Integrand size = 26, antiderivative size = 181 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {(b d-a e)^3 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (1+2 p)}+\frac {3 e (b d-a e)^2 (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (1+p)}+\frac {3 e^2 (b d-a e) (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (3+2 p)}+\frac {e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (2+p)} \]
(-a*e+b*d)^3*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^p/b^4/(1+2*p)+3/2*e*(-a*e+b*d)^ 2*(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2)^p/b^4/(p+1)+3*e^2*(-a*e+b*d)*(b*x+a)^3*( b^2*x^2+2*a*b*x+a^2)^p/b^4/(3+2*p)+1/2*e^3*(b*x+a)^4*(b^2*x^2+2*a*b*x+a^2) ^p/b^4/(2+p)
Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.59 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {(a+b x) \left ((a+b x)^2\right )^p \left (\frac {2 (b d-a e)^3}{1+2 p}+\frac {3 e (b d-a e)^2 (a+b x)}{1+p}+\frac {6 e^2 (b d-a e) (a+b x)^2}{3+2 p}+\frac {e^3 (a+b x)^3}{2+p}\right )}{2 b^4} \]
((a + b*x)*((a + b*x)^2)^p*((2*(b*d - a*e)^3)/(1 + 2*p) + (3*e*(b*d - a*e) ^2*(a + b*x))/(1 + p) + (6*e^2*(b*d - a*e)*(a + b*x)^2)/(3 + 2*p) + (e^3*( a + b*x)^3)/(2 + p)))/(2*b^4)
Time = 0.33 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1102, 53, 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.
| \(\displaystyle \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \int \left (x b^2+a b\right )^{2 p} (d+e x)^3dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \int \left (\frac {(b d-a e)^3 \left (x b^2+a b\right )^{2 p}}{b^3}+\frac {3 e (b d-a e)^2 \left (x b^2+a b\right )^{2 p+1}}{b^4}+\frac {3 e^2 (b d-a e) \left (x b^2+a b\right )^{2 p+2}}{b^5}+\frac {e^3 \left (x b^2+a b\right )^{2 p+3}}{b^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \left (\frac {e^3 \left (a b+b^2 x\right )^{2 (p+2)}}{2 b^8 (p+2)}+\frac {3 e^2 (b d-a e) \left (a b+b^2 x\right )^{2 p+3}}{b^7 (2 p+3)}+\frac {3 e (b d-a e)^2 \left (a b+b^2 x\right )^{2 (p+1)}}{2 b^6 (p+1)}+\frac {(b d-a e)^3 \left (a b+b^2 x\right )^{2 p+1}}{b^5 (2 p+1)}\right )\) |
((a^2 + 2*a*b*x + b^2*x^2)^p*((3*e*(b*d - a*e)^2*(a*b + b^2*x)^(2*(1 + p)) )/(2*b^6*(1 + p)) + (e^3*(a*b + b^2*x)^(2*(2 + p)))/(2*b^8*(2 + p)) + ((b* d - a*e)^3*(a*b + b^2*x)^(1 + 2*p))/(b^5*(1 + 2*p)) + (3*e^2*(b*d - a*e)*( a*b + b^2*x)^(3 + 2*p))/(b^7*(3 + 2*p))))/(a*b + b^2*x)^(2*p)
3.18.44.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(404\) vs. \(2(177)=354\).
