Integrand size = 26, antiderivative size = 142 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^4 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 b (b d-a e)^3 (d+e x)^{2+m}}{e^5 (2+m)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 b^3 (b d-a e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {b^4 (d+e x)^{5+m}}{e^5 (5+m)} \]
(-a*e+b*d)^4*(e*x+d)^(1+m)/e^5/(1+m)-4*b*(-a*e+b*d)^3*(e*x+d)^(2+m)/e^5/(2 +m)+6*b^2*(-a*e+b*d)^2*(e*x+d)^(3+m)/e^5/(3+m)-4*b^3*(-a*e+b*d)*(e*x+d)^(4 +m)/e^5/(4+m)+b^4*(e*x+d)^(5+m)/e^5/(5+m)
Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (\frac {(b d-a e)^4}{1+m}-\frac {4 b (b d-a e)^3 (d+e x)}{2+m}+\frac {6 b^2 (b d-a e)^2 (d+e x)^2}{3+m}-\frac {4 b^3 (b d-a e) (d+e x)^3}{4+m}+\frac {b^4 (d+e x)^4}{5+m}\right )}{e^5} \]
((d + e*x)^(1 + m)*((b*d - a*e)^4/(1 + m) - (4*b*(b*d - a*e)^3*(d + e*x))/ (2 + m) + (6*b^2*(b*d - a*e)^2*(d + e*x)^2)/(3 + m) - (4*b^3*(b*d - a*e)*( d + e*x)^3)/(4 + m) + (b^4*(d + e*x)^4)/(5 + m)))/e^5
Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 27, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int b^4 (a+b x)^4 (d+e x)^mdx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^4 (d+e x)^mdx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {4 b^3 (b d-a e) (d+e x)^{m+3}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{m+2}}{e^4}+\frac {(a e-b d)^4 (d+e x)^m}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{m+1}}{e^4}+\frac {b^4 (d+e x)^{m+4}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}+\frac {(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac {b^4 (d+e x)^{m+5}}{e^5 (m+5)}\) |
((b*d - a*e)^4*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*b*(b*d - a*e)^3*(d + e*x)^(2 + m))/(e^5*(2 + m)) + (6*b^2*(b*d - a*e)^2*(d + e*x)^(3 + m))/(e^5 *(3 + m)) - (4*b^3*(b*d - a*e)*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (b^4*(d + e*x)^(5 + m))/(e^5*(5 + m))
3.18.32.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(734\) vs. \(2(142)=284\).
Time = 2.30 (sec) , antiderivative size = 735, normalized size of antiderivative = 5.18
| method | result | size |
| norman | \(\frac {b^{4} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {d \left (a^{4} e^{4} m^{4}+14 a^{4} e^{4} m^{3}-4 a^{3} b d \,e^{3} m^{3}+71 a^{4} e^{4} m^{2}-48 a^{3} b d \,e^{3} m^{2}+12 a^{2} b^{2} d^{2} e^{2} m^{2}+154 a^{4} e^{4} m -188 a^{3} b d \,e^{3} m +108 a^{2} b^{2} d^{2} e^{2} m -24 a \,b^{3} d^{3} e m +120 e^{4} a^{4}-240 b \,e^{3} d \,a^{3}+240 b^{2} e^{2} d^{2} a^{2}-120 a \,b^{3} d^{3} e +24 b^{4} d^{4}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {\left (a^{4} e^{4} m^{4}+4 a^{3} b d \,e^{3} m^{4}+14 