Integrand size = 29, antiderivative size = 166 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {f^4 x}{b^2 d^2}-\frac {(b e-a f)^4}{b^3 (b c-a d)^2 (a+b x)}-\frac {(d e-c f)^4}{d^3 (b c-a d)^2 (c+d x)}-\frac {2 (b e-a f)^3 (b d e-2 b c f+a d f) \log (a+b x)}{b^3 (b c-a d)^3}+\frac {2 (d e-c f)^3 (b d e+b c f-2 a d f) \log (c+d x)}{d^3 (b c-a d)^3} \] Output:
f^4*x/b^2/d^2-(-a*f+b*e)^4/b^3/(-a*d+b*c)^2/(b*x+a)-(-c*f+d*e)^4/d^3/(-a*d +b*c)^2/(d*x+c)-2*(-a*f+b*e)^3*(a*d*f-2*b*c*f+b*d*e)*ln(b*x+a)/b^3/(-a*d+b *c)^3+2*(-c*f+d*e)^3*(-2*a*d*f+b*c*f+b*d*e)*ln(d*x+c)/d^3/(-a*d+b*c)^3
Time = 0.12 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {f^4 x}{b^2 d^2}-\frac {(b e-a f)^4}{b^3 (b c-a d)^2 (a+b x)}-\frac {(d e-c f)^4}{d^3 (b c-a d)^2 (c+d x)}-\frac {2 (b e-a f)^3 (b d e-2 b c f+a d f) \log (a+b x)}{b^3 (b c-a d)^3}-\frac {2 (d e-c f)^3 (b d e+b c f-2 a d f) \log (c+d x)}{d^3 (-b c+a d)^3} \] Input:
Integrate[(e + f*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
Output:
(f^4*x)/(b^2*d^2) - (b*e - a*f)^4/(b^3*(b*c - a*d)^2*(a + b*x)) - (d*e - c *f)^4/(d^3*(b*c - a*d)^2*(c + d*x)) - (2*(b*e - a*f)^3*(b*d*e - 2*b*c*f + a*d*f)*Log[a + b*x])/(b^3*(b*c - a*d)^3) - (2*(d*e - c*f)^3*(b*d*e + b*c*f - 2*a*d*f)*Log[c + d*x])/(d^3*(-(b*c) + a*d)^3)
Time = 0.47 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.11, 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.
| \(\displaystyle \int \frac {(e+f x)^4}{\left (x (a d+b c)+a c+b d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1141 |
| \(\displaystyle b^2 d^2 \int \left (\frac {f^4}{b^4 d^4}-\frac {2 (b e-a f)^3 (b d e-2 b c f+a d f)}{b^4 d^2 (b c-a d)^3 (a+b x)}+\frac {2 (d e-c f)^3 (b d e+b c f-2 a d f)}{b^2 d^4 (b c-a d)^3 (c+d x)}+\frac {(b e-a f)^4}{b^4 d^2 (b c-a d)^2 (a+b x)^2}+\frac {(d e-c f)^4}{b^2 d^4 (b c-a d)^2 (c+d x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b^2 d^2 \left (-\frac {(b e-a f)^4}{b^5 d^2 (a+b x) (b c-a d)^2}-\frac {2 (b e-a f)^3 \log (a+b x) (a d f-2 b c f+b d e)}{b^5 d^2 (b c-a d)^3}-\frac {(d e-c f)^4}{b^2 d^5 (c+d x) (b c-a d)^2}+\frac {2 (d e-c f)^3 \log (c+d x) (-2 a d f+b c f+b d e)}{b^2 d^5 (b c-a d)^3}+\frac {f^4 x}{b^4 d^4}\right )\) |
Input:
Int[(e + f*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
Output:
b^2*d^2*((f^4*x)/(b^4*d^4) - (b*e - a*f)^4/(b^5*d^2*(b*c - a*d)^2*(a + b*x )) - (d*e - c*f)^4/(b^2*d^5*(b*c - a*d)^2*(c + d*x)) - (2*(b*e - a*f)^3*(b *d*e - 2*b*c*f + a*d*f)*Log[a + b*x])/(b^5*d^2*(b*c - a*d)^3) + (2*(d*e - c*f)^3*(b*d*e + b*c*f - 2*a*d*f)*Log[c + d*x])/(b^2*d^5*(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. \(368\) vs. \(2(166)=332\).
