Integrand size = 29, antiderivative size = 201 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {f^4 (5 b d e-2 b c f-2 a d f) x}{b^3 d^3}+\frac {f^5 x^2}{2 b^2 d^2}-\frac {(b e-a f)^5}{b^4 (b c-a d)^2 (a+b x)}-\frac {(d e-c f)^5}{d^4 (b c-a d)^2 (c+d x)}-\frac {(b e-a f)^4 (2 b d e-5 b c f+3 a d f) \log (a+b x)}{b^4 (b c-a d)^3}+\frac {(d e-c f)^4 (2 b d e+3 b c f-5 a d f) \log (c+d x)}{d^4 (b c-a d)^3} \] Output:
f^4*(-2*a*d*f-2*b*c*f+5*b*d*e)*x/b^3/d^3+1/2*f^5*x^2/b^2/d^2-(-a*f+b*e)^5/ b^4/(-a*d+b*c)^2/(b*x+a)-(-c*f+d*e)^5/d^4/(-a*d+b*c)^2/(d*x+c)-(-a*f+b*e)^ 4*(3*a*d*f-5*b*c*f+2*b*d*e)*ln(b*x+a)/b^4/(-a*d+b*c)^3+(-c*f+d*e)^4*(-5*a* d*f+3*b*c*f+2*b*d*e)*ln(d*x+c)/d^4/(-a*d+b*c)^3
Time = 0.14 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {f^4 (5 b d e-2 b c f-2 a d f) x}{b^3 d^3}+\frac {f^5 x^2}{2 b^2 d^2}-\frac {(b e-a f)^5}{b^4 (b c-a d)^2 (a+b x)}-\frac {(d e-c f)^5}{d^4 (b c-a d)^2 (c+d x)}+\frac {(b e-a f)^4 (-2 b d e+5 b c f-3 a d f) \log (a+b x)}{b^4 (b c-a d)^3}-\frac {(d e-c f)^4 (2 b d e+3 b c f-5 a d f) \log (c+d x)}{d^4 (-b c+a d)^3} \] Input:
Integrate[(e + f*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
Output:
(f^4*(5*b*d*e - 2*b*c*f - 2*a*d*f)*x)/(b^3*d^3) + (f^5*x^2)/(2*b^2*d^2) - (b*e - a*f)^5/(b^4*(b*c - a*d)^2*(a + b*x)) - (d*e - c*f)^5/(d^4*(b*c - a* d)^2*(c + d*x)) + ((b*e - a*f)^4*(-2*b*d*e + 5*b*c*f - 3*a*d*f)*Log[a + b* x])/(b^4*(b*c - a*d)^3) - ((d*e - c*f)^4*(2*b*d*e + 3*b*c*f - 5*a*d*f)*Log [c + d*x])/(d^4*(-(b*c) + a*d)^3)
Time = 0.58 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1141, 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.
| \(\displaystyle \int \frac {(e+f x)^5}{\left (x (a d+b c)+a c+b d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1141 |
| \(\displaystyle b^2 d^2 \int \left (\frac {x f^5}{b^4 d^4}+\frac {(5 b d e-2 b c f-2 a d f) f^4}{b^5 d^5}-\frac {(b e-a f)^4 (2 b d e-5 b c f+3 a d f)}{b^5 d^2 (b c-a d)^3 (a+b x)}+\frac {(d e-c f)^4 (2 b d e+3 b c f-5 a d f)}{b^2 d^5 (b c-a d)^3 (c+d x)}+\frac {(b e-a f)^5}{b^5 d^2 (b c-a d)^2 (a+b x)^2}+\frac {(d e-c f)^5}{b^2 d^5 (b c-a d)^2 (c+d x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b^2 d^2 \left (-\frac {(b e-a f)^5}{b^6 d^2 (a+b x) (b c-a d)^2}-\frac {(b e-a f)^4 \log (a+b x) (3 a d f-5 b c f+2 b d e)}{b^6 d^2 (b c-a d)^3}+\frac {f^4 x (-2 a d f-2 b c f+5 b d e)}{b^5 d^5}-\frac {(d e-c f)^5}{b^2 d^6 (c+d x) (b c-a d)^2}+\frac {(d e-c f)^4 \log (c+d x) (-5 a d f+3 b c f+2 b d e)}{b^2 d^6 (b c-a d)^3}+\frac {f^5 x^2}{2 b^4 d^4}\right )\) |
Input:
Int[(e + f*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
Output:
b^2*d^2*((f^4*(5*b*d*e - 2*b*c*f - 2*a*d*f)*x)/(b^5*d^5) + (f^5*x^2)/(2*b^ 4*d^4) - (b*e - a*f)^5/(b^6*d^2*(b*c - a*d)^2*(a + b*x)) - (d*e - c*f)^5/( b^2*d^6*(b*c - a*d)^2*(c + d*x)) - ((b*e - a*f)^4*(2*b*d*e - 5*b*c*f + 3*a *d*f)*Log[a + b*x])/(b^6*d^2*(b*c - a*d)^3) + ((d*e - c*f)^4*(2*b*d*e + 3* b*c*f - 5*a*d*f)*Log[c + d*x])/(b^2*d^6*(b*c - a*d)^3))
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Leaf count of result is larger than twice the leaf count of optimal. \(485\) vs. \(2(199)=398\).