a^{4} e^{4} m^{3}+48 a^{3} b d \,e^{3} m^{3}-12 a^{2} b^{2} d^{2} e^{2} m^{3}+71 a^{4} e^{4} m^{2}+188 a^{3} b d \,e^{3} m^{2}-108 a^{2} b^{2} d^{2} e^{2} m^{2}+24 a \,b^{3} d^{3} e \,m^{2}+154 a^{4} e^{4} m +240 a^{3} b d \,e^{3} m -240 a^{2} b^{2} d^{2} e^{2} m +120 a \,b^{3} d^{3} e m -24 b^{4} d^{4} m +120 e^{4} a^{4}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {b^{3} \left (4 a e m +b d m +20 a e \right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+9 m +20\right )}+\frac {2 \left (3 a^{2} e^{2} m^{2}+2 a b d e \,m^{2}+27 a^{2} e^{2} m +10 a b d e m -2 b^{2} d^{2} m +60 a^{2} e^{2}\right ) b^{2} x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {2 \left (2 a^{3} e^{3} m^{3}+3 a^{2} b d \,e^{2} m^{3}+24 a^{3} e^{3} m^{2}+27 a^{2} b d \,e^{2} m^{2}-6 a \,b^{2} d^{2} e \,m^{2}+94 a^{3} e^{3} m +60 a^{2} b d \,e^{2} m -30 a \,b^{2} d^{2} e m +6 b^{3} d^{3} m +120 a^{3} e^{3}\right ) b \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}\) | \(735\) |
| gosper | \(\frac {\left (e x +d \right )^{1+m} \left (b^{4} e^{4} m^{4} x^{4}+4 a \,b^{3} e^{4} m^{4} x^{3}+10 b^{4} e^{4} m^{3} x^{4}+6 a^{2} b^{2} e^{4} m^{4} x^{2}+44 a \,b^{3} e^{4} m^{3} x^{3}-4 b^{4} d \,e^{3} m^{3} x^{3}+35 b^{4} e^{4} m^{2} x^{4}+4 a^{3} b \,e^{4} m^{4} x +72 a^{2} b^{2} e^{4} m^{3} x^{2}-12 a \,b^{3} d \,e^{3} m^{3} x^{2}+164 a \,b^{3} e^{4} m^{2} x^{3}-24 b^{4} d \,e^{3} m^{2} x^{3}+50 b^{4} e^{4} m \,x^{4}+a^{4} e^{4} m^{4}+52 a^{3} b \,e^{4} m^{3} x -12 a^{2} b^{2} d \,e^{3} m^{3} x +294 a^{2} b^{2} e^{4} m^{2} x^{2}-96 a \,b^{3} d \,e^{3} m^{2} x^{2}+244 a \,b^{3} e^{4} m \,x^{3}+12 b^{4} d^{2} e^{2} m^{2} x^{2}-44 b^{4} d \,e^{3} m \,x^{3}+24 b^{4} x^{4} e^{4}+14 a^{4} e^{4} m^{3}-4 a^{3} b d \,e^{3} m^{3}+236 a^{3} b \,e^{4} m^{2} x -120 a^{2} b^{2} d \,e^{3} m^{2} x +468 a^{2} b^{2} e^{4} m \,x^{2}+24 a \,b^{3} d^{2} e^{2} m^{2} x -204 a \,b^{3} d \,e^{3} m \,x^{2}+120 x^{3} a \,b^{3} e^{4}+36 b^{4} d^{2} e^{2} m \,x^{2}-24 x^{3} b^{4} d \,e^{3}+71 a^{4} e^{4} m^{2}-48 a^{3} b d \,e^{3} m^{2}+428 a^{3} b \,e^{4} m x +12 a^{2} b^{2} d^{2} e^{2} m^{2}-348 a^{2} b^{2} d \,e^{3} m x +240 x^{2} a^{2} b^{2} e^{4}+144 a \,b^{3} d^{2} e^{2} m x -120 x^{2} a \,b^{3} d \,e^{3}-24 b^{4} d^{3} e m x +24 x^{2} b^{4} d^{2} e^{2}+154 a^{4} e^{4} m -188 a^{3} b d \,e^{3} m +240 x \,a^{3} b \,e^{4}+108 a^{2} b^{2} d^{2} e^{2} m -240 x \,a^{2} b^{2} d \,e^{3}-24 a \,b^{3} d^{3} e m +120 x a \,b^{3} d^{2} e^{2}-24 x \,b^{4} d^{3} e +120 e^{4} a^{4}-240 b \,e^{3} d \,a^{3}+240 b^{2} e^{2} d^{2} a^{2}-120 a \,b^{3} d^{3} e +24 b^{4} d^{4}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\) | \(768\) |
