Integrand size = 29, antiderivative size = 221 \[ \int (e+f x)^m \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {(b e-a f)^2 (d e-c f)^2 (e+f x)^{1+m}}{f^5 (1+m)}-\frac {2 (b e-a f) (d e-c f) (2 b d e-b c f-a d f) (e+f x)^{2+m}}{f^5 (2+m)}+\frac {\left (a^2 d^2 f^2-2 a b d f (3 d e-2 c f)+b^2 \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) (e+f x)^{3+m}}{f^5 (3+m)}-\frac {2 b d (2 b d e-b c f-a d f) (e+f x)^{4+m}}{f^5 (4+m)}+\frac {b^2 d^2 (e+f x)^{5+m}}{f^5 (5+m)} \] Output:
(-a*f+b*e)^2*(-c*f+d*e)^2*(f*x+e)^(1+m)/f^5/(1+m)-2*(-a*f+b*e)*(-c*f+d*e)* (-a*d*f-b*c*f+2*b*d*e)*(f*x+e)^(2+m)/f^5/(2+m)+(a^2*d^2*f^2-2*a*b*d*f*(-2* c*f+3*d*e)+b^2*(c^2*f^2-6*c*d*e*f+6*d^2*e^2))*(f*x+e)^(3+m)/f^5/(3+m)-2*b* d*(-a*d*f-b*c*f+2*b*d*e)*(f*x+e)^(4+m)/f^5/(4+m)+b^2*d^2*(f*x+e)^(5+m)/f^5 /(5+m)
Time = 0.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.90 \[ \int (e+f x)^m \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {(e+f x)^{1+m} \left (\frac {(b e-a f)^2 (d e-c f)^2}{1+m}-\frac {2 (b e-a f) (d e-c f) (2 b d e-b c f-a d f) (e+f x)}{2+m}+\frac {\left (a^2 d^2 f^2+2 a b d f (-3 d e+2 c f)+b^2 \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) (e+f x)^2}{3+m}-\frac {2 b d (2 b d e-b c f-a d f) (e+f x)^3}{4+m}+\frac {b^2 d^2 (e+f x)^4}{5+m}\right )}{f^5} \] Input:
Integrate[(e + f*x)^m*(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
Output:
((e + f*x)^(1 + m)*(((b*e - a*f)^2*(d*e - c*f)^2)/(1 + m) - (2*(b*e - a*f) *(d*e - c*f)*(2*b*d*e - b*c*f - a*d*f)*(e + f*x))/(2 + m) + ((a^2*d^2*f^2 + 2*a*b*d*f*(-3*d*e + 2*c*f) + b^2*(6*d^2*e^2 - 6*c*d*e*f + c^2*f^2))*(e + f*x)^2)/(3 + m) - (2*b*d*(2*b*d*e - b*c*f - a*d*f)*(e + f*x)^3)/(4 + m) + (b^2*d^2*(e + f*x)^4)/(5 + m)))/f^5
Time = 0.44 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1140, 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.