Time = 1.08 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.22
| method | result | size |
| default | \(\frac {f^{4} x}{b^{2} d^{2}}-\frac {a^{4} f^{4}-4 a^{3} b e \,f^{3}+6 a^{2} b^{2} e^{2} f^{2}-4 a \,b^{3} e^{3} f +b^{4} e^{4}}{b^{3} \left (a d -b c \right )^{2} \left (b x +a \right )}+\frac {\left (-2 a^{4} d \,f^{4}+4 a^{3} b c \,f^{4}+4 a^{3} b d e \,f^{3}-12 a^{2} b^{2} c e \,f^{3}+12 a \,b^{3} c \,e^{2} f^{2}-4 a \,b^{3} d \,e^{3} f -4 b^{4} c \,e^{3} f +2 b^{4} d \,e^{4}\right ) \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )^{3}}-\frac {c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}}{d^{3} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {\left (-4 a \,c^{3} d \,f^{4}+12 a \,c^{2} d^{2} e \,f^{3}-12 a c \,d^{3} e^{2} f^{2}+4 a \,d^{4} e^{3} f +2 b \,c^{4} f^{4}-4 b \,c^{3} d e \,f^{3}+4 b c \,d^{3} e^{3} f -2 b \,d^{4} e^{4}\right ) \ln \left (d x +c \right )}{d^{3} \left (a d -b c \right )^{3}}\) | \(369\) |
| norman | \(\frac {\frac {f^{4} x^{3}}{b d}-\frac {2 a^{4} c \,d^{3} f^{4}-a^{3} b \,c^{2} d^{2} f^{4}-4 a^{3} b c \,d^{3} e \,f^{3}-a^{2} b^{2} c^{3} d \,f^{4}+6 a^{2} b^{2} c \,d^{3} e^{2} f^{2}+2 a \,b^{3} c^{4} f^{4}-4 a \,b^{3} c^{3} d e \,f^{3}+6 a \,b^{3} c^{2} d^{2} e^{2} f^{2}-8 a \,b^{3} c \,d^{3} e^{3} f +a \,b^{3} d^{4} e^{4}+b^{4} c \,d^{3} e^{4}}{d^{3} b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 a^{4} d^{4} f^{4}-a^{3} b c \,d^{3} f^{4}-4 a^{3} b \,d^{4} e \,f^{3}+6 a^{2} b^{2} d^{4} e^{2} f^{2}-a \,b^{3} c^{3} d \,f^{4}-4 a \,b^{3} d^{4} e^{3} f +2 b^{4} c^{4} f^{4}-4 b^{4} c^{3} d e \,f^{3}+6 b^{4} c^{2} d^{2} e^{2} f^{2}-4 b^{4} c \,d^{3} e^{3} f +2 b^{4} d^{4} e^{4}\right ) x}{d^{3} b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}-\frac {2 \left (2 a \,c^{3} d \,f^{4}-6 a \,c^{2} d^{2} e \,f^{3}+6 a c \,d^{3} e^{2} f^{2}-2 a \,d^{4} e^{3} f -b \,c^{4} f^{4}+2 b \,c^{3} d e \,f^{3}-2 b c \,d^{3} e^{3} f +b \,d^{4} e^{4}\right ) \ln \left (d x +c \right )}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 \left (a^{4} d \,f^{4}-2 a^{3} b c \,f^{4}-2 a^{3} b d e \,f^{3}+6 a^{2} b^{2} c e \,f^{3}-6 a \,b^{3} c \,e^{2} f^{2}+2 a \,b^{3} d \,e^{3} f +2 b^{4} c \,e^{3} f -b^{4} d \,e^{4}\right ) \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3}}\) | \(665\) |
| risch | \(\text {Expression too large to display}\) | \(1239\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1907\) |
