - 240*B*a^3*b*d*e^3*x + 240*A*a^2*b^2*d*e^3*x + 60*B*a^4*e^4*x - 60*A*a^3*b*e^4*x)/b^5 - (B*a*b^4*d^4 - A*b^5*d^4 - 4*B*a ^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 6*B*a^3*b^2*d^2*e^2 - 6*A*a^2*b^3*d^2*e^2 - 4*B*a^4*b*d*e^3 + 4*A*a^3*b^2*d*e^3 + B*a^5*e^4 - A*a^4*b*e^4)*log(abs( b*x + a))/b^6
Time = 0.06 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.65 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=x\,\left (\frac {B\,d^4+4\,A\,e\,d^3}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b}\right )}{b}+\frac {2\,d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{b}\right )}{b}\right )-x^3\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{3\,b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{3\,b}\right )+x^4\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{4\,b}-\frac {B\,a\,e^4}{4\,b^2}\right )+x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b}-\frac {B\,a\,e^4}{b^2}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b}\right )}{2\,b}+\frac {d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4\right )}{b^6}+\frac {B\,e^4\,x^5}{5\,b} \] Input:
int(((A + B*x)*(d + e*x)^4)/(a + b*x),x)
Output:
x*((B*d^4 + 4*A*d^3*e)/b - (a*((a*((a*((A*e^4 + 4*B*d*e^3)/b - (B*a*e^4)/b ^2))/b - (2*d*e^2*(2*A*e + 3*B*d))/b))/b + (2*d^2*e*(3*A*e + 2*B*d))/b))/b ) - x^3*((a*((A*e^4 + 4*B*d*e^3)/b - (B*a*e^4)/b^2))/(3*b) - (2*d*e^2*(2*A *e + 3*B*d))/(3*b)) + x^4*((A*e^4 + 4*B*d*e^3)/(4*b) - (B*a*e^4)/(4*b^2)) + x^2*((a*((a*((A*e^4 + 4*B*d*e^3)/b - (B*a*e^4)/b^2))/b - (2*d*e^2*(2*A*e + 3*B*d))/b))/(2*b) + (d^2*e*(3*A*e + 2*B*d))/b) + (log(a + b*x)*(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^6 + (B*e^4*x^5)/(5*b)
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.28 \[ \int \frac {(A+B x) (d+e x)^4}{a+b x} \, dx=\frac {x \left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right )}{5} \] Input:
int((B*x+A)*(e*x+d)^4/(b*x+a),x)
Output:
(x*(5*d**4 + 10*d**3*e*x + 10*d**2*e**2*x**2 + 5*d*e**3*x**3 + e**4*x**4)) /5