| \(\displaystyle \int (e+f x)^m \left (x (a d+b c)+a c+b d x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(e+f x)^{m+2} \left (a^2 d^2 f^2-2 a b d f (3 d e-2 c f)+b^2 \left (c^2 f^2-6 c d e f+6 d^2 e^2\right )\right )}{f^4}+\frac {(a f-b e)^2 (c f-d e)^2 (e+f x)^m}{f^4}+\frac {2 (a f-b e) (c f-d e) (e+f x)^{m+1} (a d f+b c f-2 b d e)}{f^4}-\frac {2 b d (e+f x)^{m+3} (-a d f-b c f+2 b d e)}{f^4}+\frac {b^2 d^2 (e+f x)^{m+4}}{f^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(e+f x)^{m+3} \left (a^2 d^2 f^2-2 a b d f (3 d e-2 c f)+b^2 \left (c^2 f^2-6 c d e f+6 d^2 e^2\right )\right )}{f^5 (m+3)}+\frac {(b e-a f)^2 (d e-c f)^2 (e+f x)^{m+1}}{f^5 (m+1)}-\frac {2 (b e-a f) (d e-c f) (e+f x)^{m+2} (-a d f-b c f+2 b d e)}{f^5 (m+2)}-\frac {2 b d (e+f x)^{m+4} (-a d f-b c f+2 b d e)}{f^5 (m+4)}+\frac {b^2 d^2 (e+f x)^{m+5}}{f^5 (m+5)}\) |
Input:
Int[(e + f*x)^m*(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
Output:
((b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^(1 + m))/(f^5*(1 + m)) - (2*(b*e - a*f)*(d*e - c*f)*(2*b*d*e - b*c*f - a*d*f)*(e + f*x)^(2 + m))/(f^5*(2 + m) ) + ((a^2*d^2*f^2 - 2*a*b*d*f*(3*d*e - 2*c*f) + b^2*(6*d^2*e^2 - 6*c*d*e*f + c^2*f^2))*(e + f*x)^(3 + m))/(f^5*(3 + m)) - (2*b*d*(2*b*d*e - b*c*f - a*d*f)*(e + f*x)^(4 + m))/(f^5*(4 + m)) + (b^2*d^2*(e + f*x)^(5 + m))/(f^5 *(5 + m))
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1400\) vs. \(2(221)=442\).
Time = 1.00 (sec) , antiderivative size = 1401, normalized size of antiderivative = 6.34
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1401\) |
| gosper | \(\text {Expression too large to display}\) | \(1518\) |
| orering | \(\text {Expression too large to display}\) | \(1556\) |
| risch | \(\text {Expression too large to display}\) | \(2041\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(3059\) |
Input:
int((f*x+e)^m*(a*c+(a*d+b*c)*x+b*d*x^2)^2,x,method=_RETURNVERBOSE)
Output:
b^2*d^2/(5+m)*x^5*exp(m*ln(f*x+e))+e*(a^2*c^2*f^4*m^4+14*a^2*c^2*f^4*m^3-2 *a^2*c*d*e*f^3*m^3-2*a*b*c^2*e*f^3*m^3+71*a^2*c^2*f^4*m^2-24*a^2*c*d*e*f^3 *m^2+2*a^2*d^2*e^2*f^2*m^2-24*a*b*c^2*e*f^3*m^2+8*a*b*c*d*e^2*f^2*m^2+2*b^ 2*c^2*e^2*f^2*m^2+154*a^2*c^2*f^4*m-94*a^2*c*d*e*f^3*m+18*a^2*d^2*e^2*f^2* m-94*a*b*c^2*e*f^3*m+72*a*b*c*d*e^2*f^2*m-12*a*b*d^2*e^3*f*m+18*b^2*c^2*e^ 2*f^2*m-12*b^2*c*d*e^3*f*m+120*a^2*c^2*f^4-120*a^2*c*d*e*f^3+40*a^2*d^2*e^ 2*f^2-120*a*b*c^2*e*f^3+160*a*b*c*d*e^2*f^2-60*a*b*d^2*e^3*f+40*b^2*c^2*e^ 2*f^2-60*b^2*c*d*e^3*f+24*b^2*d^2*e^4)/f^5/(m^5+15*m^4+85*m^3+225*m^2+274* m+120)*exp(m*ln(f*x+e))+(a^2*d^2*f^2*m^2+4*a*b*c*d*f^2*m^2+2*a*b*d^2*e*f*m ^2+b^2*c^2*f^2*m^2+2*b^2*c*d*e*f*m^2+9*a^2*d^2*f^2*m+36*a*b*c*d*f^2*m+10*a *b*d^2*e*f*m+9*b^2*c^2*f^2*m+10*b^2*c*d*e*f*m-4*b^2*d^2*e^2*m+20*a^2*d^2*f ^2+80*a*b*c*d*f^2+20*b^2*c^2*f^2)/f^2/(m^3+12*m^2+47*m+60)*x^3*exp(m*ln(f* x+e))+(2*a^2*c*d*f^3*m^3+a^2*d^2*e*f^2*m^3+2*a*b*c^2*f^3*m^3+4*a*b*c*d*e*f ^2*m^3+b^2*c^2*e*f^2*m^3+24*a^2*c*d*f^3*m^2+9*a^2*d^2*e*f^2*m^2+24*a*b*c^2 *f^3*m^2+36*a*b*c*d*e*f^2*m^2-6*a*b*d^2*e^2*f*m^2+9*b^2*c^2*e*f^2*m^2-6*b^ 2*c*d*e^2*f*m^2+94*a^2*c*d*f^3*m+20*a^2*d^2*e*f^2*m+94*a*b*c^2*f^3*m+80*a* b*c*d*e*f^2*m-30*a*b*d^2*e^2*f*m+20*b^2*c^2*e*f^2*m-30*b^2*c*d*e^2*f*m+12* b^2*d^2*e^3*m+120*a^2*c*d*f^3+120*a*b*c^2*f^3)/f^3/(m^4+14*m^3+71*m^2+154* m+120)*x^2*exp(m*ln(f*x+e))+(a^2*c^2*f^4*m^4+2*a^2*c*d*e*f^3*m^4+2*a*b*c^2 *e*f^3*m^4+14*a^2*c^2*f^4*m^3+24*a^2*c*d*e*f^3*m^3-2*a^2*d^2*e^2*f^2*m^...
Leaf count of result is larger than twice the leaf count of optimal. 1461 vs. \(2 (221) = 442\).
Time = 0.11 (sec) , antiderivative size = 1461, normalized size of antiderivative = 6.61 \[ \int (e+f x)^m \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^m*(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas")
Output:
(a^2*c^2*e*f^4*m^4 + 24*b^2*d^2*e^5 + 120*a^2*c^2*e*f^4 - 60*(b^2*c*d + a* b*d^2)*e^4*f + 40*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^2 - 120*(a*b*c^2 + a^2*c*d)*e^2*f^3 + (b^2*d^2*f^5*m^4 + 10*b^2*d^2*f^5*m^3 + 35*b^2*d^2*f^5 *m^2 + 50*b^2*d^2*f^5*m + 24*b^2*d^2*f^5)*x^5 + (60*(b^2*c*d + a*b*d^2)*f^ 5 + (b^2*d^2*e*f^4 + 2*(b^2*c*d + a*b*d^2)*f^5)*m^4 + 2*(3*b^2*d^2*e*f^4 + 11*(b^2*c*d + a*b*d^2)*f^5)*m^3 + (11*b^2*d^2*e*f^4 + 82*(b^2*c*d + a*b*d ^2)*f^5)*m^2 + 2*(3*b^2*d^2*e*f^4 + 61*(b^2*c*d + a*b*d^2)*f^5)*m)*x^4 + 2 *(7*a^2*c^2*e*f^4 - (a*b*c^2 + a^2*c*d)*e^2*f^3)*m^3 + (40*(b^2*c^2 + 4*a* b*c*d + a^2*d^2)*f^5 + (2*(b^2*c*d + a*b*d^2)*e*f^4 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^5)*m^4 - 4*(b^2*d^2*e^2*f^3 - 4*(b^2*c*d + a*b*d^2)*e*f^4 - 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^5)*m^3 - (12*b^2*d^2*e^2*f^3 - 34*(b^2 *c*d + a*b*d^2)*e*f^4 - 49*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^5)*m^2 - 2*(4 *b^2*d^2*e^2*f^3 - 10*(b^2*c*d + a*b*d^2)*e*f^4 - 39*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^5)*m)*x^3 + (71*a^2*c^2*e*f^4 + 2*(b^2*c^2 + 4*a*b*c*d + a^2* d^2)*e^3*f^2 - 24*(a*b*c^2 + a^2*c*d)*e^2*f^3)*m^2 + (120*(a*b*c^2 + a^2*c *d)*f^5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e*f^4 + 2*(a*b*c^2 + a^2*c*d)*f ^5)*m^4 - 2*(3*(b^2*c*d + a*b*d^2)*e^2*f^3 - 5*(b^2*c^2 + 4*a*b*c*d + a^2* d^2)*e*f^4 - 13*(a*b*c^2 + a^2*c*d)*f^5)*m^3 + (12*b^2*d^2*e^3*f^2 - 36*(b ^2*c*d + a*b*d^2)*e^2*f^3 + 29*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e*f^4 + 118 *(a*b*c^2 + a^2*c*d)*f^5)*m^2 + 2*(6*b^2*d^2*e^3*f^2 - 15*(b^2*c*d + a*...
