Integrand size = 29, antiderivative size = 339 \[ \int \frac {(e+f x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {f^6 x}{b^3 d^3}-\frac {(b e-a f)^6}{2 b^4 (b c-a d)^3 (a+b x)^2}+\frac {3 (b e-a f)^5 (b d e-2 b c f+a d f)}{b^4 (b c-a d)^4 (a+b x)}+\frac {(d e-c f)^6}{2 d^4 (b c-a d)^3 (c+d x)^2}+\frac {3 (d e-c f)^5 (b d e+b c f-2 a d f)}{d^4 (b c-a d)^4 (c+d x)}+\frac {3 (b e-a f)^4 \left (a^2 d^2 f^2+2 a b d f (d e-2 c f)+b^2 \left (2 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right ) \log (a+b x)}{b^4 (b c-a d)^5}-\frac {3 (d e-c f)^4 \left (5 a^2 d^2 f^2-2 a b d f (3 d e+2 c f)+b^2 \left (2 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \log (c+d x)}{d^4 (b c-a d)^5} \] Output:
f^6*x/b^3/d^3-1/2*(-a*f+b*e)^6/b^4/(-a*d+b*c)^3/(b*x+a)^2+3*(-a*f+b*e)^5*( a*d*f-2*b*c*f+b*d*e)/b^4/(-a*d+b*c)^4/(b*x+a)+1/2*(-c*f+d*e)^6/d^4/(-a*d+b *c)^3/(d*x+c)^2+3*(-c*f+d*e)^5*(-2*a*d*f+b*c*f+b*d*e)/d^4/(-a*d+b*c)^4/(d* x+c)+3*(-a*f+b*e)^4*(a^2*d^2*f^2+2*a*b*d*f*(-2*c*f+d*e)+b^2*(5*c^2*f^2-6*c *d*e*f+2*d^2*e^2))*ln(b*x+a)/b^4/(-a*d+b*c)^5-3*(-c*f+d*e)^4*(5*a^2*d^2*f^ 2-2*a*b*d*f*(2*c*f+3*d*e)+b^2*(c^2*f^2+2*c*d*e*f+2*d^2*e^2))*ln(d*x+c)/d^4 /(-a*d+b*c)^5
Time = 0.40 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00 \[ \int \frac {(e+f x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {f^6 x}{b^3 d^3}-\frac {(b e-a f)^6}{2 b^4 (b c-a d)^3 (a+b x)^2}+\frac {3 (b e-a f)^5 (b d e-2 b c f+a d f)}{b^4 (b c-a d)^4 (a+b x)}-\frac {(d e-c f)^6}{2 d^4 (-b c+a d)^3 (c+d x)^2}+\frac {3 (d e-c f)^5 (b d e+b c f-2 a d f)}{d^4 (b c-a d)^4 (c+d x)}+\frac {3 (b e-a f)^4 \left (a^2 d^2 f^2+2 a b d f (d e-2 c f)+b^2 \left (2 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right ) \log (a+b x)}{b^4 (b c-a d)^5}+\frac {3 (d e-c f)^4 \left (5 a^2 d^2 f^2-2 a b d f (3 d e+2 c f)+b^2 \left (2 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \log (c+d x)}{d^4 (-b c+a d)^5} \] Input:
Integrate[(e + f*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
Output:
(f^6*x)/(b^3*d^3) - (b*e - a*f)^6/(2*b^4*(b*c - a*d)^3*(a + b*x)^2) + (3*( b*e - a*f)^5*(b*d*e - 2*b*c*f + a*d*f))/(b^4*(b*c - a*d)^4*(a + b*x)) - (d *e - c*f)^6/(2*d^4*(-(b*c) + a*d)^3*(c + d*x)^2) + (3*(d*e - c*f)^5*(b*d*e + b*c*f - 2*a*d*f))/(d^4*(b*c - a*d)^4*(c + d*x)) + (3*(b*e - a*f)^4*(a^2 *d^2*f^2 + 2*a*b*d*f*(d*e - 2*c*f) + b^2*(2*d^2*e^2 - 6*c*d*e*f + 5*c^2*f^ 2))*Log[a + b*x])/(b^4*(b*c - a*d)^5) + (3*(d*e - c*f)^4*(5*a^2*d^2*f^2 - 2*a*b*d*f*(3*d*e + 2*c*f) + b^2*(2*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*Log[c + d*x])/(d^4*(-(b*c) + a*d)^5)
Time = 1.02 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.07, 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.