Time = 1.11 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.42
| method | result | size |
| default | \(-\frac {f^{4} \left (-\frac {1}{2} b d f \,x^{2}+2 a d f x +2 b c f x -5 b d x e \right )}{b^{3} d^{3}}-\frac {-a^{5} f^{5}+5 a^{4} b e \,f^{4}-10 a^{3} b^{2} e^{2} f^{3}+10 a^{2} b^{3} e^{3} f^{2}-5 a \,b^{4} e^{4} f +b^{5} e^{5}}{b^{4} \left (a d -b c \right )^{2} \left (b x +a \right )}+\frac {\left (3 a^{5} d \,f^{5}-5 a^{4} b c \,f^{5}-10 a^{4} b d e \,f^{4}+20 a^{3} b^{2} c e \,f^{4}+10 a^{3} b^{2} d \,e^{2} f^{3}-30 a^{2} b^{3} c \,e^{2} f^{3}+20 a \,b^{4} c \,e^{3} f^{2}-5 a \,b^{4} d \,e^{4} f -5 b^{5} c \,e^{4} f +2 b^{5} d \,e^{5}\right ) \ln \left (b x +a \right )}{b^{4} \left (a d -b c \right )^{3}}-\frac {-c^{5} f^{5}+5 c^{4} d e \,f^{4}-10 c^{3} d^{2} e^{2} f^{3}+10 c^{2} d^{3} e^{3} f^{2}-5 c \,d^{4} e^{4} f +d^{5} e^{5}}{d^{4} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {\left (5 a \,c^{4} d \,f^{5}-20 a \,c^{3} d^{2} e \,f^{4}+30 a \,c^{2} d^{3} e^{2} f^{3}-20 a c \,d^{4} e^{3} f^{2}+5 a \,d^{5} e^{4} f -3 b \,c^{5} f^{5}+10 b \,c^{4} d e \,f^{4}-10 b \,c^{3} d^{2} e^{2} f^{3}+5 b c \,d^{4} e^{4} f -2 b \,d^{5} e^{5}\right ) \ln \left (d x +c \right )}{d^{4} \left (a d -b c \right )^{3}}\) | \(486\) |
| norman | \(\frac {\frac {f^{5} x^{4}}{2 b d}+\frac {6 a^{5} c \,d^{4} f^{5}-a^{4} b \,c^{2} d^{3} f^{5}-20 a^{4} b c \,d^{4} e \,f^{4}-6 a^{3} b^{2} c^{3} d^{2} f^{5}+10 a^{3} b^{2} c^{2} d^{3} e \,f^{4}+20 a^{3} b^{2} c \,d^{4} e^{2} f^{3}-a^{2} b^{3} c^{4} d \,f^{5}+10 a^{2} b^{3} c^{3} d^{2} e \,f^{4}-20 a^{2} b^{3} c \,d^{4} e^{3} f^{2}+6 a \,b^{4} c^{5} f^{5}-20 a \,b^{4} c^{4} d e \,f^{4}+20 a \,b^{4} c^{3} d^{2} e^{2} f^{3}-20 a \,b^{4} c^{2} d^{3} e^{3} f^{2}+20 a \,b^{4} c \,d^{4} e^{4} f -2 a \,b^{4} d^{5} e^{5}-2 b^{5} c \,d^{4} e^{5}}{2 b^{4} d^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {f^{4} \left (3 a d f +3 b c f -10 b d e \right ) x^{3}}{2 b^{2} d^{2}}+\frac {\left (6 a^{5} d^{5} f^{5}-a^{4} b c \,d^{4} f^{5}-20 a^{4} b \,d^{5} e \,f^{4}-3 a^{3} b^{2} c^{2} d^{3} f^{5}+10 a^{3} b^{2} c \,d^{4} e \,f^{4}+20 a^{3} b^{2} d^{5} e^{2} f^{3}-3 a^{2} b^{3} c^{3} d^{2} f^{5}-20 a^{2} b^{3} d^{5} e^{3} f^{2}-a \,b^{4} c^{4} d \,f^{5}+10 a \,b^{4} c^{3} d^{2} e \,f^{4}+10 a \,b^{4} d^{5} e^{4} f +6 b^{5} c^{5} f^{5}-20 b^{5} c^{4} d e \,f^{4}+20 b^{5} c^{3} d^{2} e^{2} f^{3}-20 b^{5} c^{2} d^{3} e^{3} f^{2}+10 b^{5} c \,d^{4} e^{4} f -4 b^{5} d^{5} e^{5}\right ) x}{2 b^{4} d^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {\left (5 a \,c^{4} d \,f^{5}-20 a \,c^{3} d^{2} e \,f^{4}+30 a \,c^{2} d^{3} e^{2} f^{3}-20 a c \,d^{4} e^{3} f^{2}+5 a \,d^{5} e^{4} f -3 b \,c^{5} f^{5}+10 b \,c^{4} d e \,f^{4}-10 b \,c^{3} d^{2} e^{2} f^{3}+5 b c \,d^{4} e^{4} f -2 b \,d^{5} e^{5}\right ) \ln \left (d x +c \right )}{d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (3 a^{5} d \,f^{5}-5 a^{4} b c \,f^{5}-10 a^{4} b d e \,f^{4}+20 a^{3} b^{2} c e \,f^{4}+10 a^{3} b^{2} d \,e^{2} f^{3}-30 a^{2} b^{3} c \,e^{2} f^{3}+20 a \,b^{4} c \,e^{3} f^{2}-5 a \,b^{4} d \,e^{4} f -5 b^{5} c \,e^{4} f +2 b^{5} d \,e^{5}\right ) \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4}}\) | \(947\) |
| risch | \(\text {Expression too large to display}\) | \(1593\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2633\) |
Input:
int((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x,method=_RETURNVERBOSE)
Output:
-f^4/b^3/d^3*(-1/2*b*d*f*x^2+2*a*d*f*x+2*b*c*f*x-5*b*d*x*e)-(-a^5*f^5+5*a^ 4*b*e*f^4-10*a^3*b^2*e^2*f^3+10*a^2*b^3*e^3*f^2-5*a*b^4*e^4*f+b^5*e^5)/b^4 /(a*d-b*c)^2/(b*x+a)+1/b^4*(3*a^5*d*f^5-5*a^4*b*c*f^5-10*a^4*b*d*e*f^4+20* a^3*b^2*c*e*f^4+10*a^3*b^2*d*e^2*f^3-30*a^2*b^3*c*e^2*f^3+20*a*b^4*c*e^3*f ^2-5*a*b^4*d*e^4*f-5*b^5*c*e^4*f+2*b^5*d*e^5)/(a*d-b*c)^3*ln(b*x+a)-(-c^5* f^5+5*c^4*d*e*f^4-10*c^3*d^2*e^2*f^3+10*c^2*d^3*e^3*f^2-5*c*d^4*e^4*f+d^5* e^5)/d^4/(a*d-b*c)^2/(d*x+c)+1/d^4*(5*a*c^4*d*f^5-20*a*c^3*d^2*e*f^4+30*a* c^2*d^3*e^2*f^3-20*a*c*d^4*e^3*f^2+5*a*d^5*e^4*f-3*b*c^5*f^5+10*b*c^4*d*e* f^4-10*b*c^3*d^2*e^2*f^3+5*b*c*d^4*e^4*f-2*b*d^5*e^5)/(a*d-b*c)^3*ln(d*x+c )
Leaf count of result is larger than twice the leaf count of optimal. 2046 vs. \(2 (199) = 398\).