Time = 2.34 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.24
| method | result | size |
| gosper | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} \left (-4 b^{3} e^{3} p^{3} x^{3}-12 b^{3} d \,e^{2} p^{3} x^{2}-12 b^{3} e^{3} p^{2} x^{3}+6 a \,b^{2} e^{3} p^{2} x^{2}-12 b^{3} d^{2} e \,p^{3} x -42 b^{3} d \,e^{2} p^{2} x^{2}-11 b^{3} e^{3} p \,x^{3}+12 a \,b^{2} d \,e^{2} p^{2} x +9 a \,b^{2} e^{3} p \,x^{2}-4 b^{3} d^{3} p^{3}-48 b^{3} d^{2} e \,p^{2} x -42 b^{3} d \,e^{2} p \,x^{2}-3 e^{3} x^{3} b^{3}-6 a^{2} b \,e^{3} p x +6 a \,b^{2} d^{2} e \,p^{2}+30 a \,b^{2} d \,e^{2} p x +3 x^{2} a \,b^{2} e^{3}-18 b^{3} d^{3} p^{2}-57 b^{3} d^{2} e p x -12 x^{2} b^{3} d \,e^{2}-6 a^{2} b d \,e^{2} p -3 a^{2} b \,e^{3} x +21 a \,b^{2} d^{2} e p +12 x a \,b^{2} d \,e^{2}-26 b^{3} d^{3} p -18 b^{3} d^{2} e x +3 a^{3} e^{3}-12 a^{2} b d \,e^{2}+18 a \,b^{2} d^{2} e -12 b^{3} d^{3}\right ) \left (b x +a \right )}{2 b^{4} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}\) | \(405\) |
| norman | \(\frac {\left (6 a \,b^{2} d^{2} e \,p^{3}+2 b^{3} d^{3} p^{3}-6 a^{2} b d \,e^{2} p^{2}+21 a \,b^{2} d^{2} e \,p^{2}+9 b^{3} d^{3} p^{2}+3 a^{3} e^{3} p -12 a^{2} b d \,e^{2} p +18 a \,b^{2} d^{2} e p +13 b^{3} d^{3} p +6 b^{3} d^{3}\right ) x \,{\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b^{3} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}+\frac {e^{2} \left (a e p +3 b d p +6 b d \right ) x^{3} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b \left (2 p^{2}+7 p +6\right )}+\frac {e^{3} x^{4} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{4+2 p}-\frac {a \left (-4 b^{3} d^{3} p^{3}+6 a \,b^{2} d^{2} e \,p^{2}-18 b^{3} d^{3} p^{2}-6 a^{2} b d \,e^{2} p +21 a \,b^{2} d^{2} e p -26 b^{3} d^{3} p +3 a^{3} e^{3}-12 a^{2} b d \,e^{2}+18 a \,b^{2} d^{2} e -12 b^{3} d^{3}\right ) {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 b^{4} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}-\frac {3 \left (-2 a b d e \,p^{2}-2 b^{2} d^{2} p^{2}+a^{2} e^{2} p -4 a b d e p -7 b^{2} d^{2} p -6 b^{2} d^{2}\right ) e \,x^{2} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 b^{2} \left (2 p^{3}+9 p^{2}+13 p +6\right )}\) | \(498\) |