\(\displaystyle \int \frac {(e+f x)^6}{\left (x (a d+b c)+a c+b d x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle b^3 d^3 \int \left (\frac {f^6}{b^6 d^6}+\frac {3 (b e-a f)^4 \left (\left (2 d^2 e^2-6 c d f e+5 c^2 f^2\right ) b^2+2 a d f (d e-2 c f) b+a^2 d^2 f^2\right )}{b^6 d^3 (b c-a d)^5 (a+b x)}-\frac {3 (d e-c f)^4 \left (\left (2 d^2 e^2+2 c d f e+c^2 f^2\right ) b^2-2 a d f (3 d e+2 c f) b+5 a^2 d^2 f^2\right )}{b^3 d^6 (b c-a d)^5 (c+d x)}-\frac {3 (b e-a f)^5 (b d e-2 b c f+a d f)}{b^6 d^3 (b c-a d)^4 (a+b x)^2}-\frac {3 (d e-c f)^5 (b d e+b c f-2 a d f)}{b^3 d^6 (b c-a d)^4 (c+d x)^2}+\frac {(b e-a f)^6}{b^6 d^3 (b c-a d)^3 (a+b x)^3}-\frac {(d e-c f)^6}{b^3 d^6 (b c-a d)^3 (c+d x)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle b^3 d^3 \left (\frac {3 (b e-a f)^4 \log (a+b x) \left (a^2 d^2 f^2+2 a b d f (d e-2 c f)+b^2 \left (5 c^2 f^2-6 c d e f+2 d^2 e^2\right )\right )}{b^7 d^3 (b c-a d)^5}-\frac {3 (d e-c f)^4 \log (c+d x) \left (5 a^2 d^2 f^2-2 a b d f (2 c f+3 d e)+b^2 \left (c^2 f^2+2 c d e f+2 d^2 e^2\right )\right )}{b^3 d^7 (b c-a d)^5}+\frac {3 (b e-a f)^5 (a d f-2 b c f+b d e)}{b^7 d^3 (a+b x) (b c-a d)^4}-\frac {(b e-a f)^6}{2 b^7 d^3 (a+b x)^2 (b c-a d)^3}+\frac {3 (d e-c f)^5 (-2 a d f+b c f+b d e)}{b^3 d^7 (c+d x) (b c-a d)^4}+\frac {(d e-c f)^6}{2 b^3 d^7 (c+d x)^2 (b c-a d)^3}+\frac {f^6 x}{b^6 d^6}\right )\) |
Input:
Int[(e + f*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
Output:
b^3*d^3*((f^6*x)/(b^6*d^6) - (b*e - a*f)^6/(2*b^7*d^3*(b*c - a*d)^3*(a + b *x)^2) + (3*(b*e - a*f)^5*(b*d*e - 2*b*c*f + a*d*f))/(b^7*d^3*(b*c - a*d)^ 4*(a + b*x)) + (d*e - c*f)^6/(2*b^3*d^7*(b*c - a*d)^3*(c + d*x)^2) + (3*(d *e - c*f)^5*(b*d*e + b*c*f - 2*a*d*f))/(b^3*d^7*(b*c - a*d)^4*(c + d*x)) + (3*(b*e - a*f)^4*(a^2*d^2*f^2 + 2*a*b*d*f*(d*e - 2*c*f) + b^2*(2*d^2*e^2 - 6*c*d*e*f + 5*c^2*f^2))*Log[a + b*x])/(b^7*d^3*(b*c - a*d)^5) - (3*(d*e - c*f)^4*(5*a^2*d^2*f^2 - 2*a*b*d*f*(3*d*e + 2*c*f) + b^2*(2*d^2*e^2 + 2*c *d*e*f + c^2*f^2))*Log[c + d*x])/(b^3*d^7*(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. \(1004\) vs. \(2(335)=670\).