Leaf count of result is larger than twice the leaf count of optimal. 16288 vs. \(2 (211) = 422\).
Time = 3.63 (sec) , antiderivative size = 16288, normalized size of antiderivative = 73.70 \[ \int (e+f x)^m \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)**m*(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
Output:
Piecewise((e**m*(a**2*c**2*x + a**2*c*d*x**2 + a**2*d**2*x**3/3 + a*b*c**2 *x**2 + 4*a*b*c*d*x**3/3 + a*b*d**2*x**4/2 + b**2*c**2*x**3/3 + b**2*c*d*x **4/2 + b**2*d**2*x**5/5), Eq(f, 0)), (-3*a**2*c**2*f**4/(12*e**4*f**5 + 4 8*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8*x**3 + 12*f**9*x**4) - 2*a** 2*c*d*e*f**3/(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f** 8*x**3 + 12*f**9*x**4) - 8*a**2*c*d*f**4*x/(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8*x**3 + 12*f**9*x**4) - a**2*d**2*e**2*f**2 /(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8*x**3 + 12* f**9*x**4) - 4*a**2*d**2*e*f**3*x/(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2 *f**7*x**2 + 48*e*f**8*x**3 + 12*f**9*x**4) - 6*a**2*d**2*f**4*x**2/(12*e* *4*f**5 + 48*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8*x**3 + 12*f**9*x* *4) - 2*a*b*c**2*e*f**3/(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8*x**3 + 12*f**9*x**4) - 8*a*b*c**2*f**4*x/(12*e**4*f**5 + 48*e **3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8*x**3 + 12*f**9*x**4) - 4*a*b*c* d*e**2*f**2/(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8 *x**3 + 12*f**9*x**4) - 16*a*b*c*d*e*f**3*x/(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8*x**3 + 12*f**9*x**4) - 24*a*b*c*d*f**4*x* *2/(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2*f**7*x**2 + 48*e*f**8*x**3 + 1 2*f**9*x**4) - 6*a*b*d**2*e**3*f/(12*e**4*f**5 + 48*e**3*f**6*x + 72*e**2* f**7*x**2 + 48*e*f**8*x**3 + 12*f**9*x**4) - 24*a*b*d**2*e**2*f**2*x/(1...
Leaf count of result is larger than twice the leaf count of optimal. 701 vs. \(2 (221) = 442\).