| risch | \(\text {Expression too large to display}\) | \(1029\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1582\) |
b^4/(5+m)*x^5*exp(m*ln(e*x+d))+d*(a^4*e^4*m^4+14*a^4*e^4*m^3-4*a^3*b*d*e^3 *m^3+71*a^4*e^4*m^2-48*a^3*b*d*e^3*m^2+12*a^2*b^2*d^2*e^2*m^2+154*a^4*e^4* m-188*a^3*b*d*e^3*m+108*a^2*b^2*d^2*e^2*m-24*a*b^3*d^3*e*m+120*a^4*e^4-240 *a^3*b*d*e^3+240*a^2*b^2*d^2*e^2-120*a*b^3*d^3*e+24*b^4*d^4)/e^5/(m^5+15*m ^4+85*m^3+225*m^2+274*m+120)*exp(m*ln(e*x+d))+(a^4*e^4*m^4+4*a^3*b*d*e^3*m ^4+14*a^4*e^4*m^3+48*a^3*b*d*e^3*m^3-12*a^2*b^2*d^2*e^2*m^3+71*a^4*e^4*m^2 +188*a^3*b*d*e^3*m^2-108*a^2*b^2*d^2*e^2*m^2+24*a*b^3*d^3*e*m^2+154*a^4*e^ 4*m+240*a^3*b*d*e^3*m-240*a^2*b^2*d^2*e^2*m+120*a*b^3*d^3*e*m-24*b^4*d^4*m +120*a^4*e^4)/e^4/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*x*exp(m*ln(e*x+d)) +b^3*(4*a*e*m+b*d*m+20*a*e)/e/(m^2+9*m+20)*x^4*exp(m*ln(e*x+d))+2*(3*a^2*e ^2*m^2+2*a*b*d*e*m^2+27*a^2*e^2*m+10*a*b*d*e*m-2*b^2*d^2*m+60*a^2*e^2)*b^2 /e^2/(m^3+12*m^2+47*m+60)*x^3*exp(m*ln(e*x+d))+2*(2*a^3*e^3*m^3+3*a^2*b*d* e^2*m^3+24*a^3*e^3*m^2+27*a^2*b*d*e^2*m^2-6*a*b^2*d^2*e*m^2+94*a^3*e^3*m+6 0*a^2*b*d*e^2*m-30*a*b^2*d^2*e*m+6*b^3*d^3*m+120*a^3*e^3)*b/e^3/(m^4+14*m^ 3+71*m^2+154*m+120)*x^2*exp(m*ln(e*x+d))
Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (142) = 284\).
Time = 0.30 (sec) , antiderivative size = 901, normalized size of antiderivative = 6.35 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {{\left (a^{4} d e^{4} m^{4} + 24 \, b^{4} d^{5} - 120 \, a b^{3} d^{4} e + 240 \, a^{2} b^{2} d^{3} e^{2} - 240 \, a^{3} b d^{2} e^{3} + 120 \, a^{4} d e^{4} + {\left (b^{4} e^{5} m^{4} + 10 \, b^{4} e^{5} m^{3} + 35 \, b^{4} e^{5} m^{2} + 50 \, b^{4} e^{5} m + 24 \, b^{4} e^{5}\right )} x^{5} + {\left (120 \, a b^{3} e^{5} + {\left (b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} m^{4} + 2 \, {\left (3 \, b^{4} d e^{4} + 22 \, a b^{3} e^{5}\right )} m^{3} + {\left (11 \, b^{4} d e^{4} + 164 \, a b^{3} e^{5}\right )} m^{2} + 2 \, {\left (3 \, b^{4} d e^{4} + 122 \, a b^{3} e^{5}\right )} m\right )} x^{4} - 2 \, {\left (2 \, a^{3} b d^{2} e^{3} - 7 \, a^{4} d e^{4}\right )} m^{3} + 2 \, {\left (120 \, a^{2} b^{2} e^{5} + {\left (2 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} m^{4} - 2 \, {\left (b^{4} d^{2} e^{3} - 8 \, a b^{3} d e^{4} - 18 \, a^{2} b^{2} e^{5}\right )} m^{3} - {\left (6 \, b^{4} d^{2} e^{3} - 34 \, a b^{3} d e^{4} - 147 \, a^{2} b^{2} e^{5}\right )} m^{2} - 2 \, {\left (2 \, b^{4} d^{2} e^{3} - 10 \, a b^{3} d e^{4} - 117 \, a^{2} b^{2} e^{5}\right )} m\right )} x^{3} + {\left (12 \, a^{2} b^{2} d^{3} e^{2} - 48 \, a^{3} b d^{2} e^{3} + 71 \, a^{4} d e^{4}\right )} m^{2} + 