Time = 1.14 (sec) , antiderivative size = 1005, normalized size of antiderivative = 2.96
method | result | size |
default | \(\text {Expression too large to display}\) | \(1005\) |
norman | \(\text {Expression too large to display}\) | \(2705\) |
risch | \(\text {Expression too large to display}\) | \(4623\) |
parallelrisch | \(\text {Expression too large to display}\) | \(7811\) |
Input:
int((f*x+e)^6/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x,method=_RETURNVERBOSE)
Output:
f^6*x/b^3/d^3-1/2*(-a^6*f^6+6*a^5*b*e*f^5-15*a^4*b^2*e^2*f^4+20*a^3*b^3*e^ 3*f^3-15*a^2*b^4*e^4*f^2+6*a*b^5*e^5*f-b^6*e^6)/b^4/(a*d-b*c)^3/(b*x+a)^2- (3*a^6*d*f^6-6*a^5*b*c*f^6-12*a^5*b*d*e*f^5+30*a^4*b^2*c*e*f^5+15*a^4*b^2* d*e^2*f^4-60*a^3*b^3*c*e^2*f^4+60*a^2*b^4*c*e^3*f^3-15*a^2*b^4*d*e^4*f^2-3 0*a*b^5*c*e^4*f^2+12*a*b^5*d*e^5*f+6*b^6*c*e^5*f-3*b^6*d*e^6)/b^4/(a*d-b*c )^4/(b*x+a)+(-3*a^6*d^2*f^6+12*a^5*b*c*d*f^6+6*a^5*b*d^2*e*f^5-15*a^4*b^2* c^2*f^6-30*a^4*b^2*c*d*e*f^5+60*a^3*b^3*c^2*e*f^5-90*a^2*b^4*c^2*e^2*f^4+6 0*a^2*b^4*c*d*e^3*f^3-15*a^2*b^4*d^2*e^4*f^2+60*a*b^5*c^2*e^3*f^3-60*a*b^5 *c*d*e^4*f^2+18*a*b^5*d^2*e^5*f-15*b^6*c^2*e^4*f^2+18*b^6*c*d*e^5*f-6*b^6* d^2*e^6)/b^4/(a*d-b*c)^5*ln(b*x+a)-1/2/d^4*(c^6*f^6-6*c^5*d*e*f^5+15*c^4*d ^2*e^2*f^4-20*c^3*d^3*e^3*f^3+15*c^2*d^4*e^4*f^2-6*c*d^5*e^5*f+d^6*e^6)/(a *d-b*c)^3/(d*x+c)^2-(-6*a*c^5*d*f^6+30*a*c^4*d^2*e*f^5-60*a*c^3*d^3*e^2*f^ 4+60*a*c^2*d^4*e^3*f^3-30*a*c*d^5*e^4*f^2+6*a*d^6*e^5*f+3*b*c^6*f^6-12*b*c ^5*d*e*f^5+15*b*c^4*d^2*e^2*f^4-15*b*c^2*d^4*e^4*f^2+12*b*c*d^5*e^5*f-3*b* d^6*e^6)/d^4/(a*d-b*c)^4/(d*x+c)+(15*a^2*c^4*d^2*f^6-60*a^2*c^3*d^3*e*f^5+ 90*a^2*c^2*d^4*e^2*f^4-60*a^2*c*d^5*e^3*f^3+15*a^2*d^6*e^4*f^2-12*a*b*c^5* d*f^6+30*a*b*c^4*d^2*e*f^5-60*a*b*c^2*d^4*e^3*f^3+60*a*b*c*d^5*e^4*f^2-18* a*b*d^6*e^5*f+3*b^2*c^6*f^6-6*b^2*c^5*d*e*f^5+15*b^2*c^2*d^4*e^4*f^2-18*b^ 2*c*d^5*e^5*f+6*b^2*d^6*e^6)/d^4/(a*d-b*c)^5*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 5430 vs. \(2 (335) = 670\).
Time = 4.43 (sec) , antiderivative size = 5430, normalized size of antiderivative = 16.02 \[ \int \frac {(e+f x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^6/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**6/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 2337 vs. \(2 (335) = 670\).