Time = 0.07 (sec) , antiderivative size = 701, normalized size of antiderivative = 3.17 \[ \int (e+f x)^m \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((f*x+e)^m*(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="maxima")
Output:
2*(f^2*(m + 1)*x^2 + e*f*m*x - e^2)*(f*x + e)^m*a*b*c^2/((m^2 + 3*m + 2)*f ^2) + 2*(f^2*(m + 1)*x^2 + e*f*m*x - e^2)*(f*x + e)^m*a^2*c*d/((m^2 + 3*m + 2)*f^2) + (f*x + e)^(m + 1)*a^2*c^2/(f*(m + 1)) + ((m^2 + 3*m + 2)*f^3*x ^3 + (m^2 + m)*e*f^2*x^2 - 2*e^2*f*m*x + 2*e^3)*(f*x + e)^m*b^2*c^2/((m^3 + 6*m^2 + 11*m + 6)*f^3) + 4*((m^2 + 3*m + 2)*f^3*x^3 + (m^2 + m)*e*f^2*x^ 2 - 2*e^2*f*m*x + 2*e^3)*(f*x + e)^m*a*b*c*d/((m^3 + 6*m^2 + 11*m + 6)*f^3 ) + ((m^2 + 3*m + 2)*f^3*x^3 + (m^2 + m)*e*f^2*x^2 - 2*e^2*f*m*x + 2*e^3)* (f*x + e)^m*a^2*d^2/((m^3 + 6*m^2 + 11*m + 6)*f^3) + 2*((m^3 + 6*m^2 + 11* m + 6)*f^4*x^4 + (m^3 + 3*m^2 + 2*m)*e*f^3*x^3 - 3*(m^2 + m)*e^2*f^2*x^2 + 6*e^3*f*m*x - 6*e^4)*(f*x + e)^m*b^2*c*d/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*f^4) + 2*((m^3 + 6*m^2 + 11*m + 6)*f^4*x^4 + (m^3 + 3*m^2 + 2*m)*e*f^ 3*x^3 - 3*(m^2 + m)*e^2*f^2*x^2 + 6*e^3*f*m*x - 6*e^4)*(f*x + e)^m*a*b*d^2 /((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*f^4) + ((m^4 + 10*m^3 + 35*m^2 + 50* m + 24)*f^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*e*f^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*e^2*f^3*x^3 + 12*(m^2 + m)*e^3*f^2*x^2 - 24*e^4*f*m*x + 24*e^5)*(f* x + e)^m*b^2*d^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*f^5)
Leaf count of result is larger than twice the leaf count of optimal. 2953 vs. \(2 (221) = 442\).
Time = 0.40 (sec) , antiderivative size = 2953, normalized size of antiderivative = 13.36 \[ \int (e+f x)^m \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^m*(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")
Output:
((f*x + e)^m*b^2*d^2*f^5*m^4*x^5 + (f*x + e)^m*b^2*d^2*e*f^4*m^4*x^4 + 2*( f*x + e)^m*b^2*c*d*f^5*m^4*x^4 + 2*(f*x + e)^m*a*b*d^2*f^5*m^4*x^4 + 10*(f *x + e)^m*b^2*d^2*f^5*m^3*x^5 + 2*(f*x + e)^m*b^2*c*d*e*f^4*m^4*x^3 + 2*(f *x + e)^m*a*b*d^2*e*f^4*m^4*x^3 + (f*x + e)^m*b^2*c^2*f^5*m^4*x^3 + 4*(f*x + e)^m*a*b*c*d*f^5*m^4*x^3 + (f*x + e)^m*a^2*d^2*f^5*m^4*x^3 + 6*(f*x + e )^m*b^2*d^2*e*f^4*m^3*x^4 + 22*(f*x + e)^m*b^2*c*d*f^5*m^3*x^4 + 22*(f*x + e)^m*a*b*d^2*f^5*m^3*x^4 + 35*(f*x + e)^m*b^2*d^2*f^5*m^2*x^5 + (f*x + e) ^m*b^2*c^2*e*f^4*m^4*x^2 + 4*(f*x + e)^m*a*b*c*d*e*f^4*m^4*x^2 + (f*x + e) ^m*a^2*d^2*e*f^4*m^4*x^2 + 2*(f*x + e)^m*a*b*c^2*f^5*m^4*x^2 + 2*(f*x + e) ^m*a^2*c*d*f^5*m^4*x^2 - 4*(f*x + e)^m*b^2*d^2*e^2*f^3*m^3*x^3 + 16*(f*x + e)^m*b^2*c*d*e*f^4*m^3*x^3 + 16*(f*x + e)^m*a*b*d^2*e*f^4*m^3*x^3 + 12*(f *x + e)^m*b^2*c^2*f^5*m^3*x^3 + 48*(f*x + e)^m*a*b*c*d*f^5*m^3*x^3 + 12*(f *x + e)^m*a^2*d^2*f^5*m^3*x^3 + 11*(f*x + e)^m*b^2*d^2*e*f^4*m^2*x^4 + 82* (f*x + e)^m*b^2*c*d*f^5*m^2*x^4 + 82*(f*x + e)^m*a*b*d^2*f^5*m^2*x^4 + 50* (f*x + e)^m*b^2*d^2*f^5*m*x^5 + 2*(f*x + e)^m*a*b*c^2*e*f^4*m^4*x + 2*(f*x + e)^m*a^2*c*d*e*f^4*m^4*x + (f*x + e)^m*a^2*c^2*f^5*m^4*x - 6*(f*x + e)^ m*b^2*c*d*e^2*f^3*m^3*x^2 - 6*(f*x + e)^m*a*b*d^2*e^2*f^3*m^3*x^2 + 10*(f* x + e)^m*b^2*c^2*e*f^4*m^3*x^2 + 40*(f*x + e)^m*a*b*c*d*e*f^4*m^3*x^2 + 10 *(f*x + e)^m*a^2*d^2*e*f^4*m^3*x^2 + 26*(f*x + e)^m*a*b*c^2*f^5*m^3*x^2 + 26*(f*x + e)^m*a^2*c*d*f^5*m^3*x^2 - 12*(f*x + e)^m*b^2*d^2*e^2*f^3*m^2...
