Integrand size = 29, antiderivative size = 237 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {(b e-a f)^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac {(b e-a f)^3 (3 b d e-4 b c f+a d f)}{b^2 (b c-a d)^4 (a+b x)}+\frac {(d e-c f)^4}{2 d^2 (b c-a d)^3 (c+d x)^2}+\frac {(d e-c f)^3 (3 b d e+b c f-4 a d f)}{d^2 (b c-a d)^4 (c+d x)}+\frac {6 (b e-a f)^2 (d e-c f)^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 (b e-a f)^2 (d e-c f)^2 \log (c+d x)}{(b c-a d)^5} \] Output:
-1/2*(-a*f+b*e)^4/b^2/(-a*d+b*c)^3/(b*x+a)^2+(-a*f+b*e)^3*(a*d*f-4*b*c*f+3 *b*d*e)/b^2/(-a*d+b*c)^4/(b*x+a)+1/2*(-c*f+d*e)^4/d^2/(-a*d+b*c)^3/(d*x+c) ^2+(-c*f+d*e)^3*(-4*a*d*f+b*c*f+3*b*d*e)/d^2/(-a*d+b*c)^4/(d*x+c)+6*(-a*f+ b*e)^2*(-c*f+d*e)^2*ln(b*x+a)/(-a*d+b*c)^5-6*(-a*f+b*e)^2*(-c*f+d*e)^2*ln( d*x+c)/(-a*d+b*c)^5
Time = 0.17 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {(b e-a f)^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac {(b e-a f)^3 (3 b d e-4 b c f+a d f)}{b^2 (b c-a d)^4 (a+b x)}-\frac {(d e-c f)^4}{2 d^2 (-b c+a d)^3 (c+d x)^2}+\frac {(d e-c f)^3 (3 b d e+b c f-4 a d f)}{d^2 (b c-a d)^4 (c+d x)}+\frac {6 (b e-a f)^2 (d e-c f)^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 (b e-a f)^2 (d e-c f)^2 \log (c+d x)}{(b c-a d)^5} \] Input:
Integrate[(e + f*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
Output:
-1/2*(b*e - a*f)^4/(b^2*(b*c - a*d)^3*(a + b*x)^2) + ((b*e - a*f)^3*(3*b*d *e - 4*b*c*f + a*d*f))/(b^2*(b*c - a*d)^4*(a + b*x)) - (d*e - c*f)^4/(2*d^ 2*(-(b*c) + a*d)^3*(c + d*x)^2) + ((d*e - c*f)^3*(3*b*d*e + b*c*f - 4*a*d* f))/(d^2*(b*c - a*d)^4*(c + d*x)) + (6*(b*e - a*f)^2*(d*e - c*f)^2*Log[a + b*x])/(b*c - a*d)^5 - (6*(b*e - a*f)^2*(d*e - c*f)^2*Log[c + d*x])/(b*c - a*d)^5
Time = 0.67 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.13, 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.
\(\displaystyle \int \frac {(e+f x)^4}{\left (x (a d+b c)+a c+b d x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle b^3 d^3 \int \left (\frac {(b e-a f)^4}{b^4 d^3 (b c-a d)^3 (a+b x)^3}-\frac {(3 b d e-4 b c f+a d f) (b e-a f)^3}{b^4 d^3 (b c-a d)^4 (a+b x)^2}+\frac {6 (d e-c f)^2 (b e-a f)^2}{b^2 d^3 (b c-a d)^5 (a+b x)}-\frac {6 (d e-c f)^2 (b e-a f)^2}{b^3 d^2 (b c-a d)^5 (c+d x)}-\frac {(d e-c f)^3 (3 b d e+b c f-4 a d f)}{b^3 d^4 (b c-a d)^4 (c+d x)^2}-\frac {(d e-c f)^4}{b^3 d^4 (b c-a d)^3 (c+d x)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle b^3 d^3 \left (-\frac {(b e-a f)^4}{2 b^5 d^3 (a+b x)^2 (b c-a d)^3}+\frac {(b e-a f)^3 (a d f-4 b c f+3 b d e)}{b^5 d^3 (a+b x) (b c-a d)^4}+\frac {(d e-c f)^3 (-4 a d f+b c f+3 b d e)}{b^3 d^5 (c+d x) (b c-a d)^4}+\frac {(d e-c f)^4}{2 b^3 d^5 (c+d x)^2 (b c-a d)^3}+\frac {6 (b e-a f)^2 \log (a+b x) (d e-c f)^2}{b^3 d^3 (b c-a d)^5}-\frac {6 (b e-a f)^2 (d e-c f)^2 \log (c+d x)}{b^3 d^3 (b c-a d)^5}\right )\) |
Input:
Int[(e + f*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
Output:
b^3*d^3*(-1/2*(b*e - a*f)^4/(b^5*d^3*(b*c - a*d)^3*(a + b*x)^2) + ((b*e - a*f)^3*(3*b*d*e - 4*b*c*f + a*d*f))/(b^5*d^3*(b*c - a*d)^4*(a + b*x)) + (d *e - c*f)^4/(2*b^3*d^5*(b*c - a*d)^3*(c + d*x)^2) + ((d*e - c*f)^3*(3*b*d* e + b*c*f - 4*a*d*f))/(b^3*d^5*(b*c - a*d)^4*(c + d*x)) + (6*(b*e - a*f)^2 *(d*e - c*f)^2*Log[a + b*x])/(b^3*d^3*(b*c - a*d)^5) - (6*(b*e - a*f)^2*(d *e - c*f)^2*Log[c + d*x])/(b^3*d^3*(b*c - a*d)^5))
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. \(611\) vs. \(2(233)=466\).
Time = 1.04 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.58
method | result | size |
default | \(-\frac {-a^{4} f^{4}+4 e \,f^{3} a^{3} b -6 a^{2} b^{2} e^{2} f^{2}+4 a \,b^{3} e^{3} f -b^{4} e^{4}}{2 b^{2} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {a^{4} d \,f^{4}-4 a^{3} b c \,f^{4}+12 a^{2} b^{2} c e \,f^{3}-6 a^{2} b^{2} d \,e^{2} f^{2}-12 a \,b^{3} c \,e^{2} f^{2}+8 a \,b^{3} d \,e^{3} f +4 b^{4} c \,e^{3} f -3 b^{4} d \,e^{4}}{\left (a d -b c \right )^{4} b^{2} \left (b x +a \right )}-\frac {6 \left (a^{2} c^{2} f^{4}-2 c d \,f^{3} a^{2} e +a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}+4 a b c d \,e^{2} f^{2}-2 b \,d^{2} f a \,e^{3}+b^{2} c^{2} f^{2} e^{2}-2 b^{2} c d \,e^{3} f +b^{2} d^{2} e^{4}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}-\frac {c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} e^{2} f^{2} d^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}}{2 d^{2} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}-\frac {-4 a \,c^{3} d \,f^{4}+12 a \,c^{2} d^{2} e \,f^{3}-12 a c \,d^{3} e^{2} f^{2}+4 a \,d^{4} e^{3} f +b \,c^{4} f^{4}-6 b \,c^{2} d^{2} e^{2} f^{2}+8 b c \,d^{3} e^{3} f -3 b \,d^{4} e^{4}}{\left (a d -b c \right )^{4} d^{2} \left (d x +c \right )}+\frac {6 \left (a^{2} c^{2} f^{4}-2 c d \,f^{3} a^{2} e +a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}+4 a b c d \,e^{2} f^{2}-2 b \,d^{2} f a \,e^{3}+b^{2} c^{2} f^{2} e^{2}-2 b^{2} c d \,e^{3} f +b^{2} d^{2} e^{4}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}\) | \(612\) |
norman | \(\text {Expression too large to display}\) | \(1630\) |
risch | \(\text {Expression too large to display}\) | \(2809\) |
parallelrisch | \(\text {Expression too large to display}\) | \(4669\) |
Input:
int((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(-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^2/ (a*d-b*c)^3/(b*x+a)^2-(a^4*d*f^4-4*a^3*b*c*f^4+12*a^2*b^2*c*e*f^3-6*a^2*b^ 2*d*e^2*f^2-12*a*b^3*c*e^2*f^2+8*a*b^3*d*e^3*f+4*b^4*c*e^3*f-3*b^4*d*e^4)/ (a*d-b*c)^4/b^2/(b*x+a)-6*(a^2*c^2*f^4-2*a^2*c*d*e*f^3+a^2*d^2*e^2*f^2-2*a *b*c^2*e*f^3+4*a*b*c*d*e^2*f^2-2*a*b*d^2*e^3*f+b^2*c^2*e^2*f^2-2*b^2*c*d*e ^3*f+b^2*d^2*e^4)/(a*d-b*c)^5*ln(b*x+a)-1/2*(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^2/(a*d-b*c)^3/(d*x+c)^2-(-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+b*c^4*f^4-6*b*c^2*d^ 2*e^2*f^2+8*b*c*d^3*e^3*f-3*b*d^4*e^4)/(a*d-b*c)^4/d^2/(d*x+c)+6*(a^2*c^2* f^4-2*a^2*c*d*e*f^3+a^2*d^2*e^2*f^2-2*a*b*c^2*e*f^3+4*a*b*c*d*e^2*f^2-2*a* b*d^2*e^3*f+b^2*c^2*e^2*f^2-2*b^2*c*d*e^3*f+b^2*d^2*e^4)/(a*d-b*c)^5*ln(d* x+c)
Leaf count of result is larger than twice the leaf count of optimal. 3352 vs. \(2 (233) = 466\).
