Integrand size = 16, antiderivative size = 125 \[ \int x^m (a+b x)^4 (A+B x) \, dx=\frac {a^4 A x^{1+m}}{1+m}+\frac {a^3 (4 A b+a B) x^{2+m}}{2+m}+\frac {2 a^2 b (3 A b+2 a B) x^{3+m}}{3+m}+\frac {2 a b^2 (2 A b+3 a B) x^{4+m}}{4+m}+\frac {b^3 (A b+4 a B) x^{5+m}}{5+m}+\frac {b^4 B x^{6+m}}{6+m} \]
a^4*A*x^(1+m)/(1+m)+a^3*(4*A*b+B*a)*x^(2+m)/(2+m)+2*a^2*b*(3*A*b+2*B*a)*x^ (3+m)/(3+m)+2*a*b^2*(2*A*b+3*B*a)*x^(4+m)/(4+m)+b^3*(A*b+4*B*a)*x^(5+m)/(5 +m)+b^4*B*x^(6+m)/(6+m)
Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.82 \[ \int x^m (a+b x)^4 (A+B x) \, dx=\frac {x^{1+m} \left (B (a+b x)^5+(-a B (1+m)+A b (6+m)) \left (\frac {a^4}{1+m}+\frac {4 a^3 b x}{2+m}+\frac {6 a^2 b^2 x^2}{3+m}+\frac {4 a b^3 x^3}{4+m}+\frac {b^4 x^4}{5+m}\right )\right )}{b (6+m)} \]
(x^(1 + m)*(B*(a + b*x)^5 + (-(a*B*(1 + m)) + A*b*(6 + m))*(a^4/(1 + m) + (4*a^3*b*x)/(2 + m) + (6*a^2*b^2*x^2)/(3 + m) + (4*a*b^3*x^3)/(4 + m) + (b ^4*x^4)/(5 + m))))/(b*(6 + m))
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {85, 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.
| \(\displaystyle \int x^m (a+b x)^4 (A+B x) \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (a^4 A x^m+a^3 x^{m+1} (a B+4 A b)+2 a^2 b x^{m+2} (2 a B+3 A b)+b^3 x^{m+4} (4 a B+A b)+2 a b^2 x^{m+3} (3 a B+2 A b)+b^4 B x^{m+5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^4 A x^{m+1}}{m+1}+\frac {a^3 x^{m+2} (a B+4 A b)}{m+2}+\frac {2 a^2 b x^{m+3} (2 a B+3 A b)}{m+3}+\frac {b^3 x^{m+5} (4 a B+A b)}{m+5}+\frac {2 a b^2 x^{m+4} (3 a B+2 A b)}{m+4}+\frac {b^4 B x^{m+6}}{m+6}\) |
(a^4*A*x^(1 + m))/(1 + m) + (a^3*(4*A*b + a*B)*x^(2 + m))/(2 + m) + (2*a^2 *b*(3*A*b + 2*a*B)*x^(3 + m))/(3 + m) + (2*a*b^2*(2*A*b + 3*a*B)*x^(4 + m) )/(4 + m) + (b^3*(A*b + 4*a*B)*x^(5 + m))/(5 + m) + (b^4*B*x^(6 + m))/(6 + m)
3.4.71.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Time = 1.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14
| method | result | size |
| norman | \(\frac {A \,a^{4} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {B \,b^{4} x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}+\frac {a^{3} \left (4 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {b^{3} \left (A b +4 B a \right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {2 a \,b^{2} \left (2 A b +3 B a \right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {2 a^{2} b \left (3 A b +2 B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}\) | \(142\) |