2 \, {\left (120 \, a^{3} b e^{5} + {\left (3 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} m^{4} - 2 \, {\left (3 \, a b^{3} d^{2} e^{3} - 15 \, a^{2} b^{2} d e^{4} - 13 \, a^{3} b e^{5}\right )} m^{3} + {\left (6 \, b^{4} d^{3} e^{2} - 36 \, a b^{3} d^{2} e^{3} + 87 \, a^{2} b^{2} d e^{4} + 118 \, a^{3} b e^{5}\right )} m^{2} + 2 \, {\left (3 \, b^{4} d^{3} e^{2} - 15 \, a b^{3} d^{2} e^{3} + 30 \, a^{2} b^{2} d e^{4} + 107 \, a^{3} b e^{5}\right )} m\right )} x^{2} - 2 \, {\left (12 \, a b^{3} d^{4} e - 54 \, a^{2} b^{2} d^{3} e^{2} + 94 \, a^{3} b d^{2} e^{3} - 77 \, a^{4} d e^{4}\right )} m + {\left (120 \, a^{4} e^{5} + {\left (4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} m^{4} - 2 \, {\left (6 \, a^{2} b^{2} d^{2} e^{3} - 24 \, a^{3} b d e^{4} - 7 \, a^{4} e^{5}\right )} m^{3} + {\left (24 \, a b^{3} d^{3} e^{2} - 108 \, a^{2} b^{2} d^{2} e^{3} + 188 \, a^{3} b d e^{4} + 71 \, a^{4} e^{5}\right )} m^{2} - 2 \, {\left (12 \, b^{4} d^{4} e - 60 \, a b^{3} d^{3} e^{2} + 120 \, a^{2} b^{2} d^{2} e^{3} - 120 \, a^{3} b d e^{4} - 77 \, a^{4} e^{5}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \]
(a^4*d*e^4*m^4 + 24*b^4*d^5 - 120*a*b^3*d^4*e + 240*a^2*b^2*d^3*e^2 - 240* a^3*b*d^2*e^3 + 120*a^4*d*e^4 + (b^4*e^5*m^4 + 10*b^4*e^5*m^3 + 35*b^4*e^5 *m^2 + 50*b^4*e^5*m + 24*b^4*e^5)*x^5 + (120*a*b^3*e^5 + (b^4*d*e^4 + 4*a* b^3*e^5)*m^4 + 2*(3*b^4*d*e^4 + 22*a*b^3*e^5)*m^3 + (11*b^4*d*e^4 + 164*a* b^3*e^5)*m^2 + 2*(3*b^4*d*e^4 + 122*a*b^3*e^5)*m)*x^4 - 2*(2*a^3*b*d^2*e^3 - 7*a^4*d*e^4)*m^3 + 2*(120*a^2*b^2*e^5 + (2*a*b^3*d*e^4 + 3*a^2*b^2*e^5) *m^4 - 2*(b^4*d^2*e^3 - 8*a*b^3*d*e^4 - 18*a^2*b^2*e^5)*m^3 - (6*b^4*d^2*e ^3 - 34*a*b^3*d*e^4 - 147*a^2*b^2*e^5)*m^2 - 2*(2*b^4*d^2*e^3 - 10*a*b^3*d *e^4 - 117*a^2*b^2*e^5)*m)*x^3 + (12*a^2*b^2*d^3*e^2 - 48*a^3*b*d^2*e^3 + 71*a^4*d*e^4)*m^2 + 2*(120*a^3*b*e^5 + (3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*m^4 - 2*(3*a*b^3*d^2*e^3 - 15*a^2*b^2*d*e^4 - 13*a^3*b*e^5)*m^3 + (6*b^4*d^3* e^2 - 36*a*b^3*d^2*e^3 + 87*a^2*b^2*d*e^4 + 118*a^3*b*e^5)*m^2 + 2*(3*b^4* d^3*e^2 - 15*a*b^3*d^2*e^3 + 30*a^2*b^2*d*e^4 + 107*a^3*b*e^5)*m)*x^2 - 2* (12*a*b^3*d^4*e - 54*a^2*b^2*d^3*e^2 + 94*a^3*b*d^2*e^3 - 77*a^4*d*e^4)*m + (120*a^4*e^5 + (4*a^3*b*d*e^4 + a^4*e^5)*m^4 - 2*(6*a^2*b^2*d^2*e^3 - 24 *a^3*b*d*e^4 - 7*a^4*e^5)*m^3 + (24*a*b^3*d^3*e^2 - 108*a^2*b^2*d^2*e^3 + 188*a^3*b*d*e^4 + 71*a^4*e^5)*m^2 - 2*(12*b^4*d^4*e - 60*a*b^3*d^3*e^2 + 1 20*a^2*b^2*d^2*e^3 - 120*a^3*b*d*e^4 - 77*a^4*e^5)*m)*x)*(e*x + d)^m/(e^5* m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e^5*m^2 + 274*e^5*m + 120*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 8719 vs. \(2 (124) = 248\).