Time = 0.15 (sec) , antiderivative size = 3352, normalized size of antiderivative = 14.14 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 2864 vs. \(2 (212) = 424\).
Time = 52.72 (sec) , antiderivative size = 2864, normalized size of antiderivative = 12.08 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
Output:
(-a**5*c**2*d**3*f**4 + 7*a**4*b*c**3*d**2*f**4 - 4*a**4*b*c**2*d**3*e*f** 3 + 7*a**3*b**2*c**4*d*f**4 - 40*a**3*b**2*c**3*d**2*e*f**3 + 36*a**3*b**2 *c**2*d**3*e**2*f**2 - 4*a**3*b**2*c*d**4*e**3*f - a**3*b**2*d**5*e**4 - a **2*b**3*c**5*f**4 - 4*a**2*b**3*c**4*d*e*f**3 + 36*a**2*b**3*c**3*d**2*e* *2*f**2 - 40*a**2*b**3*c**2*d**3*e**3*f + 7*a**2*b**3*c*d**4*e**4 - 4*a*b* *4*c**3*d**2*e**3*f + 7*a*b**4*c**2*d**3*e**4 - b**5*c**3*d**2*e**4 + x**3 *(-2*a**4*b*d**5*f**4 + 8*a**3*b**2*c*d**4*f**4 - 24*a**2*b**3*c*d**4*e*f* *3 + 12*a**2*b**3*d**5*e**2*f**2 + 8*a*b**4*c**3*d**2*f**4 - 24*a*b**4*c** 2*d**3*e*f**3 + 48*a*b**4*c*d**4*e**2*f**2 - 24*a*b**4*d**5*e**3*f - 2*b** 5*c**4*d*f**4 + 12*b**5*c**2*d**3*e**2*f**2 - 24*b**5*c*d**4*e**3*f + 12*b **5*d**5*e**4) + x**2*(-a**5*d**5*f**4 + 3*a**4*b*c*d**4*f**4 - 4*a**4*b*d **5*e*f**3 + 16*a**3*b**2*c**2*d**3*f**4 - 20*a**3*b**2*c*d**4*e*f**3 + 18 *a**3*b**2*d**5*e**2*f**2 + 16*a**2*b**3*c**3*d**2*f**4 - 96*a**2*b**3*c** 2*d**3*e*f**3 + 90*a**2*b**3*c*d**4*e**2*f**2 - 36*a**2*b**3*d**5*e**3*f + 3*a*b**4*c**4*d*f**4 - 20*a*b**4*c**3*d**2*e*f**3 + 90*a*b**4*c**2*d**3*e **2*f**2 - 72*a*b**4*c*d**4*e**3*f + 18*a*b**4*d**5*e**4 - b**5*c**5*f**4 - 4*b**5*c**4*d*e*f**3 + 18*b**5*c**3*d**2*e**2*f**2 - 36*b**5*c**2*d**3*e **3*f + 18*b**5*c*d**4*e**4) + x*(-2*a**5*c*d**4*f**4 + 12*a**4*b*c**2*d** 3*f**4 - 8*a**4*b*c*d**4*e*f**3 + 16*a**3*b**2*c**3*d**2*f**4 - 64*a**3*b* *2*c**2*d**3*e*f**3 + 60*a**3*b**2*c*d**4*e**2*f**2 - 8*a**3*b**2*d**5*...