Time = 1.24 (sec) , antiderivative size = 2046, normalized size of antiderivative = 10.18 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas")
Output:
1/2*((b^6*c^3*d^3 - 3*a*b^5*c^2*d^4 + 3*a^2*b^4*c*d^5 - a^3*b^3*d^6)*f^5*x ^4 - 2*(b^6*c^2*d^4 - a^2*b^4*d^6)*e^5 + 20*(a*b^5*c^2*d^4 - a^2*b^4*c*d^5 )*e^4*f - 20*(a*b^5*c^3*d^3 - a^3*b^3*c*d^5)*e^3*f^2 + 20*(a*b^5*c^4*d^2 - a^2*b^4*c^3*d^3 + a^3*b^3*c^2*d^4 - a^4*b^2*c*d^5)*e^2*f^3 - 10*(a*b^5*c^ 5*d - a^2*b^4*c^4*d^2 + a^4*b^2*c^2*d^4 - a^5*b*c*d^5)*e*f^4 + 2*(a*b^5*c^ 6 - a^2*b^4*c^5*d + a^5*b*c^2*d^4 - a^6*c*d^5)*f^5 + (10*(b^6*c^3*d^3 - 3* a*b^5*c^2*d^4 + 3*a^2*b^4*c*d^5 - a^3*b^3*d^6)*e*f^4 - 3*(b^6*c^4*d^2 - 2* a*b^5*c^3*d^3 + 2*a^3*b^3*c*d^5 - a^4*b^2*d^6)*f^5)*x^3 + (10*(b^6*c^4*d^2 - 2*a*b^5*c^3*d^3 + 2*a^3*b^3*c*d^5 - a^4*b^2*d^6)*e*f^4 - (4*b^6*c^5*d - 5*a*b^5*c^4*d^2 - 5*a^2*b^4*c^3*d^3 + 5*a^3*b^3*c^2*d^4 + 5*a^4*b^2*c*d^5 - 4*a^5*b*d^6)*f^5)*x^2 - 2*(2*(b^6*c*d^5 - a*b^5*d^6)*e^5 - 5*(b^6*c^2*d ^4 - a^2*b^4*d^6)*e^4*f + 10*(b^6*c^3*d^3 - a*b^5*c^2*d^4 + a^2*b^4*c*d^5 - a^3*b^3*d^6)*e^3*f^2 - 10*(b^6*c^4*d^2 - a*b^5*c^3*d^3 + a^3*b^3*c*d^5 - a^4*b^2*d^6)*e^2*f^3 + 5*(b^6*c^5*d - 2*a*b^5*c^4*d^2 + 3*a^2*b^4*c^3*d^3 - 3*a^3*b^3*c^2*d^4 + 2*a^4*b^2*c*d^5 - a^5*b*d^6)*e*f^4 - (b^6*c^6 - 3*a *b^5*c^5*d + 4*a^2*b^4*c^4*d^2 - 4*a^4*b^2*c^2*d^4 + 3*a^5*b*c*d^5 - a^6*d ^6)*f^5)*x - 2*(2*a*b^5*c*d^5*e^5 + 20*a^2*b^4*c^2*d^4*e^3*f^2 - 5*(a*b^5* c^2*d^4 + a^2*b^4*c*d^5)*e^4*f - 10*(3*a^3*b^3*c^2*d^4 - a^4*b^2*c*d^5)*e^ 2*f^3 + 10*(2*a^4*b^2*c^2*d^4 - a^5*b*c*d^5)*e*f^4 - (5*a^5*b*c^2*d^4 - 3* a^6*c*d^5)*f^5 + (2*b^6*d^6*e^5 + 20*a*b^5*c*d^5*e^3*f^2 - 5*(b^6*c*d^5...
Timed out. \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**5/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (199) = 398\).
Time = 0.05 (sec) , antiderivative size = 767, normalized size of antiderivative = 3.82 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {{\left (2 \, b^{5} d e^{5} + 20 \, a b^{4} c e^{3} f^{2} - 5 \, {\left (b^{5} c + a b^{4} d\right )} e^{4} f - 10 \, {\left (3 \, a^{2} b^{3} c - a^{3} b^{2} d\right )} e^{2} f^{3} + 10 \, {\left (2 \, a^{3} b^{2} c - a^{4} b d\right )} e f^{4} - {\left (5 \, a^{4} b c - 3 \, a^{5} d\right )} f^{5}\right )} \log \left (b x + a\right )}{b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}} + \frac {{\left (2 \, b d^{5} e^{5} + 20 \, a c d^{4} e^{3} f^{2} - 5 \, {\left (b c d^{4} + a d^{5}\right )} e^{4} f + 10 \, {\left (b c^{3} d^{2} - 3 \, a c^{2} d^{3}\right )} e^{2} f^{3} - 10 \, {\left (b c^{4} d - 2 \, a c^{3} d^{2}\right )} e f^{4} + {\left (3 \, b c^{5} - 5 \, a c^{4} d\right )} f^{5}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} + \frac {10 \, a b^{4} c d^{4} e^{4} f - {\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} e^{5} - 10 \, {\left (a b^{4} c^{2} d^{3} + a^{2} b^{3} c d^{4}\right )} e^{3} f^{2} + 10 \, {\left (a b^{4} c^{3} d^{2} + a^{3} b^{2} c d^{4}\right )} e^{2} f^{3} - 5 \, {\left (a b^{4} c^{4} d + a^{4} b c d^{4}\right )} e f^{4} + {\left (a b^{4} c^{5} + a^{5} c d^{4}\right )} f^{5} - {\left (2 \, b^{5} d^{5} e^{5} - 5 \, {\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} e^{4} f + 10 \, {\left (b^{5} c^{2} d^{3} + a^{2} b^{3} d^{5}\right )} e^{3} f^{2} - 10 \, {\left (b^{5} c^{3} d^{2} + a^{3} b^{2} d^{5}\right )} e^{2} f^{3} + 5 \, {\left (b^{5} c^{4} d + a^{4} b d^{5}\right )} e f^{4} - {\left (b^{5} c^{5} + a^{5} d^{5}\right )} f^{5}\right )} x}{a b^{6} c^{3} d^{4} - 2 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{2} + {\left (b^{7} c^{3} d^{4} - a b^{6} c^{2} d^{5} - a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x} + \frac {b d f^{5} x^{2} + 2 \, {\left (5 \, b d e f^{4} - 2 \, {\left (b c + a d\right )} f^{5}\right )} x}{2 \, b^{3} d^{3}} \] Input:
integrate((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="maxima")
Output:
-(2*b^5*d*e^5 + 20*a*b^4*c*e^3*f^2 - 5*(b^5*c + a*b^4*d)*e^4*f - 10*(3*a^2 *b^3*c - a^3*b^2*d)*e^2*f^3 + 10*(2*a^3*b^2*c - a^4*b*d)*e*f^4 - (5*a^4*b* c - 3*a^5*d)*f^5)*log(b*x + a)/(b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3) + (2*b*d^5*e^5 + 20*a*c*d^4*e^3*f^2 - 5*(b*c*d^4 + a*d^5)*e ^4*f + 10*(b*c^3*d^2 - 3*a*c^2*d^3)*e^2*f^3 - 10*(b*c^4*d - 2*a*c^3*d^2)*e *f^4 + (3*b*c^5 - 5*a*c^4*d)*f^5)*log(d*x + c)/(b^3*c^3*d^4 - 3*a*b^2*c^2* d^5 + 3*a^2*b*c*d^6 - a^3*d^7) + (10*a*b^4*c*d^4*e^4*f - (b^5*c*d^4 + a*b^ 4*d^5)*e^5 - 10*(a*b^4*c^2*d^3 + a^2*b^3*c*d^4)*e^3*f^2 + 10*(a*b^4*c^3*d^ 2 + a^3*b^2*c*d^4)*e^2*f^3 - 5*(a*b^4*c^4*d + a^4*b*c*d^4)*e*f^4 + (a*b^4* c^5 + a^5*c*d^4)*f^5 - (2*b^5*d^5*e^5 - 5*(b^5*c*d^4 + a*b^4*d^5)*e^4*f + 10*(b^5*c^2*d^3 + a^2*b^3*d^5)*e^3*f^2 - 10*(b^5*c^3*d^2 + a^3*b^2*d^5)*e^ 2*f^3 + 5*(b^5*c^4*d + a^4*b*d^5)*e*f^4 - (b^5*c^5 + a^5*d^5)*f^5)*x)/(a*b ^6*c^3*d^4 - 2*a^2*b^5*c^2*d^5 + a^3*b^4*c*d^6 + (b^7*c^2*d^5 - 2*a*b^6*c* d^6 + a^2*b^5*d^7)*x^2 + (b^7*c^3*d^4 - a*b^6*c^2*d^5 - a^2*b^5*c*d^6 + a^ 3*b^4*d^7)*x) + 1/2*(b*d*f^5*x^2 + 2*(5*b*d*e*f^4 - 2*(b*c + a*d)*f^5)*x)/ (b^3*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (199) = 398\).