Time = 0.12 (sec) , antiderivative size = 2337, normalized size of antiderivative = 6.89 \[ \int \frac {(e+f x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^6/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")
Output:
f^6*x/(b^3*d^3) + 3*(2*b^6*d^2*e^6 + 30*a^2*b^4*c^2*e^2*f^4 - 6*(b^6*c*d + a*b^5*d^2)*e^5*f + 5*(b^6*c^2 + 4*a*b^5*c*d + a^2*b^4*d^2)*e^4*f^2 - 20*( a*b^5*c^2 + a^2*b^4*c*d)*e^3*f^3 - 2*(10*a^3*b^3*c^2 - 5*a^4*b^2*c*d + a^5 *b*d^2)*e*f^5 + (5*a^4*b^2*c^2 - 4*a^5*b*c*d + a^6*d^2)*f^6)*log(b*x + a)/ (b^9*c^5 - 5*a*b^8*c^4*d + 10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4 *b^5*c*d^4 - a^5*b^4*d^5) - 3*(2*b^2*d^6*e^6 + 30*a^2*c^2*d^4*e^2*f^4 - 6* (b^2*c*d^5 + a*b*d^6)*e^5*f + 5*(b^2*c^2*d^4 + 4*a*b*c*d^5 + a^2*d^6)*e^4* f^2 - 20*(a*b*c^2*d^4 + a^2*c*d^5)*e^3*f^3 - 2*(b^2*c^5*d - 5*a*b*c^4*d^2 + 10*a^2*c^3*d^3)*e*f^5 + (b^2*c^6 - 4*a*b*c^5*d + 5*a^2*c^4*d^2)*f^6)*log (d*x + c)/(b^5*c^5*d^4 - 5*a*b^4*c^4*d^5 + 10*a^2*b^3*c^3*d^6 - 10*a^3*b^2 *c^2*d^7 + 5*a^4*b*c*d^8 - a^5*d^9) - 1/2*((b^7*c^3*d^4 - 7*a*b^6*c^2*d^5 - 7*a^2*b^5*c*d^6 + a^3*b^4*d^7)*e^6 + 6*(a*b^6*c^3*d^4 + 10*a^2*b^5*c^2*d ^5 + a^3*b^4*c*d^6)*e^5*f - 90*(a^2*b^5*c^3*d^4 + a^3*b^4*c^2*d^5)*e^4*f^2 + 20*(a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 + a^4*b^3*c^2*d^5)*e^3*f^3 + 1 5*(a^2*b^5*c^5*d^2 - 7*a^3*b^4*c^4*d^3 - 7*a^4*b^3*c^3*d^4 + a^5*b^2*c^2*d ^5)*e^2*f^4 - 18*(a^2*b^5*c^6*d - 3*a^3*b^4*c^5*d^2 - 3*a^5*b^2*c^3*d^4 + a^6*b*c^2*d^5)*e*f^5 + (5*a^2*b^5*c^7 - 11*a^3*b^4*c^6*d - 11*a^6*b*c^3*d^ 4 + 5*a^7*c^2*d^5)*f^6 - 6*(2*b^7*d^7*e^6 - 6*(b^7*c*d^6 + a*b^6*d^7)*e^5* f + 5*(b^7*c^2*d^5 + 4*a*b^6*c*d^6 + a^2*b^5*d^7)*e^4*f^2 - 20*(a*b^6*c^2* d^5 + a^2*b^5*c*d^6)*e^3*f^3 - 5*(b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 - 4*a^3...
Leaf count of result is larger than twice the leaf count of optimal. 2387 vs. \(2 (335) = 670\).
Time = 0.31 (sec) , antiderivative size = 2387, normalized size of antiderivative = 7.04 \[ \int \frac {(e+f x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^6/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")
Output:
f^6*x/(b^3*d^3) + 3*(2*b^6*d^2*e^6 - 6*b^6*c*d*e^5*f - 6*a*b^5*d^2*e^5*f + 5*b^6*c^2*e^4*f^2 + 20*a*b^5*c*d*e^4*f^2 + 5*a^2*b^4*d^2*e^4*f^2 - 20*a*b ^5*c^2*e^3*f^3 - 20*a^2*b^4*c*d*e^3*f^3 + 30*a^2*b^4*c^2*e^2*f^4 - 20*a^3* b^3*c^2*e*f^5 + 10*a^4*b^2*c*d*e*f^5 - 2*a^5*b*d^2*e*f^5 + 5*a^4*b^2*c^2*f ^6 - 4*a^5*b*c*d*f^6 + a^6*d^2*f^6)*log(abs(b*x + a))/(b^9*c^5 - 5*a*b^8*c ^4*d + 10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4 *d^5) - 3*(2*b^2*d^6*e^6 - 6*b^2*c*d^5*e^5*f - 6*a*b*d^6*e^5*f + 5*b^2*c^2 *d^4*e^4*f^2 + 20*a*b*c*d^5*e^4*f^2 + 5*a^2*d^6*e^4*f^2 - 20*a*b*c^2*d^4*e ^3*f^3 - 20*a^2*c*d^5*e^3*f^3 + 30*a^2*c^2*d^4*e^2*f^4 - 2*b^2*c^5*d*e*f^5 + 10*a*b*c^4*d^2*e*f^5 - 20*a^2*c^3*d^3*e*f^5 + b^2*c^6*f^6 - 4*a*b*c^5*d *f^6 + 5*a^2*c^4*d^2*f^6)*log(abs(d*x + c))/(b^5*c^5*d^4 - 5*a*b^4*c^4*d^5 + 10*a^2*b^3*c^3*d^6 - 10*a^3*b^2*c^2*d^7 + 5*a^4*b*c*d^8 - a^5*d^9) - 1/ 2*(b^7*c^3*d^4*e^6 - 7*a*b^6*c^2*d^5*e^6 - 7*a^2*b^5*c*d^6*e^6 + a^3*b^4*d ^7*e^6 + 6*a*b^6*c^3*d^4*e^5*f + 60*a^2*b^5*c^2*d^5*e^5*f + 6*a^3*b^4*c*d^ 6*e^5*f - 90*a^2*b^5*c^3*d^4*e^4*f^2 - 90*a^3*b^4*c^2*d^5*e^4*f^2 + 20*a^2 *b^5*c^4*d^3*e^3*f^3 + 200*a^3*b^4*c^3*d^4*e^3*f^3 + 20*a^4*b^3*c^2*d^5*e^ 3*f^3 + 15*a^2*b^5*c^5*d^2*e^2*f^4 - 105*a^3*b^4*c^4*d^3*e^2*f^4 - 105*a^4 *b^3*c^3*d^4*e^2*f^4 + 15*a^5*b^2*c^2*d^5*e^2*f^4 - 18*a^2*b^5*c^6*d*e*f^5 + 54*a^3*b^4*c^5*d^2*e*f^5 + 54*a^5*b^2*c^3*d^4*e*f^5 - 18*a^6*b*c^2*d^5* e*f^5 + 5*a^2*b^5*c^7*f^6 - 11*a^3*b^4*c^6*d*f^6 - 11*a^6*b*c^3*d^4*f^6...
Time = 7.31 (sec) , antiderivative size = 2656, normalized size of antiderivative = 7.83 \[ \int \frac {(e+f x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:
int((e + f*x)^6/(a*c + x*(a*d + b*c) + b*d*x^2)^3,x)
Output:
(log(a + b*x)*(b^2*(15*a^4*c^2*f^6 + 30*a^4*c*d*e*f^5) - b^5*(60*a*c^2*e^3 *f^3 + 18*a*d^2*e^5*f - 60*a*c*d*e^4*f^2) + b^6*(6*d^2*e^6 + 15*c^2*e^4*f^ 2 - 18*c*d*e^5*f) + b^4*(90*a^2*c^2*e^2*f^4 + 15*a^2*d^2*e^4*f^2 - 60*a^2* c*d*e^3*f^3) - b*(6*a^5*d^2*e*f^5 + 12*a^5*c*d*f^6) + 3*a^6*d^2*f^6 - 60*a ^3*b^3*c^2*e*f^5))/(b^9*c^5 - a^5*b^4*d^5 + 5*a^4*b^5*c*d^4 + 10*a^2*b^7*c ^3*d^2 - 10*a^3*b^6*c^2*d^3 - 5*a*b^8*c^4*d) - ((3*x^3*(a^6*d^6*f^6 + b^6* c^6*f^6 - 2*b^6*d^6*e^6 - 5*a^2*b^4*d^6*e^4*f^2 + 5*a^4*b^2*d^6*e^2*f^4 - 5*b^6*c^2*d^4*e^4*f^2 + 5*b^6*c^4*d^2*e^2*f^4 - 2*a*b^5*c^5*d*f^6 - 2*a^5* b*c*d^5*f^6 + 6*a*b^5*d^6*e^5*f - 4*a^5*b*d^6*e*f^5 + 6*b^6*c*d^5*e^5*f - 4*b^6*c^5*d*e*f^5 - 20*a*b^5*c*d^5*e^4*f^2 + 10*a*b^5*c^4*d^2*e*f^5 + 10*a ^4*b^2*c*d^5*e*f^5 + 20*a*b^5*c^2*d^4*e^3*f^3 - 20*a*b^5*c^3*d^3*e^2*f^4 + 20*a^2*b^4*c*d^5*e^3*f^3 - 