| risch | \(\frac {x \left (B \,b^{4} m^{5} x^{5}+A \,b^{4} m^{5} x^{4}+4 B a \,b^{3} m^{5} x^{4}+15 B \,b^{4} m^{4} x^{5}+4 A a \,b^{3} m^{5} x^{3}+16 A \,b^{4} m^{4} x^{4}+6 B \,a^{2} b^{2} m^{5} x^{3}+64 B a \,b^{3} m^{4} x^{4}+85 B \,b^{4} m^{3} x^{5}+6 A \,a^{2} b^{2} m^{5} x^{2}+68 A a \,b^{3} m^{4} x^{3}+95 A \,b^{4} m^{3} x^{4}+4 B \,a^{3} b \,m^{5} x^{2}+102 B \,a^{2} b^{2} m^{4} x^{3}+380 B a \,b^{3} m^{3} x^{4}+225 B \,b^{4} m^{2} x^{5}+4 A \,a^{3} b \,m^{5} x +108 A \,a^{2} b^{2} m^{4} x^{2}+428 A a \,b^{3} m^{3} x^{3}+260 A \,b^{4} m^{2} x^{4}+B \,a^{4} m^{5} x +72 B \,a^{3} b \,m^{4} x^{2}+642 B \,a^{2} b^{2} m^{3} x^{3}+1040 B a \,b^{3} m^{2} x^{4}+274 m \,x^{5} B \,b^{4}+A \,a^{4} m^{5}+76 A \,a^{3} b \,m^{4} x +726 A \,a^{2} b^{2} m^{3} x^{2}+1228 A a \,b^{3} m^{2} x^{3}+324 A \,b^{4} x^{4} m +19 B \,a^{4} m^{4} x +484 B \,a^{3} b \,m^{3} x^{2}+1842 B \,a^{2} b^{2} m^{2} x^{3}+1296 B a \,b^{3} x^{4} m +120 B \,b^{4} x^{5}+20 A \,a^{4} m^{4}+548 A \,a^{3} b \,m^{3} x +2232 A \,a^{2} b^{2} m^{2} x^{2}+1584 A a \,b^{3} x^{3} m +144 A \,b^{4} x^{4}+137 B \,a^{4} m^{3} x +1488 B \,a^{3} b \,m^{2} x^{2}+2376 B \,a^{2} b^{2} x^{3} m +576 B a \,b^{3} x^{4}+155 A \,a^{4} m^{3}+1844 A \,a^{3} b \,m^{2} x +3048 A \,a^{2} b^{2} x^{2} m +720 A a \,b^{3} x^{3}+461 B \,a^{4} m^{2} x +2032 B \,a^{3} b \,x^{2} m +1080 B \,a^{2} b^{2} x^{3}+580 A \,a^{4} m^{2}+2808 A \,a^{3} b x m +1440 A \,a^{2} b^{2} x^{2}+702 B \,a^{4} x m +960 B \,a^{3} b \,x^{2}+1044 A \,a^{4} m +1440 A \,a^{3} b x +360 B \,a^{4} x +720 A \,a^{4}\right ) x^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(721\) |
| gosper | \(\frac {x^{1+m} \left (B \,b^{4} m^{5} x^{5}+A \,b^{4} m^{5} x^{4}+4 B a \,b^{3} m^{5} x^{4}+15 B \,b^{4} m^{4} x^{5}+4 A a \,b^{3} m^{5} x^{3}+16 A \,b^{4} m^{4} x^{4}+6 B \,a^{2} b^{2} m^{5} x^{3}+64 B a \,b^{3} m^{4} x^{4}+85 B \,b^{4} m^{3} x^{5}+6 A \,a^{2} b^{2} m^{5} x^{2}+68 A a \,b^{3} m^{4} x^{3}+95 A \,b^{4} m^{3} x^{4}+4 B \,a^{3} b \,m^{5} x^{2}+102 B \,a^{2} b^{2} m^{4} x^{3}+380 B a \,b^{3} m^{3} x^{4}+225 B \,b^{4} m^{2} x^{5}+4 A \,a^{3} b \,m^{5} x +108 A \,a^{2} b^{2} m^{4} x^{2}+428 A a \,b^{3} m^{3} x^{3}+260 A \,b^{4} m^{2} x^{4}+B \,a^{4} m^{5} x +72 B \,a^{3} b \,m^{4} x^{2}+642 B \,a^{2} b^{2} m^{3} x^{3}+1040 B a \,b^{3} m^{2} x^{4}+274 m \,x^{5} B \,b^{4}+A \,a^{4} m^{5}+76 A \,a^{3} b \,m^{4} x +726 A \,a^{2} b^{2} m^{3} x^{2}+1228 A a \,b^{3} m^{2} x^{3}+324 A \,b^{4} x^{4} m +19 B \,a^{4} m^{4} x +484 B \,a^{3} b \,m^{3} x^{2}+1842 B \,a^{2} b^{2} m^{2} x^{3}+1296 B a \,b^{3} x^{4} m +120 B \,b^{4} x^{5}+20 A \,a^{4} m^{4}+548 A \,a^{3} b \,m^{3} x +2232 A \,a^{2} b^{2} m^{2} x^{2}+1584 A a \,b^{3} x^{3} m +144 A \,b^{4} x^{4}+137 B \,a^{4} m^{3} x +1488 B \,a^{3} b \,m^{2} x^{2}+2376 B \,a^{2} b^{2} x^{3} m +576 B a \,b^{3} x^{4}+155 A \,a^{4} m^{3}+1844 A \,a^{3} b \,m^{2} x +3048 A \,a^{2} b^{2} x^{2} m +720 A a \,b^{3} x^{3}+461 B \,a^{4} m^{2} x +2032 B \,a^{3} b \,x^{2} m +1080 B \,a^{2} b^{2} x^{3}+580 A \,a^{4} m^{2}+2808 A \,a^{3} b x m +1440 A \,a^{2} b^{2} x^{2}+702 B \,a^{4} x m +960 B \,a^{3} b \,x^{2}+1044 A \,a^{4} m +1440 A \,a^{3} b x +360 B \,a^{4} x +720 A \,a^{4}\right )}{\left (1+m \right ) \left (2+m \right ) \left (3+m \right ) \left (4+m \right ) \left (5+m \right ) \left (6+m \right )}\) | \(722\) |
| parallelrisch | \(\frac {120 B \,x^{6} x^{m} b^{4}+144 A \,x^{5} x^{m} b^{4}+360 B \,x^{2} x^{m} a^{4}+720 A x \,x^{m} a^{4}+1296 B \,x^{5} x^{m} a \,b^{3} m +1842 B \,x^{4} x^{m} a^{2} b^{2} m^{2}+484 B \,x^{3} x^{m} a^{3} b \,m^{3}+1584 A \,x^{4} x^{m} a \,b^{3} m +2232 A \,x^{3} x^{m} a^{2} b^{2} m^{2}+548 A \,x^{2} x^{m} a^{3} b \,m^{3}+2376 B \,x^{4} x^{m} a^{2} b^{2} m +1488 B \,x^{3} x^{m} a^{3} b \,m^{2}+726 A \,x^{3} x^{m} a^{2} b^{2} m^{3}+76 A \,x^{2} x^{m} a^{3} b \,m^{4}+102 B \,x^{4} x^{m} a^{2} b^{2} m^{4}+4 B \,x^{3} x^{m} a^{3} b \,m^{5}+428 A \,x^{4} x^{m} a \,b^{3} m^{3}+108 A \,x^{3} x^{m} a^{2} b^{2} m^{4}+4 A \,x^{2} x^{m} a^{3} b \,m^{5}+1040 B \,x^{5} x^{m} a \,b^{3} m^{2}+642 B \,x^{4} x^{m} a^{2} b^{2} m^{3}+72 B \,x^{3} x^{m} a^{3} b \,m^{4}+1228 A \,x^{4} x^{m} a \,b^{3} m^{2}+6 B \,x^{4} x^{m} a^{2} b^{2} m^{5}+68 A \,x^{4} x^{m} a \,b^{3} m^{4}+6 A \,x^{3} x^{m} a^{2} b^{2} m^{5}+380 B \,x^{5} x^{m} a \,b^{3} m^{3}+64 B \,x^{5} x^{m} a \,b^{3} m^{4}+4 B \,x^{5} x^{m} a \,b^{3} m^{5}+4 A \,x^{4} x^{m} a \,b^{3} m^{5}+B \,x^{6} x^{m} b^{4} m^{5}+A \,x^{5} x^{m} b^{4} m^{5}+15 B \,x^{6} x^{m} b^{4} m^{4}+16 A \,x^{5} x^{m} b^{4} m^{4}+85 B \,x^{6} x^{m} b^{4} m^{3}+95 A \,x^{5} x^{m} b^{4} m^{3}+225 B \,x^{6} x^{m} b^{4} m^{2}+260 A \,x^{5} x^{m} b^{4} m^{2}+274 B \,x^{6} x^{m} b^{4} m +B \,x^{2} x^{m} a^{4} m^{5}+324 A \,x^{5} x^{m} b^{4} m +A x \,x^{m} a^{4} m^{5}+19 B \,x^{2} x^{m} a^{4} m^{4}+20 A x \,x^{m} a^{4} m^{4}+576 B \,x^{5} x^{m} a \,b^{3}+137 B \,x^{2} x^{m} a^{4} m^{3}+720 A \,x^{4} x^{m} a \,b^{3}+155 A x \,x^{m} a^{4} m^{3}+1080 B \,x^{4} x^{m} a^{2} b^{2}+461 B \,x^{2} x^{m} a^{4} m^{2}+3048 A \,x^{3} x^{m} a^{2} b^{2} m +1844 A \,x^{2} x^{m} a^{3} b \,m^{2}+2032 B \,x^{3} x^{m} a^{3} b m +2808 A \,x^{2} x^{m} a^{3} b m +1440 A \,x^{3} x^{m} a^{2} b^{2}+580 A x \,x^{m} a^{4} m^{2}+960 B \,x^{3} x^{m} a^{3} b +702 B \,x^{2} x^{m} a^{4} m +1440 A \,x^{2} x^{m} a^{3} b +1044 A x \,x^{m} a^{4} m}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(927\) |
A*a^4/(1+m)*x*exp(m*ln(x))+B*b^4/(6+m)*x^6*exp(m*ln(x))+a^3*(4*A*b+B*a)/(2 +m)*x^2*exp(m*ln(x))+b^3*(A*b+4*B*a)/(5+m)*x^5*exp(m*ln(x))+2*a*b^2*(2*A*b +3*B*a)/(4+m)*x^4*exp(m*ln(x))+2*a^2*b*(3*A*b+2*B*a)/(3+m)*x^3*exp(m*ln(x) )
Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (125) = 250\).
Time = 0.24 (sec) , antiderivative size = 607, normalized size of antiderivative = 4.86 \[ \int x^m (a+b x)^4 (A+B x) \, dx=\frac {{\left ({\left (B b^{4} m^{5} + 15 \, B b^{4} m^{4} + 85 \, B b^{4} m^{3} + 225 \, B b^{4} m^{2} + 274 \, B b^{4} m + 120 \, B b^{4}\right )} x^{6} + {\left ({\left (4 \, B a b^{3} + A b^{4}\right )} m^{5} + 576 \, B a b^{3} + 144 \, A b^{4} + 16 \, {\left (4 \, B a b^{3} + A b^{4}\right )} m^{4} + 95 \, {\left (4 \, B a b^{3} + A b^{4}\right )} m^{3} + 260 \, {\left (4 \, B a b^{3} + A b^{4}\right )} m^{2} + 324 \, {\left (4 \, B a b^{3} + A b^{4}\right )} m\right )} x^{5} + 2 \, {\left ({\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m^{5} + 540 \, B a^{2} b^{2} + 360 \, A a b^{3} + 17 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m^{4} + 107 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m^{3} + 307 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m^{2} + 396 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m\right )} x^{4} + 2 \, {\left ({\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m^{5} + 480 \, B a^{3} b + 720 \, A a^{2} b^{2} + 18 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m^{4} + 121 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m^{3} + 372 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m^{2} + 508 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{4} + 4 \, A a^{3} b\right )} m^{5} + 360 \, B a^{4} + 1440 \, A a^{3} b + 19 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} m^{4} + 137 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} m^{3} + 461 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} m^{2} + 702 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} m\right )} x^{2} + {\left (A a^{4} m^{5} + 20 \, A a^{4} m^{4} + 155 \, A a^{4} m^{3} + 580 \, A a^{4} m^{2} + 1044 \, A a^{4} m + 720 \, A a^{4}\right )} x\right )} x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
((B*b^4*m^5 + 15*B*b^4*m^4 + 85*B*b^4*m^3 + 225*B*b^4*m^2 + 274*B*b^4*m + 120*B*b^4)*x^6 + ((4*B*a*b^3 + A*b^4)*m^5 + 576*B*a*b^3 + 144*A*b^4 + 16*( 4*B*a*b^3 + A*b^4)*m^4 + 95*(4*B*a*b^3 + A*b^4)*m^3 + 260*(4*B*a*b^3 + A*b ^4)*m^2 + 324*(4*B*a*b^3 + A*b^4)*m)*x^5 + 2*((3*B*a^2*b^2 + 2*A*a*b^3)*m^ 5 + 540*B*a^2*b^2 + 360*A*a*b^3 + 17*(3*B*a^2*b^2 + 2*A*a*b^3)*m^4 + 107*( 3*B*a^2*b^2 + 2*A*a*b^3)*m^3 + 307*(3*B*a^2*b^2 + 2*A*a*b^3)*m^2 + 396*(3* B*a^2*b^2 + 2*A*a*b^3)*m)*x^4 + 2*((2*B*a^3*b + 3*A*a^2*b^2)*m^5 + 480*B*a ^3*b + 720*A*a^2*b^2 + 18*(2*B*a^3*b + 3*A*a^2*b^2)*m^4 + 121*(2*B*a^3*b + 3*A*a^2*b^2)*m^3 + 372*(2*B*a^3*b + 3*A*a^2*b^2)*m^2 + 508*(2*B*a^3*b + 3 *A*a^2*b^2)*m)*x^3 + ((B*a^4 + 4*A*a^3*b)*m^5 + 360*B*a^4 + 1440*A*a^3*b + 19*(B*a^4 + 4*A*a^3*b)*m^4 + 137*(B*a^4 + 4*A*a^3*b)*m^3 + 461*(B*a^4 + 4 *A*a^3*b)*m^2 + 702*(B*a^4 + 4*A*a^3*b)*m)*x^2 + (A*a^4*m^5 + 20*A*a^4*m^4 + 155*A*a^4*m^3 + 580*A*a^4*m^2 + 1044*A*a^4*m + 720*A*a^4)*x)*x^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)
Leaf count of result is larger than twice the leaf count of optimal. 3417 vs. \(2 (117) = 234\).
Time = 0.55 (sec) , antiderivative size = 3417, normalized size of antiderivative = 27.34 \[ \int x^m (a+b x)^4 (A+B x) \, dx=\text {Too large to display} \]