Time = 1.91 (sec) , antiderivative size = 8719, normalized size of antiderivative = 61.40 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
Piecewise((d**m*(a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2*x**3 + a*b**3*x**4 + b**4*x**5/5), Eq(e, 0)), (-3*a**4*e**4/(12*d**4*e**5 + 48*d**3*e**6*x + 7 2*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*a**3*b*d*e**3/(12*d* *4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x* *4) - 16*a**3*b*e**4*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*a**2*b**2*d**2*e**2/(12*d**4*e**5 + 4 8*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 24*a* *2*b**2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d *e**8*x**3 + 12*e**9*x**4) - 36*a**2*b**2*e**4*x**2/(12*d**4*e**5 + 48*d** 3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12*a*b**3* d**3*e/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 48*a*b**3*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 72*a*b**3*d*e**3*x**2 /(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12* e**9*x**4) - 48*a*b**3*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2* e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*b**4*d**4*log(d/e + x)/(12 *d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9 *x**4) + 25*b**4*d**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*b**4*d**3*e*x*log(d/e + x)/(12*d**4*e **5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**...
Leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (142) = 284\).
Time = 0.22 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.76 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {4 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{3} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{4}}{e {\left (m + 1\right )}} + \frac {6 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a^{2} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {4 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} a b^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} b^{4}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \]
4*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^3*b/((m^2 + 3*m + 2)*e^2 ) + (e*x + d)^(m + 1)*a^4/(e*(m + 1)) + 6*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a^2*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 4*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m) *d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*a* b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m ^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)* (e*x + d)^m*b^4/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 1528 vs. \(2 (142) = 284\).