Time = 0.38 (sec) , antiderivative size = 744, normalized size of antiderivative = 3.70 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {{\left (2 \, b^{5} d e^{5} - 5 \, b^{5} c e^{4} f - 5 \, a b^{4} d e^{4} f + 20 \, a b^{4} c e^{3} f^{2} - 30 \, a^{2} b^{3} c e^{2} f^{3} + 10 \, a^{3} b^{2} d e^{2} f^{3} + 20 \, a^{3} b^{2} c e f^{4} - 10 \, a^{4} b d e f^{4} - 5 \, a^{4} b c f^{5} + 3 \, a^{5} d f^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}} + \frac {{\left (2 \, b d^{5} e^{5} - 5 \, b c d^{4} e^{4} f - 5 \, a d^{5} e^{4} f + 20 \, a c d^{4} e^{3} f^{2} + 10 \, b c^{3} d^{2} e^{2} f^{3} - 30 \, a c^{2} d^{3} e^{2} f^{3} - 10 \, b c^{4} d e f^{4} + 20 \, a c^{3} d^{2} e f^{4} + 3 \, b c^{5} f^{5} - 5 \, a c^{4} d f^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} + \frac {b^{2} d^{2} f^{5} x^{2} + 10 \, b^{2} d^{2} e f^{4} x - 4 \, b^{2} c d f^{5} x - 4 \, a b d^{2} f^{5} x}{2 \, b^{4} d^{4}} - \frac {b^{5} c d^{4} e^{5} + a b^{4} d^{5} e^{5} - 10 \, a b^{4} c d^{4} e^{4} f + 10 \, a b^{4} c^{2} d^{3} e^{3} f^{2} + 10 \, a^{2} b^{3} c d^{4} e^{3} f^{2} - 10 \, a b^{4} c^{3} d^{2} e^{2} f^{3} - 10 \, a^{3} b^{2} c d^{4} e^{2} f^{3} + 5 \, a b^{4} c^{4} d e f^{4} + 5 \, a^{4} b c d^{4} e f^{4} - a b^{4} c^{5} f^{5} - a^{5} c d^{4} f^{5} + {\left (2 \, b^{5} d^{5} e^{5} - 5 \, b^{5} c d^{4} e^{4} f - 5 \, a b^{4} d^{5} e^{4} f + 10 \, b^{5} c^{2} d^{3} e^{3} f^{2} + 10 \, a^{2} b^{3} d^{5} e^{3} f^{2} - 10 \, b^{5} c^{3} d^{2} e^{2} f^{3} - 10 \, a^{3} b^{2} d^{5} e^{2} f^{3} + 5 \, b^{5} c^{4} d e f^{4} + 5 \, a^{4} b d^{5} e f^{4} - b^{5} c^{5} f^{5} - a^{5} d^{5} f^{5}\right )} x}{{\left (b c - a d\right )}^{2} {\left (b x + a\right )} {\left (d x + c\right )} b^{4} d^{4}} \] Input:
integrate((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")
Output:
-(2*b^5*d*e^5 - 5*b^5*c*e^4*f - 5*a*b^4*d*e^4*f + 20*a*b^4*c*e^3*f^2 - 30* a^2*b^3*c*e^2*f^3 + 10*a^3*b^2*d*e^2*f^3 + 20*a^3*b^2*c*e*f^4 - 10*a^4*b*d *e*f^4 - 5*a^4*b*c*f^5 + 3*a^5*d*f^5)*log(abs(b*x + a))/(b^7*c^3 - 3*a*b^6 *c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3) + (2*b*d^5*e^5 - 5*b*c*d^4*e^4*f - 5*a*d^5*e^4*f + 20*a*c*d^4*e^3*f^2 + 10*b*c^3*d^2*e^2*f^3 - 30*a*c^2*d^3* e^2*f^3 - 10*b*c^4*d*e*f^4 + 20*a*c^3*d^2*e*f^4 + 3*b*c^5*f^5 - 5*a*c^4*d* f^5)*log(abs(d*x + c))/(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^ 3*d^7) + 1/2*(b^2*d^2*f^5*x^2 + 10*b^2*d^2*e*f^4*x - 4*b^2*c*d*f^5*x - 4*a *b*d^2*f^5*x)/(b^4*d^4) - (b^5*c*d^4*e^5 + a*b^4*d^5*e^5 - 10*a*b^4*c*d^4* e^4*f + 10*a*b^4*c^2*d^3*e^3*f^2 + 