| risch | \(-\frac {\left (-4 b^{4} e^{3} p^{3} x^{4}-4 a \,b^{3} e^{3} p^{3} x^{3}-12 b^{4} d \,e^{2} p^{3} x^{3}-12 b^{4} e^{3} p^{2} x^{4}-12 a \,b^{3} d \,e^{2} p^{3} x^{2}-6 a \,b^{3} e^{3} p^{2} x^{3}-12 b^{4} d^{2} e \,p^{3} x^{2}-42 b^{4} d \,e^{2} p^{2} x^{3}-11 b^{4} e^{3} p \,x^{4}+6 a^{2} b^{2} e^{3} p^{2} x^{2}-12 a \,b^{3} d^{2} e \,p^{3} x -30 a \,b^{3} d \,e^{2} p^{2} x^{2}-2 a \,b^{3} e^{3} p \,x^{3}-4 b^{4} d^{3} p^{3} x -48 b^{4} d^{2} e \,p^{2} x^{2}-42 b^{4} d \,e^{2} p \,x^{3}-3 e^{3} x^{4} b^{4}+12 a^{2} b^{2} d \,e^{2} p^{2} x +3 a^{2} b^{2} e^{3} p \,x^{2}-4 a \,b^{3} d^{3} p^{3}-42 a \,b^{3} d^{2} e \,p^{2} x -12 a \,b^{3} d \,e^{2} p \,x^{2}-18 b^{4} d^{3} p^{2} x -57 b^{4} d^{2} e p \,x^{2}-12 b^{4} d \,e^{2} x^{3}-6 a^{3} b \,e^{3} p x +6 a^{2} b^{2} d^{2} e \,p^{2}+24 a^{2} b^{2} d \,e^{2} p x -18 a \,b^{3} d^{3} p^{2}-36 a \,b^{3} d^{2} e p x -26 b^{4} d^{3} p x -18 b^{4} d^{2} e \,x^{2}-6 a^{3} b d \,e^{2} p +21 a^{2} b^{2} d^{2} e p -26 a \,b^{3} d^{3} p -12 b^{4} d^{3} x +3 a^{4} e^{3}-12 a^{3} b d \,e^{2}+18 a^{2} b^{2} d^{2} e -12 a \,b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{p}}{2 \left (3+2 p \right ) \left (2+p \right ) \left (1+p \right ) \left (1+2 p \right ) b^{4}}\) | \(558\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1314\) |
-1/2*(b^2*x^2+2*a*b*x+a^2)^p*(-4*b^3*e^3*p^3*x^3-12*b^3*d*e^2*p^3*x^2-12*b ^3*e^3*p^2*x^3+6*a*b^2*e^3*p^2*x^2-12*b^3*d^2*e*p^3*x-42*b^3*d*e^2*p^2*x^2 -11*b^3*e^3*p*x^3+12*a*b^2*d*e^2*p^2*x+9*a*b^2*e^3*p*x^2-4*b^3*d^3*p^3-48* b^3*d^2*e*p^2*x-42*b^3*d*e^2*p*x^2-3*b^3*e^3*x^3-6*a^2*b*e^3*p*x+6*a*b^2*d ^2*e*p^2+30*a*b^2*d*e^2*p*x+3*a*b^2*e^3*x^2-18*b^3*d^3*p^2-57*b^3*d^2*e*p* x-12*b^3*d*e^2*x^2-6*a^2*b*d*e^2*p-3*a^2*b*e^3*x+21*a*b^2*d^2*e*p+12*a*b^2 *d*e^2*x-26*b^3*d^3*p-18*b^3*d^2*e*x+3*a^3*e^3-12*a^2*b*d*e^2+18*a*b^2*d^2 *e-12*b^3*d^3)*(b*x+a)/b^4/(4*p^4+20*p^3+35*p^2+25*p+6)
Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (177) = 354\).