Leaf count of result is larger than twice the leaf count of optimal. 1558 vs. \(2 (233) = 466\).
Time = 0.07 (sec) , antiderivative size = 1558, normalized size of antiderivative = 6.57 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")
Output:
6*(b^2*d^2*e^4 + a^2*c^2*f^4 - 2*(b^2*c*d + a*b*d^2)*e^3*f + (b^2*c^2 + 4* a*b*c*d + a^2*d^2)*e^2*f^2 - 2*(a*b*c^2 + a^2*c*d)*e*f^3)*log(b*x + a)/(b^ 5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b* c*d^4 - a^5*d^5) - 6*(b^2*d^2*e^4 + a^2*c^2*f^4 - 2*(b^2*c*d + a*b*d^2)*e^ 3*f + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^2 - 2*(a*b*c^2 + a^2*c*d)*e*f^ 3)*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2 *c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - 1/2*((b^5*c^3*d^2 - 7*a*b^4*c^2*d^3 - 7*a^2*b^3*c*d^4 + a^3*b^2*d^5)*e^4 + 4*(a*b^4*c^3*d^2 + 10*a^2*b^3*c^2*d ^3 + a^3*b^2*c*d^4)*e^3*f - 36*(a^2*b^3*c^3*d^2 + a^3*b^2*c^2*d^3)*e^2*f^2 + 4*(a^2*b^3*c^4*d + 10*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*e*f^3 + (a^2*b^3 *c^5 - 7*a^3*b^2*c^4*d - 7*a^4*b*c^3*d^2 + a^5*c^2*d^3)*f^4 - 2*(6*b^5*d^5 *e^4 - 12*(b^5*c*d^4 + a*b^4*d^5)*e^3*f + 6*(b^5*c^2*d^3 + 4*a*b^4*c*d^4 + a^2*b^3*d^5)*e^2*f^2 - 12*(a*b^4*c^2*d^3 + a^2*b^3*c*d^4)*e*f^3 - (b^5*c^ 4*d - 4*a*b^4*c^3*d^2 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*f^4)*x^3 - (18*(b^5*c *d^4 + a*b^4*d^5)*e^4 - 36*(b^5*c^2*d^3 + 2*a*b^4*c*d^4 + a^2*b^3*d^5)*e^3 *f + 18*(b^5*c^3*d^2 + 5*a*b^4*c^2*d^3 + 5*a^2*b^3*c*d^4 + a^3*b^2*d^5)*e^ 2*f^2 - 4*(b^5*c^4*d + 5*a*b^4*c^3*d^2 + 24*a^2*b^3*c^2*d^3 + 5*a^3*b^2*c* d^4 + a^4*b*d^5)*e*f^3 - (b^5*c^5 - 3*a*b^4*c^4*d - 16*a^2*b^3*c^3*d^2 - 1 6*a^3*b^2*c^2*d^3 - 3*a^4*b*c*d^4 + a^5*d^5)*f^4)*x^2 - 2*(2*(b^5*c^2*d^3 + 7*a*b^4*c*d^4 + a^2*b^3*d^5)*e^4 - 4*(b^5*c^3*d^2 + 8*a*b^4*c^2*d^3 +...
Leaf count of result is larger than twice the leaf count of optimal. 1599 vs. \(2 (233) = 466\).