10*a^2*b^3*c*d^4*e^3*f^2 - 10*a*b^4*c^3 *d^2*e^2*f^3 - 10*a^3*b^2*c*d^4*e^2*f^3 + 5*a*b^4*c^4*d*e*f^4 + 5*a^4*b*c* d^4*e*f^4 - a*b^4*c^5*f^5 - a^5*c*d^4*f^5 + (2*b^5*d^5*e^5 - 5*b^5*c*d^4*e ^4*f - 5*a*b^4*d^5*e^4*f + 10*b^5*c^2*d^3*e^3*f^2 + 10*a^2*b^3*d^5*e^3*f^2 - 10*b^5*c^3*d^2*e^2*f^3 - 10*a^3*b^2*d^5*e^2*f^3 + 5*b^5*c^4*d*e*f^4 + 5 *a^4*b*d^5*e*f^4 - b^5*c^5*f^5 - a^5*d^5*f^5)*x)/((b*c - a*d)^2*(b*x + a)* (d*x + c)*b^4*d^4)
Time = 6.18 (sec) , antiderivative size = 801, normalized size of antiderivative = 3.99 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=x\,\left (\frac {5\,e\,f^4}{b^2\,d^2}-\frac {2\,f^5\,\left (a\,d+b\,c\right )}{b^3\,d^3}\right )-\frac {\frac {-a^5\,c\,d^4\,f^5+5\,a^4\,b\,c\,d^4\,e\,f^4-10\,a^3\,b^2\,c\,d^4\,e^2\,f^3+10\,a^2\,b^3\,c\,d^4\,e^3\,f^2-a\,b^4\,c^5\,f^5+5\,a\,b^4\,c^4\,d\,e\,f^4-10\,a\,b^4\,c^3\,d^2\,e^2\,f^3+10\,a\,b^4\,c^2\,d^3\,e^3\,f^2-10\,a\,b^4\,c\,d^4\,e^4\,f+a\,b^4\,d^5\,e^5+b^5\,c\,d^4\,e^5}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {x\,\left (a^5\,d^5\,f^5-5\,a^4\,b\,d^5\,e\,f^4+10\,a^3\,b^2\,d^5\,e^2\,f^3-10\,a^2\,b^3\,d^5\,e^3\,f^2+5\,a\,b^4\,d^5\,e^4\,f+b^5\,c^5\,f^5-5\,b^5\,c^4\,d\,e\,f^4+10\,b^5\,c^3\,d^2\,e^2\,f^3-10\,b^5\,c^2\,d^3\,e^3\,f^2+5\,b^5\,c\,d^4\,e^4\,f-2\,b^5\,d^5\,e^5\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (c\,b^4\,d^3+a\,b^3\,d^4\right )+b^4\,d^4\,x^2+a\,b^3\,c\,d^3}+\frac {\ln \left (c+d\,x\right )\,\left (d\,\left (5\,a\,c^4\,f^5+10\,b\,e\,c^4\,f^4\right )-d^2\,\left (10\,b\,c^3\,e^2\,f^3+20\,a\,c^3\,e\,f^4\right )+d^4\,\left (5\,b\,c\,e^4\,f-20\,a\,c\,e^3\,f^2\right )-d^5\,\left (2\,b\,e^5-5\,a\,e^4\,f\right )-3\,b\,c^5\,f^5+30\,a\,c^2\,d^3\,e^2\,f^3\right )}{a^3\,d^7-3\,a^2\,b\,c\,d^6+3\,a\,b^2\,c^2\,d^5-b^3\,c^3\,d^4}+\frac {\ln \left (a+b\,x\right )\,\left (b\,\left (5\,c\,a^4\,f^5+10\,d\,e\,a^4\,f^4\right )-b^2\,\left (10\,d\,a^3\,e^2\,f^3+20\,c\,a^3\,e\,f^4\right )+b^4\,\left (5\,a\,d\,e^4\,f-20\,a\,c\,e^3\,f^2\right )-b^5\,\left (2\,d\,e^5-5\,c\,e^4\,f\right )-3\,a^5\,d\,f^5+30\,a^2\,b^3\,c\,e^2\,f^3\right )}{-a^3\,b^4\,d^3+3\,a^2\,b^5\,c\,d^2-3\,a\,b^6\,c^2\,d+b^7\,c^3}+\frac {f^5\,x^2}{2\,b^2\,d^2} \] Input:
int((e + f*x)^5/(a*c + x*(a*d + b*c) + b*d*x^2)^2,x)
Output:
x*((5*e*f^4)/(b^2*d^2) - (2*f^5*(a*d + b*c))/(b^3*d^3)) - ((a*b^4*d^5*e^5 - a*b^4*c^5*f^5 - a^5*c*d^4*f^5 + b^5*c*d^4*e^5 + 10*a*b^4*c^2*d^3*e^3*f^2 - 10*a*b^4*c^3*d^2*e^2*f^3 + 10*a^2*b^3*c*d^4*e^3*f^2 - 10*a^3*b^2*c*d^4* e^2*f^3 - 10*a*b^4*c*d^4*e^4*f + 5*a*b^4*c^4*d*e*f^4 + 5*a^4*b*c*d^4*e*f^4 )/(b*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (x*(a^5*d^5*f^5 + b^5*c^5*f^5 - 2*b^5*d^5*e^5 - 10*a^2*b^3*d^5*e^3*f^2 + 10*a^3*b^2*d^5*e^2*f^3 - 10*b^5*c ^2*d^3*e^3*f^2 + 10*b^5*c^3*d^2*e^2*f^3 + 5*a*b^4*d^5*e^4*f - 5*a^4*b*d^5* e*f^4 + 5*b^5*c*d^4*e^4*f - 5*b^5*c^4*d*e*f^4))/(b*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x*(a*b^3*d^4 + b^4*c*d^3) + b^4*d^4*x^2 + a*b^3*c*d^3) + (lo g(c + d*x)*(d*(5*a*c^4*f^5 + 10*b*c^4*e*f^4) - d^2*(10*b*c^3*e^2*f^3 + 20* a*c^3*e*f^4) + d^4*(5*b*c*e^4*f - 20*a*c*e^3*f^2) - d^5*(2*b*e^5 - 5*a*e^4 *f) - 3*b*c^5*f^5 + 30*a*c^2*d^3*e^2*f^3))/(a^3*d^7 - b^3*c^3*d^4 + 3*a*b^ 2*c^2*d^5 - 3*a^2*b*c*d^6) + (log(a + b*x)*(b*(5*a^4*c*f^5 + 10*a^4*d*e*f^ 4) - b^2*(10*a^3*d*e^2*f^3 + 20*a^3*c*e*f^4) + b^4*(5*a*d*e^4*f - 20*a*c*e ^3*f^2) - b^5*(2*d*e^5 - 5*c*e^4*f) - 3*a^5*d*f^5 + 30*a^2*b^3*c*e^2*f^3)) /(b^7*c^3 - a^3*b^4*d^3 + 3*a^2*b^5*c*d^2 - 3*a*b^6*c^2*d) + (f^5*x^2)/(2* b^2*d^2)
Time = 0.24 (sec) , antiderivative size = 3689, normalized size of antiderivative = 18.35 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)
Output:
(6*log(a + b*x)*a**7*c*d**6*f**5 + 6*log(a + b*x)*a**7*d**7*f**5*x - 4*log (a + b*x)*a**6*b*c**2*d**5*f**5 - 20*log(a + b*x)*a**6*b*c*d**6*e*f**4 + 2 *log(a + b*x)*a**6*b*c*d**6*f**5*x - 20*log(a + b*x)*a**6*b*d**7*e*f**4*x + 6*log(a + b*x)*a**6*b*d**7*f**5*x**2 - 10*log(a + b*x)*a**5*b**2*c**3*d* *4*f**5 + 20*log(a + b*x)*a**5*b**2*c**2*d**5*e*f**4 - 14*log(a + b*x)*a** 5*b**2*c**2*d**5*f**5*x + 20*log(a + b*x)*a**5*b**2*c*d**6*e**2*f**3 - 4*l og(a + b*x)*a**5*b**2*c*d**6*f**5*x**2 + 20*log(a + b*x)*a**5*b**2*d**7*e* *2*f**3*x - 20*log(a + b*x)*a**5*b**2*d**7*e*f**4*x**2 + 40*log(a + b*x)*a **4*b**3*c**3*d**4*e*f**4 - 10*log(a + b*x)*a**4*b**3*c**3*d**4*f**5*x - 4 0*log(a + b*x)*a**4*b**3*c**2*d**5*e**2*f**3 + 60*log(a + b*x)*a**4*b**3*c **2*d**5*e*f**4*x - 10*log(a + b*x)*a**4*b**3*c**2*d**5*f**5*x**2 - 20*log (a + b*x)*a**4*b**3*c*d**6*e**2*f**3*x + 20*log(a + b*x)*a**4*b**3*c*d**6* e*f**4*x**2 + 20*log(a + b*x)*a**4*b**3*d**7*e**2*f**3*x**2 - 60*log(a + b *x)*a**3*b**4*c**3*d**4*e**2*f**3 + 40*log(a + b*x)*a**3*b**4*c**3*d**4*e* f**4*x + 40*log(a + b*x)*a**3*b**4*c**2*d**5*e**3*f**2 - 100*log(a + b*x)* a**3*b**4*c**2*d**5*e**2*f**3*x + 40*log(a + b*x)*a**3*b**4*c**2*d**5*e*f* *4*x**2 - 10*log(a + b*x)*a**3*b**4*c*d**6*e**4*f + 40*log(a + b*x)*a**3*b **4*c*d**6*e**3*f**2*x - 40*log(a + b*x)*a**3*b**4*c*d**6*e**2*f**3*x**2 - 10*log(a + b*x)*a**3*b**4*d**7*e**4*f*x + 40*log(a + b*x)*a**2*b**5*c**3* d**4*e**3*f**2 - 60*log(a + b*x)*a**2*b**5*c**3*d**4*e**2*f**3*x - 20*l...