Time = 0.38 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.85 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (4 \, a b^{3} d^{3} p^{3} + 12 \, a b^{3} d^{3} - 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} - 3 \, a^{4} e^{3} + {\left (4 \, b^{4} e^{3} p^{3} + 12 \, b^{4} e^{3} p^{2} + 11 \, b^{4} e^{3} p + 3 \, b^{4} e^{3}\right )} x^{4} + 2 \, {\left (6 \, b^{4} d e^{2} + 2 \, {\left (3 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p^{3} + 3 \, {\left (7 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p^{2} + {\left (21 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p\right )} x^{3} + 6 \, {\left (3 \, a b^{3} d^{3} - a^{2} b^{2} d^{2} e\right )} p^{2} + 3 \, {\left (6 \, b^{4} d^{2} e + 4 \, {\left (b^{4} d^{2} e + a b^{3} d e^{2}\right )} p^{3} + 2 \, {\left (8 \, b^{4} d^{2} e + 5 \, a b^{3} d e^{2} - a^{2} b^{2} e^{3}\right )} p^{2} + {\left (19 \, b^{4} d^{2} e + 4 \, a b^{3} d e^{2} - a^{2} b^{2} e^{3}\right )} p\right )} x^{2} + {\left (26 \, a b^{3} d^{3} - 21 \, a^{2} b^{2} d^{2} e + 6 \, a^{3} b d e^{2}\right )} p + 2 \, {\left (6 \, b^{4} d^{3} + 2 \, {\left (b^{4} d^{3} + 3 \, a b^{3} d^{2} e\right )} p^{3} + 3 \, {\left (3 \, b^{4} d^{3} + 7 \, a b^{3} d^{2} e - 2 \, a^{2} b^{2} d e^{2}\right )} p^{2} + {\left (13 \, b^{4} d^{3} + 18 \, a b^{3} d^{2} e - 12 \, a^{2} b^{2} d e^{2} + 3 \, a^{3} b e^{3}\right )} p\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, {\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \]
1/2*(4*a*b^3*d^3*p^3 + 12*a*b^3*d^3 - 18*a^2*b^2*d^2*e + 12*a^3*b*d*e^2 - 3*a^4*e^3 + (4*b^4*e^3*p^3 + 12*b^4*e^3*p^2 + 11*b^4*e^3*p + 3*b^4*e^3)*x^ 4 + 2*(6*b^4*d*e^2 + 2*(3*b^4*d*e^2 + a*b^3*e^3)*p^3 + 3*(7*b^4*d*e^2 + a* b^3*e^3)*p^2 + (21*b^4*d*e^2 + a*b^3*e^3)*p)*x^3 + 6*(3*a*b^3*d^3 - a^2*b^ 2*d^2*e)*p^2 + 3*(6*b^4*d^2*e + 4*(b^4*d^2*e + a*b^3*d*e^2)*p^3 + 2*(8*b^4 *d^2*e + 5*a*b^3*d*e^2 - a^2*b^2*e^3)*p^2 + (19*b^4*d^2*e + 4*a*b^3*d*e^2 - a^2*b^2*e^3)*p)*x^2 + (26*a*b^3*d^3 - 21*a^2*b^2*d^2*e + 6*a^3*b*d*e^2)* p + 2*(6*b^4*d^3 + 2*(b^4*d^3 + 3*a*b^3*d^2*e)*p^3 + 3*(3*b^4*d^3 + 7*a*b^ 3*d^2*e - 2*a^2*b^2*d*e^2)*p^2 + (13*b^4*d^3 + 18*a*b^3*d^2*e - 12*a^2*b^2 *d*e^2 + 3*a^3*b*e^3)*p)*x)*(b^2*x^2 + 2*a*b*x + a^2)^p/(4*b^4*p^4 + 20*b^ 4*p^3 + 35*b^4*p^2 + 25*b^4*p + 6*b^4)
\[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\text {Too large to display} \]
Piecewise(((d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4)*(a**2)** p, Eq(b, 0)), (6*a**3*e**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18 *a*b**6*x**2 + 6*b**7*x**3) + 11*a**3*e**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 6*a**2*b*d*e**2/(6*a**3*b**4 + 18*a**2*b* *5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a**2*b*e**3*x*log(a/b + x)/(6*a* *3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 27*a**2*b*e**3* x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*a*b**2 *d**2*e/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 18 *a*b**2*d*e**2*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x **3) + 18*a*b**2*e**3*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18 *a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*e**3*x**2/(6*a**3*b**4 + 18*a**2*b **5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*b**3*d**3/(6*a**3*b**4 + 18*a**2 *b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 9*b**3*d**2*e*x/(6*a**3*b**4 + 1 8*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 18*b**3*d*e**2*x**2/(6*a** 3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3*e**3*x**3 *log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3 ), Eq(p, -2)), (Integral((d + e*x)**3/((a + b*x)**2)**(3/2), x), Eq(p, -3/ 2)), (6*a**3*e**3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3*e**3/(2*a*b* *4 + 2*b**5*x) - 12*a**2*b*d*e**2*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 12* a**2*b*d*e**2/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*e**3*x*log(a/b + x)/(2*a...
Time = 0.22 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.52 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p} d^{3}}{b {\left (2 \, p + 1\right )}} + \frac {3 \, {\left (b^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )} {\left (b x + a\right )}^{2 \, p} d^{2} e}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac {3 \, {\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} + {\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )} {\left (b x + a\right )}^{2 \, p} d e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac {{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{4} + 2 \, {\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{3} - 3 \, {\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b p x - 3 \, a^{4}\right )} {\left (b x + a\right )}^{2 \, p} e^{3}}{2 \, {\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{4}} \]
(b*x + a)*(b*x + a)^(2*p)*d^3/(b*(2*p + 1)) + 3/2*(b^2*(2*p + 1)*x^2 + 2*a *b*p*x - a^2)*(b*x + a)^(2*p)*d^2*e/((2*p^2 + 3*p + 1)*b^2) + 3*((2*p^2 + 3*p + 1)*b^3*x^3 + (2*p^2 + p)*a*b^2*x^2 - 2*a^2*b*p*x + a^3)*(b*x + a)^(2 *p)*d*e^2/((4*p^3 + 12*p^2 + 11*p + 3)*b^3) + 1/2*((4*p^3 + 12*p^2 + 11*p + 3)*b^4*x^4 + 2*(2*p^3 + 3*p^2 + p)*a*b^3*x^3 - 3*(2*p^2 + p)*a^2*b^2*x^2 + 6*a^3*b*p*x - 3*a^4)*(b*x + a)^(2*p)*e^3/((4*p^4 + 20*p^3 + 35*p^2 + 25 *p + 6)*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 1279 vs. \(2 (177) = 354\).