Piecewise((-A*a**4/(5*x**5) - A*a**3*b/x**4 - 2*A*a**2*b**2/x**3 - 2*A*a*b **3/x**2 - A*b**4/x - B*a**4/(4*x**4) - 4*B*a**3*b/(3*x**3) - 3*B*a**2*b** 2/x**2 - 4*B*a*b**3/x + B*b**4*log(x), Eq(m, -6)), (-A*a**4/(4*x**4) - 4*A *a**3*b/(3*x**3) - 3*A*a**2*b**2/x**2 - 4*A*a*b**3/x + A*b**4*log(x) - B*a **4/(3*x**3) - 2*B*a**3*b/x**2 - 6*B*a**2*b**2/x + 4*B*a*b**3*log(x) + B*b **4*x, Eq(m, -5)), (-A*a**4/(3*x**3) - 2*A*a**3*b/x**2 - 6*A*a**2*b**2/x + 4*A*a*b**3*log(x) + A*b**4*x - B*a**4/(2*x**2) - 4*B*a**3*b/x + 6*B*a**2* b**2*log(x) + 4*B*a*b**3*x + B*b**4*x**2/2, Eq(m, -4)), (-A*a**4/(2*x**2) - 4*A*a**3*b/x + 6*A*a**2*b**2*log(x) + 4*A*a*b**3*x + A*b**4*x**2/2 - B*a **4/x + 4*B*a**3*b*log(x) + 6*B*a**2*b**2*x + 2*B*a*b**3*x**2 + B*b**4*x** 3/3, Eq(m, -3)), (-A*a**4/x + 4*A*a**3*b*log(x) + 6*A*a**2*b**2*x + 2*A*a* b**3*x**2 + A*b**4*x**3/3 + B*a**4*log(x) + 4*B*a**3*b*x + 3*B*a**2*b**2*x **2 + 4*B*a*b**3*x**3/3 + B*b**4*x**4/4, Eq(m, -2)), (A*a**4*log(x) + 4*A* a**3*b*x + 3*A*a**2*b**2*x**2 + 4*A*a*b**3*x**3/3 + A*b**4*x**4/4 + B*a**4 *x + 2*B*a**3*b*x**2 + 2*B*a**2*b**2*x**3 + B*a*b**3*x**4 + B*b**4*x**5/5, Eq(m, -1)), (A*a**4*m**5*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1 624*m**2 + 1764*m + 720) + 20*A*a**4*m**4*x*x**m/(m**6 + 21*m**5 + 175*m** 4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 155*A*a**4*m**3*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*A*a**4*m** 2*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 7...
Time = 0.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.34 \[ \int x^m (a+b x)^4 (A+B x) \, dx=\frac {B b^{4} x^{m + 6}}{m + 6} + \frac {4 \, B a b^{3} x^{m + 5}}{m + 5} + \frac {A b^{4} x^{m + 5}}{m + 5} + \frac {6 \, B a^{2} b^{2} x^{m + 4}}{m + 4} + \frac {4 \, A a b^{3} x^{m + 4}}{m + 4} + \frac {4 \, B a^{3} b x^{m + 3}}{m + 3} + \frac {6 \, A a^{2} b^{2} x^{m + 3}}{m + 3} + \frac {B a^{4} x^{m + 2}}{m + 2} + \frac {4 \, A a^{3} b x^{m + 2}}{m + 2} + \frac {A a^{4} x^{m + 1}}{m + 1} \]
B*b^4*x^(m + 6)/(m + 6) + 4*B*a*b^3*x^(m + 5)/(m + 5) + A*b^4*x^(m + 5)/(m + 5) + 6*B*a^2*b^2*x^(m + 4)/(m + 4) + 4*A*a*b^3*x^(m + 4)/(m + 4) + 4*B* a^3*b*x^(m + 3)/(m + 3) + 6*A*a^2*b^2*x^(m + 3)/(m + 3) + B*a^4*x^(m + 2)/ (m + 2) + 4*A*a^3*b*x^(m + 2)/(m + 2) + A*a^4*x^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 926 vs. \(2 (125) = 250\).