Time = 0.28 (sec) , antiderivative size = 1528, normalized size of antiderivative = 10.76 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
((e*x + d)^m*b^4*e^5*m^4*x^5 + (e*x + d)^m*b^4*d*e^4*m^4*x^4 + 4*(e*x + d) ^m*a*b^3*e^5*m^4*x^4 + 10*(e*x + d)^m*b^4*e^5*m^3*x^5 + 4*(e*x + d)^m*a*b^ 3*d*e^4*m^4*x^3 + 6*(e*x + d)^m*a^2*b^2*e^5*m^4*x^3 + 6*(e*x + d)^m*b^4*d* e^4*m^3*x^4 + 44*(e*x + d)^m*a*b^3*e^5*m^3*x^4 + 35*(e*x + d)^m*b^4*e^5*m^ 2*x^5 + 6*(e*x + d)^m*a^2*b^2*d*e^4*m^4*x^2 + 4*(e*x + d)^m*a^3*b*e^5*m^4* x^2 - 4*(e*x + d)^m*b^4*d^2*e^3*m^3*x^3 + 32*(e*x + d)^m*a*b^3*d*e^4*m^3*x ^3 + 72*(e*x + d)^m*a^2*b^2*e^5*m^3*x^3 + 11*(e*x + d)^m*b^4*d*e^4*m^2*x^4 + 164*(e*x + d)^m*a*b^3*e^5*m^2*x^4 + 50*(e*x + d)^m*b^4*e^5*m*x^5 + 4*(e *x + d)^m*a^3*b*d*e^4*m^4*x + (e*x + d)^m*a^4*e^5*m^4*x - 12*(e*x + d)^m*a *b^3*d^2*e^3*m^3*x^2 + 60*(e*x + d)^m*a^2*b^2*d*e^4*m^3*x^2 + 52*(e*x + d) ^m*a^3*b*e^5*m^3*x^2 - 12*(e*x + d)^m*b^4*d^2*e^3*m^2*x^3 + 68*(e*x + d)^m *a*b^3*d*e^4*m^2*x^3 + 294*(e*x + d)^m*a^2*b^2*e^5*m^2*x^3 + 6*(e*x + d)^m *b^4*d*e^4*m*x^4 + 244*(e*x + d)^m*a*b^3*e^5*m*x^4 + 24*(e*x + d)^m*b^4*e^ 5*x^5 + (e*x + d)^m*a^4*d*e^4*m^4 - 12*(e*x + d)^m*a^2*b^2*d^2*e^3*m^3*x + 48*(e*x + d)^m*a^3*b*d*e^4*m^3*x + 14*(e*x + d)^m*a^4*e^5*m^3*x + 12*(e*x + d)^m*b^4*d^3*e^2*m^2*x^2 - 72*(e*x + d)^m*a*b^3*d^2*e^3*m^2*x^2 + 174*( e*x + d)^m*a^2*b^2*d*e^4*m^2*x^2 + 236*(e*x + d)^m*a^3*b*e^5*m^2*x^2 - 8*( e*x + d)^m*b^4*d^2*e^3*m*x^3 + 40*(e*x + d)^m*a*b^3*d*e^4*m*x^3 + 468*(e*x + d)^m*a^2*b^2*e^5*m*x^3 + 120*(e*x + d)^m*a*b^3*e^5*x^4 - 4*(e*x + d)^m* a^3*b*d^2*e^3*m^3 + 14*(e*x + d)^m*a^4*d*e^4*m^3 + 24*(e*x + d)^m*a*b^3...