Time = 0.30 (sec) , antiderivative size = 1279, normalized size of antiderivative = 7.07 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\text {Too large to display} \]
1/2*(4*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*e^3*p^3*x^4 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d*e^2*p^3*x^3 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*e^3*p^3* x^3 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*e^3*p^2*x^4 + 12*(b^2*x^2 + 2*a*b *x + a^2)^p*b^4*d^2*e*p^3*x^2 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d*e^2 *p^3*x^2 + 42*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d*e^2*p^2*x^3 + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*e^3*p^2*x^3 + 11*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*e ^3*p*x^4 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^3*p^3*x + 12*(b^2*x^2 + 2*a *b*x + a^2)^p*a*b^3*d^2*e*p^3*x + 48*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^2*e *p^2*x^2 + 30*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d*e^2*p^2*x^2 - 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*e^3*p^2*x^2 + 42*(b^2*x^2 + 2*a*b*x + a^2)^p*b ^4*d*e^2*p*x^3 + 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*e^3*p*x^3 + 3*(b^2*x^ 2 + 2*a*b*x + a^2)^p*b^4*e^3*x^4 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^3 *p^3 + 18*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^3*p^2*x + 42*(b^2*x^2 + 2*a*b* x + a^2)^p*a*b^3*d^2*e*p^2*x - 12*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d*e^ 2*p^2*x + 57*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^2*e*p*x^2 + 12*(b^2*x^2 + 2 *a*b*x + a^2)^p*a*b^3*d*e^2*p*x^2 - 3*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2* e^3*p*x^2 + 12*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d*e^2*x^3 + 18*(b^2*x^2 + 2 *a*b*x + a^2)^p*a*b^3*d^3*p^2 - 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d^2* e*p^2 + 26*(b^2*x^2 + 2*a*b*x + a^2)^p*b^4*d^3*p*x + 36*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^3*d^2*e*p*x - 24*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b^2*d*e^...
Time = 10.03 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.67 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx={\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {a\,\left (-3\,a^3\,e^3+6\,a^2\,b\,d\,e^2\,p+12\,a^2\,b\,d\,e^2-6\,a\,b^2\,d^2\,e\,p^2-21\,a\,b^2\,d^2\,e\,p-18\,a\,b^2\,d^2\,e+4\,b^3\,d^3\,p^3+18\,b^3\,d^3\,p^2+26\,b^3\,d^3\,p+12\,b^3\,d^3\right )}{2\,b^4\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {e^3\,x^4\,\left (4\,p^3+12\,p^2+11\,p+3\right )}{2\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {x\,\left (6\,a^3\,b\,e^3\,p-12\,a^2\,b^2\,d\,e^2\,p^2-24\,a^2\,b^2\,d\,e^2\,p+12\,a\,b^3\,d^2\,e\,p^3+42\,a\,b^3\,d^2\,e\,p^2+36\,a\,b^3\,d^2\,e\,p+4\,b^4\,d^3\,p^3+18\,b^4\,d^3\,p^2+26\,b^4\,d^3\,p+12\,b^4\,d^3\right )}{2\,b^4\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {3\,e\,x^2\,\left (2\,p+1\right )\,\left (-a^2\,e^2\,p+2\,a\,b\,d\,e\,p^2+4\,a\,b\,d\,e\,p+2\,b^2\,d^2\,p^2+7\,b^2\,d^2\,p+6\,b^2\,d^2\right )}{2\,b^2\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {e^2\,x^3\,\left (2\,p^2+3\,p+1\right )\,\left (6\,b\,d+a\,e\,p+3\,b\,d\,p\right )}{b\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}\right ) \]
(a^2 + b^2*x^2 + 2*a*b*x)^p*((a*(12*b^3*d^3 - 3*a^3*e^3 + 26*b^3*d^3*p + 1 8*b^3*d^3*p^2 + 4*b^3*d^3*p^3 - 18*a*b^2*d^2*e + 12*a^2*b*d*e^2 - 21*a*b^2 *d^2*e*p + 6*a^2*b*d*e^2*p - 6*a*b^2*d^2*e*p^2))/(2*b^4*(25*p + 35*p^2 + 2 0*p^3 + 4*p^4 + 6)) + (e^3*x^4*(11*p + 12*p^2 + 4*p^3 + 3))/(2*(25*p + 35* p^2 + 20*p^3 + 4*p^4 + 6)) + (x*(12*b^4*d^3 + 26*b^4*d^3*p + 18*b^4*d^3*p^ 2 + 4*b^4*d^3*p^3 + 6*a^3*b*e^3*p + 36*a*b^3*d^2*e*p - 24*a^2*b^2*d*e^2*p + 42*a*b^3*d^2*e*p^2 + 12*a*b^3*d^2*e*p^3 - 12*a^2*b^2*d*e^2*p^2))/(2*b^4* (25*p + 35*p^2 + 20*p^3 + 4*p^4 + 6)) + (3*e*x^2*(2*p + 1)*(6*b^2*d^2 - a^ 2*e^2*p + 7*b^2*d^2*p + 2*b^2*d^2*p^2 + 4*a*b*d*e*p + 2*a*b*d*e*p^2))/(2*b ^2*(25*p + 35*p^2 + 20*p^3 + 4*p^4 + 6)) + (e^2*x^3*(3*p + 2*p^2 + 1)*(6*b *d + a*e*p + 3*b*d*p))/(b*(25*p + 35*p^2 + 20*p^3 + 4*p^4 + 6)))