Time = 0.30 (sec) , antiderivative size = 926, normalized size of antiderivative = 7.41 \[ \int x^m (a+b x)^4 (A+B x) \, dx=\frac {B b^{4} m^{5} x^{6} x^{m} + 4 \, B a b^{3} m^{5} x^{5} x^{m} + A b^{4} m^{5} x^{5} x^{m} + 15 \, B b^{4} m^{4} x^{6} x^{m} + 6 \, B a^{2} b^{2} m^{5} x^{4} x^{m} + 4 \, A a b^{3} m^{5} x^{4} x^{m} + 64 \, B a b^{3} m^{4} x^{5} x^{m} + 16 \, A b^{4} m^{4} x^{5} x^{m} + 85 \, B b^{4} m^{3} x^{6} x^{m} + 4 \, B a^{3} b m^{5} x^{3} x^{m} + 6 \, A a^{2} b^{2} m^{5} x^{3} x^{m} + 102 \, B a^{2} b^{2} m^{4} x^{4} x^{m} + 68 \, A a b^{3} m^{4} x^{4} x^{m} + 380 \, B a b^{3} m^{3} x^{5} x^{m} + 95 \, A b^{4} m^{3} x^{5} x^{m} + 225 \, B b^{4} m^{2} x^{6} x^{m} + B a^{4} m^{5} x^{2} x^{m} + 4 \, A a^{3} b m^{5} x^{2} x^{m} + 72 \, B a^{3} b m^{4} x^{3} x^{m} + 108 \, A a^{2} b^{2} m^{4} x^{3} x^{m} + 642 \, B a^{2} b^{2} m^{3} x^{4} x^{m} + 428 \, A a b^{3} m^{3} x^{4} x^{m} + 1040 \, B a b^{3} m^{2} x^{5} x^{m} + 260 \, A b^{4} m^{2} x^{5} x^{m} + 274 \, B b^{4} m x^{6} x^{m} + A a^{4} m^{5} x x^{m} + 19 \, B a^{4} m^{4} x^{2} x^{m} + 76 \, A a^{3} b m^{4} x^{2} x^{m} + 484 \, B a^{3} b m^{3} x^{3} x^{m} + 726 \, A a^{2} b^{2} m^{3} x^{3} x^{m} + 1842 \, B a^{2} b^{2} m^{2} x^{4} x^{m} + 1228 \, A a b^{3} m^{2} x^{4} x^{m} + 1296 \, B a b^{3} m x^{5} x^{m} + 324 \, A b^{4} m x^{5} x^{m} + 120 \, B b^{4} x^{6} x^{m} + 20 \, A a^{4} m^{4} x x^{m} + 137 \, B a^{4} m^{3} x^{2} x^{m} + 548 \, A a^{3} b m^{3} x^{2} x^{m} + 1488 \, B a^{3} b m^{2} x^{3} x^{m} + 2232 \, A a^{2} b^{2} m^{2} x^{3} x^{m} + 2376 \, B a^{2} b^{2} m x^{4} x^{m} + 1584 \, A a b^{3} m x^{4} x^{m} + 576 \, B a b^{3} x^{5} x^{m} + 144 \, A b^{4} x^{5} x^{m} + 155 \, A a^{4} m^{3} x x^{m} + 461 \, B a^{4} m^{2} x^{2} x^{m} + 1844 \, A a^{3} b m^{2} x^{2} x^{m} + 2032 \, B a^{3} b m x^{3} x^{m} + 3048 \, A a^{2} b^{2} m x^{3} x^{m} + 1080 \, B a^{2} b^{2} x^{4} x^{m} + 720 \, A a b^{3} x^{4} x^{m} + 580 \, A a^{4} m^{2} x x^{m} + 702 \, B a^{4} m x^{2} x^{m} + 2808 \, A a^{3} b m x^{2} x^{m} + 960 \, B a^{3} b x^{3} x^{m} + 1440 \, A a^{2} b^{2} x^{3} x^{m} + 1044 \, A a^{4} m x x^{m} + 360 \, B a^{4} x^{2} x^{m} + 1440 \, A a^{3} b x^{2} x^{m} + 720 \, A a^{4} x x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