Input:
int((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x,method=_RETURNVERBOSE)
Output:
f^4*x/b^2/d^2-1/b^3*(a^4*f^4-4*a^3*b*e*f^3+6*a^2*b^2*e^2*f^2-4*a*b^3*e^3*f +b^4*e^4)/(a*d-b*c)^2/(b*x+a)+(-2*a^4*d*f^4+4*a^3*b*c*f^4+4*a^3*b*d*e*f^3- 12*a^2*b^2*c*e*f^3+12*a*b^3*c*e^2*f^2-4*a*b^3*d*e^3*f-4*b^4*c*e^3*f+2*b^4* d*e^4)/b^3/(a*d-b*c)^3*ln(b*x+a)-1/d^3*(c^4*f^4-4*c^3*d*e*f^3+6*c^2*d^2*e^ 2*f^2-4*c*d^3*e^3*f+d^4*e^4)/(a*d-b*c)^2/(d*x+c)+(-4*a*c^3*d*f^4+12*a*c^2* d^2*e*f^3-12*a*c*d^3*e^2*f^2+4*a*d^4*e^3*f+2*b*c^4*f^4-4*b*c^3*d*e*f^3+4*b *c*d^3*e^3*f-2*b*d^4*e^4)/d^3/(a*d-b*c)^3*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (166) = 332\).
Time = 0.38 (sec) , antiderivative size = 1479, normalized size of antiderivative = 8.91 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas")
Output:
((b^5*c^3*d^2 - 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 - a^3*b^2*d^5)*f^4*x^3 + (b^5*c^4*d - 2*a*b^4*c^3*d^2 + 2*a^3*b^2*c*d^4 - a^4*b*d^5)*f^4*x^2 - (b^ 5*c^2*d^3 - a^2*b^3*d^5)*e^4 + 8*(a*b^4*c^2*d^3 - a^2*b^3*c*d^4)*e^3*f - 6 *(a*b^4*c^3*d^2 - a^3*b^2*c*d^4)*e^2*f^2 + 4*(a*b^4*c^4*d - a^2*b^3*c^3*d^ 2 + a^3*b^2*c^2*d^3 - a^4*b*c*d^4)*e*f^3 - (a*b^4*c^5 - a^2*b^3*c^4*d + a^ 4*b*c^2*d^3 - a^5*c*d^4)*f^4 - (2*(b^5*c*d^4 - a*b^4*d^5)*e^4 - 4*(b^5*c^2 *d^3 - a^2*b^3*d^5)*e^3*f + 6*(b^5*c^3*d^2 - a*b^4*c^2*d^3 + a^2*b^3*c*d^4 - a^3*b^2*d^5)*e^2*f^2 - 4*(b^5*c^4*d - a*b^4*c^3*d^2 + a^3*b^2*c*d^4 - a ^4*b*d^5)*e*f^3 + (b^5*c^5 - 2*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 - 3*a^3*b^2 *c^2*d^3 + 2*a^4*b*c*d^4 - a^5*d^5)*f^4)*x - 2*(a*b^4*c*d^4*e^4 + 6*a^2*b^ 3*c^2*d^3*e^2*f^2 - 2*(a*b^4*c^2*d^3 + a^2*b^3*c*d^4)*e^3*f - 2*(3*a^3*b^2 *c^2*d^3 - a^4*b*c*d^4)*e*f^3 + (2*a^4*b*c^2*d^3 - a^5*c*d^4)*f^4 + (b^5*d ^5*e^4 + 6*a*b^4*c*d^4*e^2*f^2 - 2*(b^5*c*d^4 + a*b^4*d^5)*e^3*f - 2*(3*a^ 2*b^3*c*d^4 - a^3*b^2*d^5)*e*f^3 + (2*a^3*b^2*c*d^4 - a^4*b*d^5)*f^4)*x^2 + ((b^5*c*d^4 + a*b^4*d^5)*e^4 - 2*(b^5*c^2*d^3 + 2*a*b^4*c*d^4 + a^2*b^3* d^5)*e^3*f + 6*(a*b^4*c^2*d^3 + a^2*b^3*c*d^4)*e^2*f^2 - 2*(3*a^2*b^3*c^2* d^3 + 2*a^3*b^2*c*d^4 - a^4*b*d^5)*e*f^3 + (2*a^3*b^2*c^2*d^3 + a^4*b*c*d^ 4 - a^5*d^5)*f^4)*x)*log(b*x + a) + 2*(a*b^4*c*d^4*e^4 + 6*a^2*b^3*c^2*d^3 *e^2*f^2 - 2*(a*b^4*c^2*d^3 + a^2*b^3*c*d^4)*e^3*f + 2*(a*b^4*c^4*d - 3*a^ 2*b^3*c^3*d^2)*e*f^3 - (a*b^4*c^5 - 2*a^2*b^3*c^4*d)*f^4 + (b^5*d^5*e^4...