20*a^3*b^3*c*d^5*e^2*f^4))/(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) + (5*a^2*b^5*c^7*f^6 + a^3*b^4*d^7*e^6 + 5*a^7*c^2*d^5*f^6 + b^7*c^3*d^4*e^6 - 7*a*b^6*c^2*d^5* e^6 - 7*a^2*b^5*c*d^6*e^6 - 11*a^3*b^4*c^6*d*f^6 - 11*a^6*b*c^3*d^4*f^6 + 6*a*b^6*c^3*d^4*e^5*f - 18*a^2*b^5*c^6*d*e*f^5 + 6*a^3*b^4*c*d^6*e^5*f - 1 8*a^6*b*c^2*d^5*e*f^5 + 60*a^2*b^5*c^2*d^5*e^5*f + 54*a^3*b^4*c^5*d^2*e*f^ 5 + 54*a^5*b^2*c^3*d^4*e*f^5 - 90*a^2*b^5*c^3*d^4*e^4*f^2 + 20*a^2*b^5*c^4 *d^3*e^3*f^3 + 15*a^2*b^5*c^5*d^2*e^2*f^4 - 90*a^3*b^4*c^2*d^5*e^4*f^2 + 2 00*a^3*b^4*c^3*d^4*e^3*f^3 - 105*a^3*b^4*c^4*d^3*e^2*f^4 + 20*a^4*b^3*c...
Time = 7.83 (sec) , antiderivative size = 10338, normalized size of antiderivative = 30.50 \[ \int \frac {(e+f x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^6/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)
Output:
( - 6*log(a + b*x)*a**9*c**2*d**7*f**6 - 12*log(a + b*x)*a**9*c*d**8*f**6* x - 6*log(a + b*x)*a**9*d**9*f**6*x**2 + 18*log(a + b*x)*a**8*b*c**3*d**6* f**6 + 12*log(a + b*x)*a**8*b*c**2*d**7*e*f**5 + 24*log(a + b*x)*a**8*b*c* *2*d**7*f**6*x + 24*log(a + b*x)*a**8*b*c*d**8*e*f**5*x - 6*log(a + b*x)*a **8*b*c*d**8*f**6*x**2 + 12*log(a + b*x)*a**8*b*d**9*e*f**5*x**2 - 12*log( a + b*x)*a**8*b*d**9*f**6*x**3 - 6*log(a + b*x)*a**7*b**2*c**4*d**5*f**6 - 48*log(a + b*x)*a**7*b**2*c**3*d**6*e*f**5 + 24*log(a + b*x)*a**7*b**2*c* *3*d**6*f**6*x - 72*log(a + b*x)*a**7*b**2*c**2*d**7*e*f**5*x + 60*log(a + b*x)*a**7*b**2*c**2*d**7*f**6*x**2 + 24*log(a + b*x)*a**7*b**2*c*d**8*f** 6*x**3 + 24*log(a + b*x)*a**7*b**2*d**9*e*f**5*x**3 - 6*log(a + b*x)*a**7* b**2*d**9*f**6*x**4 - 30*log(a + b*x)*a**6*b**3*c**5*d**4*f**6 + 60*log(a + b*x)*a**6*b**3*c**4*d**5*e*f**5 - 72*log(a + b*x)*a**6*b**3*c**4*d**5*f* *6*x + 24*log(a + b*x)*a**6*b**3*c**3*d**6*e*f**5*x - 36*log(a + b*x)*a**6 *b**3*c**3*d**6*f**6*x**2 - 120*log(a + b*x)*a**6*b**3*c**2*d**7*e*f**5*x* *2 + 24*log(a + b*x)*a**6*b**3*c**2*d**7*f**6*x**3 - 72*log(a + b*x)*a**6* b**3*c*d**8*e*f**5*x**3 + 18*log(a + b*x)*a**6*b**3*c*d**8*f**6*x**4 + 12* log(a + b*x)*a**6*b**3*d**9*e*f**5*x**4 + 120*log(a + b*x)*a**5*b**4*c**5* d**4*e*f**5 - 60*log(a + b*x)*a**5*b**4*c**5*d**4*f**6*x - 180*log(a + b*x )*a**5*b**4*c**4*d**5*e**2*f**4 + 360*log(a + b*x)*a**5*b**4*c**4*d**5*e*f **5*x - 126*log(a + b*x)*a**5*b**4*c**4*d**5*f**6*x**2 + 120*log(a + b*...