Time = 6.01 (sec) , antiderivative size = 1509, normalized size of antiderivative = 6.83 \[ \int (e+f x)^m \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\text {Too large to display} \] Input:
int((e + f*x)^m*(a*c + x*(a*d + b*c) + b*d*x^2)^2,x)
Output:
((e + f*x)^m*(24*b^2*d^2*e^5 + 120*a^2*c^2*e*f^4 + 40*a^2*d^2*e^3*f^2 + 40 *b^2*c^2*e^3*f^2 - 60*a*b*d^2*e^4*f - 60*b^2*c*d*e^4*f + 2*a^2*d^2*e^3*f^2 *m^2 + 2*b^2*c^2*e^3*f^2*m^2 - 120*a*b*c^2*e^2*f^3 - 120*a^2*c*d*e^2*f^3 + 154*a^2*c^2*e*f^4*m + 71*a^2*c^2*e*f^4*m^2 + 14*a^2*c^2*e*f^4*m^3 + a^2*c ^2*e*f^4*m^4 + 18*a^2*d^2*e^3*f^2*m + 18*b^2*c^2*e^3*f^2*m - 24*a*b*c^2*e^ 2*f^3*m^2 - 2*a*b*c^2*e^2*f^3*m^3 - 24*a^2*c*d*e^2*f^3*m^2 - 2*a^2*c*d*e^2 *f^3*m^3 + 160*a*b*c*d*e^3*f^2 - 12*a*b*d^2*e^4*f*m - 12*b^2*c*d*e^4*f*m - 94*a*b*c^2*e^2*f^3*m - 94*a^2*c*d*e^2*f^3*m + 8*a*b*c*d*e^3*f^2*m^2 + 72* a*b*c*d*e^3*f^2*m))/(f^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*(e + f*x)^m*(120*a^2*c^2*f^5 + 154*a^2*c^2*f^5*m + 71*a^2*c^2*f^5*m^2 + 14*a^2*c^2*f^5*m^3 + a^2*c^2*f^5*m^4 - 18*a^2*d^2*e^2*f^3*m^2 - 18*b^2* c^2*e^2*f^3*m^2 - 2*a^2*d^2*e^2*f^3*m^3 - 2*b^2*c^2*e^2*f^3*m^3 - 24*b^2*d ^2*e^4*f*m - 40*a^2*d^2*e^2*f^3*m - 40*b^2*c^2*e^2*f^3*m + 12*a*b*d^2*e^3* f^2*m^2 + 12*b^2*c*d*e^3*f^2*m^2 + 120*a*b*c^2*e*f^4*m + 120*a^2*c*d*e*f^4 *m + 94*a*b*c^2*e*f^4*m^2 + 24*a*b*c^2*e*f^4*m^3 + 2*a*b*c^2*e*f^4*m^4 + 6 0*a*b*d^2*e^3*f^2*m + 94*a^2*c*d*e*f^4*m^2 + 24*a^2*c*d*e*f^4*m^3 + 2*a^2* c*d*e*f^4*m^4 + 60*b^2*c*d*e^3*f^2*m - 72*a*b*c*d*e^2*f^3*m^2 - 8*a*b*c*d* e^2*f^3*m^3 - 160*a*b*c*d*e^2*f^3*m))/(f^5*(274*m + 225*m^2 + 85*m^3 + 15* m^4 + m^5 + 120)) + (x^2*(e + f*x)^m*(m + 1)*(12*b^2*d^2*e^3*m + 120*a*b*c ^2*f^3 + 120*a^2*c*d*f^3 + 94*a*b*c^2*f^3*m + 94*a^2*c*d*f^3*m + 24*a*b...