Timed out. \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**4/(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. 621 vs. \(2 (166) = 332\).
Time = 0.06 (sec) , antiderivative size = 621, normalized size of antiderivative = 3.74 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {f^{4} x}{b^{2} d^{2}} - \frac {2 \, {\left (b^{4} d e^{4} + 6 \, a b^{3} c e^{2} f^{2} - 2 \, {\left (b^{4} c + a b^{3} d\right )} e^{3} f - 2 \, {\left (3 \, a^{2} b^{2} c - a^{3} b d\right )} e f^{3} + {\left (2 \, a^{3} b c - a^{4} d\right )} f^{4}\right )} \log \left (b x + a\right )}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} + \frac {2 \, {\left (b d^{4} e^{4} + 6 \, a c d^{3} e^{2} f^{2} - 2 \, {\left (b c d^{3} + a d^{4}\right )} e^{3} f + 2 \, {\left (b c^{3} d - 3 \, a c^{2} d^{2}\right )} e f^{3} - {\left (b c^{4} - 2 \, a c^{3} d\right )} f^{4}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} + \frac {8 \, a b^{3} c d^{3} e^{3} f - {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} e^{4} - 6 \, {\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} e^{2} f^{2} + 4 \, {\left (a b^{3} c^{3} d + a^{3} b c d^{3}\right )} e f^{3} - {\left (a b^{3} c^{4} + a^{4} c d^{3}\right )} f^{4} - {\left (2 \, b^{4} d^{4} e^{4} - 4 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} e^{3} f + 6 \, {\left (b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} e^{2} f^{2} - 4 \, {\left (b^{4} c^{3} d + a^{3} b d^{4}\right )} e f^{3} + {\left (b^{4} c^{4} + a^{4} d^{4}\right )} f^{4}\right )} x}{a b^{5} c^{3} d^{3} - 2 \, a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + {\left (b^{6} c^{2} d^{4} - 2 \, a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{2} + {\left (b^{6} c^{3} d^{3} - a b^{5} c^{2} d^{4} - a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x} \] Input:
integrate((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="maxima")
Output:
f^4*x/(b^2*d^2) - 2*(b^4*d*e^4 + 6*a*b^3*c*e^2*f^2 - 2*(b^4*c + a*b^3*d)*e ^3*f - 2*(3*a^2*b^2*c - a^3*b*d)*e*f^3 + (2*a^3*b*c - a^4*d)*f^4)*log(b*x + a)/(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3) + 2*(b*d^4* e^4 + 6*a*c*d^3*e^2*f^2 - 2*(b*c*d^3 + a*d^4)*e^3*f + 2*(b*c^3*d - 3*a*c^2 *d^2)*e*f^3 - (b*c^4 - 2*a*c^3*d)*f^4)*log(d*x + c)/(b^3*c^3*d^3 - 3*a*b^2 *c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6) + (8*a*b^3*c*d^3*e^3*f - (b^4*c*d^3 + a*b^3*d^4)*e^4 - 6*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*e^2*f^2 + 4*(a*b^3*c^3* d + a^3*b*c*d^3)*e*f^3 - (a*b^3*c^4 + a^4*c*d^3)*f^4 - (2*b^4*d^4*e^4 - 4* (b^4*c*d^3 + a*b^3*d^4)*e^3*f + 6*(b^4*c^2*d^2 + a^2*b^2*d^4)*e^2*f^2 - 4* (b^4*c^3*d + a^3*b*d^4)*e*f^3 + (b^4*c^4 + a^4*d^4)*f^4)*x)/(a*b^5*c^3*d^3 - 2*a^2*b^4*c^2*d^4 + a^3*b^3*c*d^5 + (b^6*c^2*d^4 - 2*a*b^5*c*d^5 + a^2* b^4*d^6)*x^2 + (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) *x)
Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (166) = 332\).