Time = 9.93 (sec) , antiderivative size = 831, normalized size of antiderivative = 5.85 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (a^4\,d\,e^4\,m^4+14\,a^4\,d\,e^4\,m^3+71\,a^4\,d\,e^4\,m^2+154\,a^4\,d\,e^4\,m+120\,a^4\,d\,e^4-4\,a^3\,b\,d^2\,e^3\,m^3-48\,a^3\,b\,d^2\,e^3\,m^2-188\,a^3\,b\,d^2\,e^3\,m-240\,a^3\,b\,d^2\,e^3+12\,a^2\,b^2\,d^3\,e^2\,m^2+108\,a^2\,b^2\,d^3\,e^2\,m+240\,a^2\,b^2\,d^3\,e^2-24\,a\,b^3\,d^4\,e\,m-120\,a\,b^3\,d^4\,e+24\,b^4\,d^5\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (a^4\,e^5\,m^4+14\,a^4\,e^5\,m^3+71\,a^4\,e^5\,m^2+154\,a^4\,e^5\,m+120\,a^4\,e^5+4\,a^3\,b\,d\,e^4\,m^4+48\,a^3\,b\,d\,e^4\,m^3+188\,a^3\,b\,d\,e^4\,m^2+240\,a^3\,b\,d\,e^4\,m-12\,a^2\,b^2\,d^2\,e^3\,m^3-108\,a^2\,b^2\,d^2\,e^3\,m^2-240\,a^2\,b^2\,d^2\,e^3\,m+24\,a\,b^3\,d^3\,e^2\,m^2+120\,a\,b^3\,d^3\,e^2\,m-24\,b^4\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {b^4\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,b^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (3\,a^2\,e^2\,m^2+27\,a^2\,e^2\,m+60\,a^2\,e^2+2\,a\,b\,d\,e\,m^2+10\,a\,b\,d\,e\,m-2\,b^2\,d^2\,m\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {b^3\,x^4\,{\left (d+e\,x\right )}^m\,\left (20\,a\,e+4\,a\,e\,m+b\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,b\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (2\,a^3\,e^3\,m^3+24\,a^3\,e^3\,m^2+94\,a^3\,e^3\,m+120\,a^3\,e^3+3\,a^2\,b\,d\,e^2\,m^3+27\,a^2\,b\,d\,e^2\,m^2+60\,a^2\,b\,d\,e^2\,m-6\,a\,b^2\,d^2\,e\,m^2-30\,a\,b^2\,d^2\,e\,m+6\,b^3\,d^3\,m\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \]
((d + e*x)^m*(24*b^4*d^5 + 120*a^4*d*e^4 - 240*a^3*b*d^2*e^3 + 71*a^4*d*e^ 4*m^2 + 14*a^4*d*e^4*m^3 + a^4*d*e^4*m^4 + 240*a^2*b^2*d^3*e^2 - 120*a*b^3 *d^4*e + 154*a^4*d*e^4*m - 24*a*b^3*d^4*e*m + 12*a^2*b^2*d^3*e^2*m^2 - 188 *a^3*b*d^2*e^3*m + 108*a^2*b^2*d^3*e^2*m - 48*a^3*b*d^2*e^3*m^2 - 4*a^3*b* d^2*e^3*m^3))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*( d + e*x)^m*(120*a^4*e^5 + 154*a^4*e^5*m + 71*a^4*e^5*m^2 + 14*a^4*e^5*m^3 + a^4*e^5*m^4 - 24*b^4*d^4*e*m + 240*a^3*b*d*e^4*m - 108*a^2*b^2*d^2*e^3*m ^2 - 12*a^2*b^2*d^2*e^3*m^3 + 120*a*b^3*d^3*e^2*m + 188*a^3*b*d*e^4*m^2 + 48*a^3*b*d*e^4*m^3 + 4*a^3*b*d*e^4*m^4 - 240*a^2*b^2*d^2*e^3*m + 24*a*b^3* d^3*e^2*m^2))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (b^4 *x^5*(d + e*x)^m*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 8 5*m^3 + 15*m^4 + m^5 + 120) + (2*b^2*x^3*(d + e*x)^m*(3*m + m^2 + 2)*(60*a ^2*e^2 + 27*a^2*e^2*m - 2*b^2*d^2*m + 3*a^2*e^2*m^2 + 10*a*b*d*e*m + 2*a*b *d*e*m^2))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (b^3*x^ 4*(d + e*x)^m*(20*a*e + 4*a*e*m + b*d*m)*(11*m + 6*m^2 + m^3 + 6))/(e*(274 *m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (2*b*x^2*(m + 1)*(d + e*x)^ m*(120*a^3*e^3 + 94*a^3*e^3*m + 6*b^3*d^3*m + 24*a^3*e^3*m^2 + 2*a^3*e^3*m ^3 - 30*a*b^2*d^2*e*m + 60*a^2*b*d*e^2*m - 6*a*b^2*d^2*e*m^2 + 27*a^2*b*d* e^2*m^2 + 3*a^2*b*d*e^2*m^3))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^ 5 + 120))