Time = 0.24 (sec) , antiderivative size = 2040, normalized size of antiderivative = 9.23 \[ \int (e+f x)^m \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx =\text {Too large to display} \] Input:
int((f*x+e)^m*(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)
Output:
((e + f*x)**m*(a**2*c**2*e*f**4*m**4 + 14*a**2*c**2*e*f**4*m**3 + 71*a**2* c**2*e*f**4*m**2 + 154*a**2*c**2*e*f**4*m + 120*a**2*c**2*e*f**4 + a**2*c* *2*f**5*m**4*x + 14*a**2*c**2*f**5*m**3*x + 71*a**2*c**2*f**5*m**2*x + 154 *a**2*c**2*f**5*m*x + 120*a**2*c**2*f**5*x - 2*a**2*c*d*e**2*f**3*m**3 - 2 4*a**2*c*d*e**2*f**3*m**2 - 94*a**2*c*d*e**2*f**3*m - 120*a**2*c*d*e**2*f* *3 + 2*a**2*c*d*e*f**4*m**4*x + 24*a**2*c*d*e*f**4*m**3*x + 94*a**2*c*d*e* f**4*m**2*x + 120*a**2*c*d*e*f**4*m*x + 2*a**2*c*d*f**5*m**4*x**2 + 26*a** 2*c*d*f**5*m**3*x**2 + 118*a**2*c*d*f**5*m**2*x**2 + 214*a**2*c*d*f**5*m*x **2 + 120*a**2*c*d*f**5*x**2 + 2*a**2*d**2*e**3*f**2*m**2 + 18*a**2*d**2*e **3*f**2*m + 40*a**2*d**2*e**3*f**2 - 2*a**2*d**2*e**2*f**3*m**3*x - 18*a* *2*d**2*e**2*f**3*m**2*x - 40*a**2*d**2*e**2*f**3*m*x + a**2*d**2*e*f**4*m **4*x**2 + 10*a**2*d**2*e*f**4*m**3*x**2 + 29*a**2*d**2*e*f**4*m**2*x**2 + 20*a**2*d**2*e*f**4*m*x**2 + a**2*d**2*f**5*m**4*x**3 + 12*a**2*d**2*f**5 *m**3*x**3 + 49*a**2*d**2*f**5*m**2*x**3 + 78*a**2*d**2*f**5*m*x**3 + 40*a **2*d**2*f**5*x**3 - 2*a*b*c**2*e**2*f**3*m**3 - 24*a*b*c**2*e**2*f**3*m** 2 - 94*a*b*c**2*e**2*f**3*m - 120*a*b*c**2*e**2*f**3 + 2*a*b*c**2*e*f**4*m **4*x + 24*a*b*c**2*e*f**4*m**3*x + 94*a*b*c**2*e*f**4*m**2*x + 120*a*b*c* *2*e*f**4*m*x + 2*a*b*c**2*f**5*m**4*x**2 + 26*a*b*c**2*f**5*m**3*x**2 + 1 18*a*b*c**2*f**5*m**2*x**2 + 214*a*b*c**2*f**5*m*x**2 + 120*a*b*c**2*f**5* x**2 + 8*a*b*c*d*e**3*f**2*m**2 + 72*a*b*c*d*e**3*f**2*m + 160*a*b*c*d*...