Time = 0.32 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.48 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {f^{4} x}{b^{2} d^{2}} - \frac {2 \, {\left (b^{4} d e^{4} - 2 \, b^{4} c e^{3} f - 2 \, a b^{3} d e^{3} f + 6 \, a b^{3} c e^{2} f^{2} - 6 \, a^{2} b^{2} c e f^{3} + 2 \, a^{3} b d e f^{3} + 2 \, a^{3} b c f^{4} - a^{4} d f^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} + \frac {2 \, {\left (b d^{4} e^{4} - 2 \, b c d^{3} e^{3} f - 2 \, a d^{4} e^{3} f + 6 \, a c d^{3} e^{2} f^{2} + 2 \, b c^{3} d e f^{3} - 6 \, a c^{2} d^{2} e f^{3} - b c^{4} f^{4} + 2 \, a c^{3} d f^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} - \frac {\frac {{\left (2 \, b^{4} d^{4} e^{4} - 4 \, b^{4} c d^{3} e^{3} f - 4 \, a b^{3} d^{4} e^{3} f + 6 \, b^{4} c^{2} d^{2} e^{2} f^{2} + 6 \, a^{2} b^{2} d^{4} e^{2} f^{2} - 4 \, b^{4} c^{3} d e f^{3} - 4 \, a^{3} b d^{4} e f^{3} + b^{4} c^{4} f^{4} + a^{4} d^{4} f^{4}\right )} x}{b d} + \frac {b^{4} c d^{3} e^{4} + a b^{3} d^{4} e^{4} - 8 \, a b^{3} c d^{3} e^{3} f + 6 \, a b^{3} c^{2} d^{2} e^{2} f^{2} + 6 \, a^{2} b^{2} c d^{3} e^{2} f^{2} - 4 \, a b^{3} c^{3} d e f^{3} - 4 \, a^{3} b c d^{3} e f^{3} + a b^{3} c^{4} f^{4} + a^{4} c d^{3} f^{4}}{b d}}{{\left (b c - a d\right )}^{2} {\left (b x + a\right )} {\left (d x + c\right )} b^{2} d^{2}} \] Input:
integrate((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")
Output:
f^4*x/(b^2*d^2) - 2*(b^4*d*e^4 - 2*b^4*c*e^3*f - 2*a*b^3*d*e^3*f + 6*a*b^3 *c*e^2*f^2 - 6*a^2*b^2*c*e*f^3 + 2*a^3*b*d*e*f^3 + 2*a^3*b*c*f^4 - a^4*d*f ^4)*log(abs(b*x + a))/(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3 *d^3) + 2*(b*d^4*e^4 - 2*b*c*d^3*e^3*f - 2*a*d^4*e^3*f + 6*a*c*d^3*e^2*f^2 + 2*b*c^3*d*e*f^3 - 6*a*c^2*d^2*e*f^3 - b*c^4*f^4 + 2*a*c^3*d*f^4)*log(ab s(d*x + c))/(b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6) - (( 2*b^4*d^4*e^4 - 4*b^4*c*d^3*e^3*f - 4*a*b^3*d^4*e^3*f + 6*b^4*c^2*d^2*e^2* f^2 + 6*a^2*b^2*d^4*e^2*f^2 - 4*b^4*c^3*d*e*f^3 - 4*a^3*b*d^4*e*f^3 + b^4* c^4*f^4 + a^4*d^4*f^4)*x/(b*d) + (b^4*c*d^3*e^4 + a*b^3*d^4*e^4 - 8*a*b^3* c*d^3*e^3*f + 6*a*b^3*c^2*d^2*e^2*f^2 + 6*a^2*b^2*c*d^3*e^2*f^2 - 4*a*b^3* c^3*d*e*f^3 - 4*a^3*b*c*d^3*e*f^3 + a*b^3*c^4*f^4 + a^4*c*d^3*f^4)/(b*d))/ ((b*c - a*d)^2*(b*x + a)*(d*x + c)*b^2*d^2)