Time = 0.32 (sec) , antiderivative size = 1599, normalized size of antiderivative = 6.75 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")
Output:
6*(b^3*d^2*e^4 - 2*b^3*c*d*e^3*f - 2*a*b^2*d^2*e^3*f + b^3*c^2*e^2*f^2 + 4 *a*b^2*c*d*e^2*f^2 + a^2*b*d^2*e^2*f^2 - 2*a*b^2*c^2*e*f^3 - 2*a^2*b*c*d*e *f^3 + a^2*b*c^2*f^4)*log(abs(b*x + a))/(b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2* b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5) - 6*(b^2*d ^3*e^4 - 2*b^2*c*d^2*e^3*f - 2*a*b*d^3*e^3*f + b^2*c^2*d*e^2*f^2 + 4*a*b*c *d^2*e^2*f^2 + a^2*d^3*e^2*f^2 - 2*a*b*c^2*d*e*f^3 - 2*a^2*c*d^2*e*f^3 + a ^2*c^2*d*f^4)*log(abs(d*x + c))/(b^5*c^5*d - 5*a*b^4*c^4*d^2 + 10*a^2*b^3* c^3*d^3 - 10*a^3*b^2*c^2*d^4 + 5*a^4*b*c*d^5 - a^5*d^6) + 1/2*(12*b^5*d^5* e^4*x^3 - 24*b^5*c*d^4*e^3*f*x^3 - 24*a*b^4*d^5*e^3*f*x^3 + 12*b^5*c^2*d^3 *e^2*f^2*x^3 + 48*a*b^4*c*d^4*e^2*f^2*x^3 + 12*a^2*b^3*d^5*e^2*f^2*x^3 - 2 4*a*b^4*c^2*d^3*e*f^3*x^3 - 24*a^2*b^3*c*d^4*e*f^3*x^3 - 2*b^5*c^4*d*f^4*x ^3 + 8*a*b^4*c^3*d^2*f^4*x^3 + 8*a^3*b^2*c*d^4*f^4*x^3 - 2*a^4*b*d^5*f^4*x ^3 + 18*b^5*c*d^4*e^4*x^2 + 18*a*b^4*d^5*e^4*x^2 - 36*b^5*c^2*d^3*e^3*f*x^ 2 - 72*a*b^4*c*d^4*e^3*f*x^2 - 36*a^2*b^3*d^5*e^3*f*x^2 + 18*b^5*c^3*d^2*e ^2*f^2*x^2 + 90*a*b^4*c^2*d^3*e^2*f^2*x^2 + 90*a^2*b^3*c*d^4*e^2*f^2*x^2 + 18*a^3*b^2*d^5*e^2*f^2*x^2 - 4*b^5*c^4*d*e*f^3*x^2 - 20*a*b^4*c^3*d^2*e*f ^3*x^2 - 96*a^2*b^3*c^2*d^3*e*f^3*x^2 - 20*a^3*b^2*c*d^4*e*f^3*x^2 - 4*a^4 *b*d^5*e*f^3*x^2 - b^5*c^5*f^4*x^2 + 3*a*b^4*c^4*d*f^4*x^2 + 16*a^2*b^3*c^ 3*d^2*f^4*x^2 + 16*a^3*b^2*c^2*d^3*f^4*x^2 + 3*a^4*b*c*d^4*f^4*x^2 - a^5*d ^5*f^4*x^2 + 4*b^5*c^2*d^3*e^4*x + 28*a*b^4*c*d^4*e^4*x + 4*a^2*b^3*d^5...
Time = 7.34 (sec) , antiderivative size = 1677, normalized size of antiderivative = 7.08 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
int((e + f*x)^4/(a*c + x*(a*d + b*c) + b*d*x^2)^3,x)
Output:
- ((a^2*b^3*c^5*f^4 + a^3*b^2*d^5*e^4 + a^5*c^2*d^3*f^4 + b^5*c^3*d^2*e^4 - 7*a*b^4*c^2*d^3*e^4 - 7*a^2*b^3*c*d^4*e^4 - 7*a^3*b^2*c^4*d*f^4 - 7*a^4* b*c^3*d^2*f^4 + 4*a*b^4*c^3*d^2*e^3*f + 4*a^2*b^3*c^4*d*e*f^3 + 4*a^3*b^2* c*d^4*e^3*f + 4*a^4*b*c^2*d^3*e*f^3 + 40*a^2*b^3*c^2*d^3*e^3*f + 40*a^3*b^ 2*c^3*d^2*e*f^3 - 36*a^2*b^3*c^3*d^2*e^2*f^2 - 36*a^3*b^2*c^2*d^3*e^2*f^2) /(2*b^2*d^2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3 *b*c*d^3)) + (x^3*(a^4*d^4*f^4 + b^4*c^4*f^4 - 6*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*c^3*d*f^4 - 4*a^3*b*c*d^3*f^4 + 12*a*b^3*d^4*e^3*f + 12*b^4*c*d^3*e^3*f - 24*a*b^3*c*d^3*e^2*f^2 + 12*a *b^3*c^2*d^2*e*f^3 + 12*a^2*b^2*c*d^3*e*f^3))/(b*d*(a^4*d^4 + b^4*c^4 + 6* a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (x^2*(a^5*d^5*f^4 + b^ 5*c^5*f^4 - 18*a*b^4*d^5*e^4 - 18*b^5*c*d^4*e^4 - 16*a^2*b^3*c^3*d^2*f^4 - 16*a^3*b^2*c^2*d^3*f^4 - 18*a^3*b^2*d^5*e^2*f^2 - 18*b^5*c^3*d^2*e^2*f^2 - 3*a*b^4*c^4*d*f^4 - 3*a^4*b*c*d^4*f^4 + 4*a^4*b*d^5*e*f^3 + 4*b^5*c^4*d* e*f^3 + 36*a^2*b^3*d^5*e^3*f + 36*b^5*c^2*d^3*e^3*f + 20*a*b^4*c^3*d^2*e*f ^3 + 20*a^3*b^2*c*d^4*e*f^3 - 90*a*b^4*c^2*d^3*e^2*f^2 - 90*a^2*b^3*c*d^4* e^2*f^2 + 96*a^2*b^3*c^2*d^3*e*f^3 + 72*a*b^4*c*d^4*e^3*f))/(2*b^2*d^2*(a^ 4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (x *(a*b^4*c^5*f^4 + a^5*c*d^4*f^4 - 2*a^2*b^3*d^5*e^4 - 2*b^5*c^2*d^3*e^4 - 8*a^3*b^2*c^3*d^2*f^4 - 14*a*b^4*c*d^4*e^4 - 6*a^2*b^3*c^4*d*f^4 - 6*a^...