(B*b^4*m^5*x^6*x^m + 4*B*a*b^3*m^5*x^5*x^m + A*b^4*m^5*x^5*x^m + 15*B*b^4* m^4*x^6*x^m + 6*B*a^2*b^2*m^5*x^4*x^m + 4*A*a*b^3*m^5*x^4*x^m + 64*B*a*b^3 *m^4*x^5*x^m + 16*A*b^4*m^4*x^5*x^m + 85*B*b^4*m^3*x^6*x^m + 4*B*a^3*b*m^5 *x^3*x^m + 6*A*a^2*b^2*m^5*x^3*x^m + 102*B*a^2*b^2*m^4*x^4*x^m + 68*A*a*b^ 3*m^4*x^4*x^m + 380*B*a*b^3*m^3*x^5*x^m + 95*A*b^4*m^3*x^5*x^m + 225*B*b^4 *m^2*x^6*x^m + B*a^4*m^5*x^2*x^m + 4*A*a^3*b*m^5*x^2*x^m + 72*B*a^3*b*m^4* x^3*x^m + 108*A*a^2*b^2*m^4*x^3*x^m + 642*B*a^2*b^2*m^3*x^4*x^m + 428*A*a* b^3*m^3*x^4*x^m + 1040*B*a*b^3*m^2*x^5*x^m + 260*A*b^4*m^2*x^5*x^m + 274*B *b^4*m*x^6*x^m + A*a^4*m^5*x*x^m + 19*B*a^4*m^4*x^2*x^m + 76*A*a^3*b*m^4*x ^2*x^m + 484*B*a^3*b*m^3*x^3*x^m + 726*A*a^2*b^2*m^3*x^3*x^m + 1842*B*a^2* b^2*m^2*x^4*x^m + 1228*A*a*b^3*m^2*x^4*x^m + 1296*B*a*b^3*m*x^5*x^m + 324* A*b^4*m*x^5*x^m + 120*B*b^4*x^6*x^m + 20*A*a^4*m^4*x*x^m + 137*B*a^4*m^3*x ^2*x^m + 548*A*a^3*b*m^3*x^2*x^m + 1488*B*a^3*b*m^2*x^3*x^m + 2232*A*a^2*b ^2*m^2*x^3*x^m + 2376*B*a^2*b^2*m*x^4*x^m + 1584*A*a*b^3*m*x^4*x^m + 576*B *a*b^3*x^5*x^m + 144*A*b^4*x^5*x^m + 155*A*a^4*m^3*x*x^m + 461*B*a^4*m^2*x ^2*x^m + 1844*A*a^3*b*m^2*x^2*x^m + 2032*B*a^3*b*m*x^3*x^m + 3048*A*a^2*b^ 2*m*x^3*x^m + 1080*B*a^2*b^2*x^4*x^m + 720*A*a*b^3*x^4*x^m + 580*A*a^4*m^2 *x*x^m + 702*B*a^4*m*x^2*x^m + 2808*A*a^3*b*m*x^2*x^m + 960*B*a^3*b*x^3*x^ m + 1440*A*a^2*b^2*x^3*x^m + 1044*A*a^4*m*x*x^m + 360*B*a^4*x^2*x^m + 1440 *A*a^3*b*x^2*x^m + 720*A*a^4*x*x^m)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 +...
Time = 0.73 (sec) , antiderivative size = 417, normalized size of antiderivative = 3.34 \[ \int x^m (a+b x)^4 (A+B x) \, dx=\frac {B\,b^4\,x^m\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^3\,x^m\,x^2\,\left (4\,A\,b+B\,a\right )\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {b^3\,x^m\,x^5\,\left (A\,b+4\,B\,a\right )\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {A\,a^4\,x\,x^m\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,a\,b^2\,x^m\,x^4\,\left (2\,A\,b+3\,B\,a\right )\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,a^2\,b\,x^m\,x^3\,\left (3\,A\,b+2\,B\,a\right )\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720} \]
(B*b^4*x^m*x^6*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (a^3*x^m*x^2*(4*A*b + B*a)*(702*m + 461*m^2 + 137*m^3 + 19*m^4 + m^5 + 360))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (b^3*x^m*x^5*(A*b + 4*B*a)*(3 24*m + 260*m^2 + 95*m^3 + 16*m^4 + m^5 + 144))/(1764*m + 1624*m^2 + 735*m^ 3 + 175*m^4 + 21*m^5 + m^6 + 720) + (A*a^4*x*x^m*(1044*m + 580*m^2 + 155*m ^3 + 20*m^4 + m^5 + 720))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (2*a*b^2*x^m*x^4*(2*A*b + 3*B*a)*(396*m + 307*m^2 + 107*m^3 + 17*m^4 + m^5 + 180))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (2*a^2*b*x^m*x^3*(3*A*b + 2*B*a)*(508*m + 372*m^2 + 121*m^3 + 18*m^4 + m^5 + 240))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^ 6 + 720)