Time = 6.22 (sec) , antiderivative size = 634, normalized size of antiderivative = 3.82 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {\ln \left (a+b\,x\right )\,\left (b^3\,\left (4\,a\,d\,e^3\,f-12\,a\,c\,e^2\,f^2\right )-b\,\left (4\,c\,a^3\,f^4+4\,d\,e\,a^3\,f^3\right )-b^4\,\left (2\,d\,e^4-4\,c\,e^3\,f\right )+2\,a^4\,d\,f^4+12\,a^2\,b^2\,c\,e\,f^3\right )}{-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}-\frac {\frac {a^4\,c\,d^3\,f^4-4\,a^3\,b\,c\,d^3\,e\,f^3+6\,a^2\,b^2\,c\,d^3\,e^2\,f^2+a\,b^3\,c^4\,f^4-4\,a\,b^3\,c^3\,d\,e\,f^3+6\,a\,b^3\,c^2\,d^2\,e^2\,f^2-8\,a\,b^3\,c\,d^3\,e^3\,f+a\,b^3\,d^4\,e^4+b^4\,c\,d^3\,e^4}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (a^4\,d^4\,f^4-4\,a^3\,b\,d^4\,e\,f^3+6\,a^2\,b^2\,d^4\,e^2\,f^2-4\,a\,b^3\,d^4\,e^3\,f+b^4\,c^4\,f^4-4\,b^4\,c^3\,d\,e\,f^3+6\,b^4\,c^2\,d^2\,e^2\,f^2-4\,b^4\,c\,d^3\,e^3\,f+2\,b^4\,d^4\,e^4\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (c\,b^3\,d^2+a\,b^2\,d^3\right )+b^3\,d^3\,x^2+a\,b^2\,c\,d^2}+\frac {\ln \left (c+d\,x\right )\,\left (d^3\,\left (4\,b\,c\,e^3\,f-12\,a\,c\,e^2\,f^2\right )-d\,\left (4\,a\,c^3\,f^4+4\,b\,e\,c^3\,f^3\right )-d^4\,\left (2\,b\,e^4-4\,a\,e^3\,f\right )+2\,b\,c^4\,f^4+12\,a\,c^2\,d^2\,e\,f^3\right )}{a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}+\frac {f^4\,x}{b^2\,d^2} \] Input:
int((e + f*x)^4/(a*c + x*(a*d + b*c) + b*d*x^2)^2,x)
Output:
(log(a + b*x)*(b^3*(4*a*d*e^3*f - 12*a*c*e^2*f^2) - b*(4*a^3*c*f^4 + 4*a^3 *d*e*f^3) - b^4*(2*d*e^4 - 4*c*e^3*f) + 2*a^4*d*f^4 + 12*a^2*b^2*c*e*f^3)) /(b^6*c^3 - a^3*b^3*d^3 + 3*a^2*b^4*c*d^2 - 3*a*b^5*c^2*d) - ((a*b^3*c^4*f ^4 + a*b^3*d^4*e^4 + a^4*c*d^3*f^4 + b^4*c*d^3*e^4 + 6*a*b^3*c^2*d^2*e^2*f ^2 + 6*a^2*b^2*c*d^3*e^2*f^2 - 8*a*b^3*c*d^3*e^3*f - 4*a*b^3*c^3*d*e*f^3 - 4*a^3*b*c*d^3*e*f^3)/(b*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(a^4*d^4* f^4 + b^4*c^4*f^4 + 2*b^4*d^4*e^4 + 6*a^2*b^2*d^4*e^2*f^2 + 