Time = 0.25 (sec) , antiderivative size = 6415, normalized size of antiderivative = 27.07 \[ \int \frac {(e+f x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)
Output:
( - 12*log(a + b*x)*a**5*b*c**4*d**2*f**4 + 24*log(a + b*x)*a**5*b*c**3*d* *3*e*f**3 - 24*log(a + b*x)*a**5*b*c**3*d**3*f**4*x - 12*log(a + b*x)*a**5 *b*c**2*d**4*e**2*f**2 + 48*log(a + b*x)*a**5*b*c**2*d**4*e*f**3*x - 12*lo g(a + b*x)*a**5*b*c**2*d**4*f**4*x**2 - 24*log(a + b*x)*a**5*b*c*d**5*e**2 *f**2*x + 24*log(a + b*x)*a**5*b*c*d**5*e*f**3*x**2 - 12*log(a + b*x)*a**5 *b*d**6*e**2*f**2*x**2 - 12*log(a + b*x)*a**4*b**2*c**5*d*f**4 + 48*log(a + b*x)*a**4*b**2*c**4*d**2*e*f**3 - 48*log(a + b*x)*a**4*b**2*c**4*d**2*f* *4*x - 60*log(a + b*x)*a**4*b**2*c**3*d**3*e**2*f**2 + 144*log(a + b*x)*a* *4*b**2*c**3*d**3*e*f**3*x - 60*log(a + b*x)*a**4*b**2*c**3*d**3*f**4*x**2 + 24*log(a + b*x)*a**4*b**2*c**2*d**4*e**3*f - 144*log(a + b*x)*a**4*b**2 *c**2*d**4*e**2*f**2*x + 144*log(a + b*x)*a**4*b**2*c**2*d**4*e*f**3*x**2 - 24*log(a + b*x)*a**4*b**2*c**2*d**4*f**4*x**3 + 48*log(a + b*x)*a**4*b** 2*c*d**5*e**3*f*x - 108*log(a + b*x)*a**4*b**2*c*d**5*e**2*f**2*x**2 + 48* log(a + b*x)*a**4*b**2*c*d**5*e*f**3*x**3 + 24*log(a + b*x)*a**4*b**2*d**6 *e**3*f*x**2 - 24*log(a + b*x)*a**4*b**2*d**6*e**2*f**2*x**3 + 24*log(a + b*x)*a**3*b**3*c**5*d*e*f**3 - 24*log(a + b*x)*a**3*b**3*c**5*d*f**4*x - 6 0*log(a + b*x)*a**3*b**3*c**4*d**2*e**2*f**2 + 144*log(a + b*x)*a**3*b**3* c**4*d**2*e*f**3*x - 60*log(a + b*x)*a**3*b**3*c**4*d**2*f**4*x**2 + 48*lo g(a + b*x)*a**3*b**3*c**3*d**3*e**3*f - 240*log(a + b*x)*a**3*b**3*c**3*d* *3*e**2*f**2*x + 240*log(a + b*x)*a**3*b**3*c**3*d**3*e*f**3*x**2 - 48*...