6*b^4*c^2*d^2* e^2*f^2 - 4*a*b^3*d^4*e^3*f - 4*a^3*b*d^4*e*f^3 - 4*b^4*c*d^3*e^3*f - 4*b^ 4*c^3*d*e*f^3))/(b*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x*(a*b^2*d^3 + b^3 *c*d^2) + b^3*d^3*x^2 + a*b^2*c*d^2) + (log(c + d*x)*(d^3*(4*b*c*e^3*f - 1 2*a*c*e^2*f^2) - d*(4*a*c^3*f^4 + 4*b*c^3*e*f^3) - d^4*(2*b*e^4 - 4*a*e^3* f) + 2*b*c^4*f^4 + 12*a*c^2*d^2*e*f^3))/(a^3*d^6 - b^3*c^3*d^3 + 3*a*b^2*c ^2*d^4 - 3*a^2*b*c*d^5) + (f^4*x)/(b^2*d^2)
Time = 0.27 (sec) , antiderivative size = 2785, normalized size of antiderivative = 16.78 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)
Output:
( - 2*log(a + b*x)*a**6*c*d**5*f**4 - 2*log(a + b*x)*a**6*d**6*f**4*x + 2* log(a + b*x)*a**5*b*c**2*d**4*f**4 + 4*log(a + b*x)*a**5*b*c*d**5*e*f**3 + 4*log(a + b*x)*a**5*b*d**6*e*f**3*x - 2*log(a + b*x)*a**5*b*d**6*f**4*x** 2 + 4*log(a + b*x)*a**4*b**2*c**3*d**3*f**4 - 8*log(a + b*x)*a**4*b**2*c** 2*d**4*e*f**3 + 6*log(a + b*x)*a**4*b**2*c**2*d**4*f**4*x - 4*log(a + b*x) *a**4*b**2*c*d**5*e*f**3*x + 2*log(a + b*x)*a**4*b**2*c*d**5*f**4*x**2 + 4 *log(a + b*x)*a**4*b**2*d**6*e*f**3*x**2 - 12*log(a + b*x)*a**3*b**3*c**3* d**3*e*f**3 + 4*log(a + b*x)*a**3*b**3*c**3*d**3*f**4*x + 12*log(a + b*x)* a**3*b**3*c**2*d**4*e**2*f**2 - 20*log(a + b*x)*a**3*b**3*c**2*d**4*e*f**3 *x + 4*log(a + b*x)*a**3*b**3*c**2*d**4*f**4*x**2 - 4*log(a + b*x)*a**3*b* *3*c*d**5*e**3*f + 12*log(a + b*x)*a**3*b**3*c*d**5*e**2*f**2*x - 8*log(a + b*x)*a**3*b**3*c*d**5*e*f**3*x**2 - 4*log(a + b*x)*a**3*b**3*d**6*e**3*f *x + 12*log(a + b*x)*a**2*b**4*c**3*d**3*e**2*f**2 - 12*log(a + b*x)*a**2* b**4*c**3*d**3*e*f**3*x - 8*log(a + b*x)*a**2*b**4*c**2*d**4*e**3*f + 24*l og(a + b*x)*a**2*b**4*c**2*d**4*e**2*f**2*x - 12*log(a + b*x)*a**2*b**4*c* *2*d**4*e*f**3*x**2 + 2*log(a + b*x)*a**2*b**4*c*d**5*e**4 - 12*log(a + b* x)*a**2*b**4*c*d**5*e**3*f*x + 12*log(a + b*x)*a**2*b**4*c*d**5*e**2*f**2* x**2 + 2*log(a + b*x)*a**2*b**4*d**6*e**4*x - 4*log(a + b*x)*a**2*b**4*d** 6*e**3*f*x**2 - 4*log(a + b*x)*a*b**5*c**3*d**3*e**3*f + 12*log(a + b*x)*a *b**5*c**3*d**3*e**2*f**2*x + 2*log(a + b*x)*a*b**5*c**2*d**4*e**4 - 12...