Integrand size = 20, antiderivative size = 71 \[ \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx=\frac {A b^2 x^{3+m}}{3+m}+\frac {b (b B+2 A c) x^{4+m}}{4+m}+\frac {c (2 b B+A c) x^{5+m}}{5+m}+\frac {B c^2 x^{6+m}}{6+m} \]
A*b^2*x^(3+m)/(3+m)+b*(2*A*c+B*b)*x^(4+m)/(4+m)+c*(A*c+2*B*b)*x^(5+m)/(5+m )+B*c^2*x^(6+m)/(6+m)
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx=\frac {x^{3+m} \left (B (b+c x)^3+(-b B (3+m)+A c (6+m)) \left (\frac {b^2}{3+m}+\frac {2 b c x}{4+m}+\frac {c^2 x^2}{5+m}\right )\right )}{c (6+m)} \]
(x^(3 + m)*(B*(b + c*x)^3 + (-(b*B*(3 + m)) + A*c*(6 + m))*(b^2/(3 + m) + (2*b*c*x)/(4 + m) + (c^2*x^2)/(5 + m))))/(c*(6 + m))
Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {9, 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.
| \(\displaystyle \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int x^{m+2} (A+B x) (b+c x)^2dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (A b^2 x^{m+2}+b x^{m+3} (2 A c+b B)+c x^{m+4} (A c+2 b B)+B c^2 x^{m+5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A b^2 x^{m+3}}{m+3}+\frac {b x^{m+4} (2 A c+b B)}{m+4}+\frac {c x^{m+5} (A c+2 b B)}{m+5}+\frac {B c^2 x^{m+6}}{m+6}\) |
(A*b^2*x^(3 + m))/(3 + m) + (b*(b*B + 2*A*c)*x^(4 + m))/(4 + m) + (c*(2*b* B + A*c)*x^(5 + m))/(5 + m) + (B*c^2*x^(6 + m))/(6 + m)
3.1.14.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
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 = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.18
| method | result | size |
| norman | \(\frac {A \,b^{2} x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {B \,c^{2} x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}+\frac {b \left (2 A c +B b \right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {c \left (A c +2 B b \right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}\) | \(84\) |
| gosper | \(\frac {x^{3+m} \left (B \,c^{2} m^{3} x^{3}+A \,c^{2} m^{3} x^{2}+2 B b c \,m^{3} x^{2}+12 B \,c^{2} m^{2} x^{3}+2 A b c \,m^{3} x +13 A \,c^{2} m^{2} x^{2}+B \,b^{2} m^{3} x +26 B b c \,m^{2} x^{2}+47 m \,x^{3} B \,c^{2}+A \,b^{2} m^{3}+28 A b c \,m^{2} x +54 A \,c^{2} m \,x^{2}+14 B \,b^{2} m^{2} x +108 B b c m \,x^{2}+60 B \,c^{2} x^{3}+15 A \,b^{2} m^{2}+126 A b c m x +72 A \,c^{2} x^{2}+63 B \,b^{2} m x +144 B b c \,x^{2}+74 A \,b^{2} m +180 A b c x +90 b^{2} B x +120 A \,b^{2}\right )}{\left (3+m \right ) \left (4+m \right ) \left (5+m \right ) \left (6+m \right )}\) | \(246\) |
| risch | \(\frac {x^{m} \left (B \,c^{2} m^{3} x^{3}+A \,c^{2} m^{3} x^{2}+2 B b c \,m^{3} x^{2}+12 B \,c^{2} m^{2} x^{3}+2 A b c \,m^{3} x +13 A \,c^{2} m^{2} x^{2}+B \,b^{2} m^{3} x +26 B b c \,m^{2} x^{2}+47 m \,x^{3} B \,c^{2}+A \,b^{2} m^{3}+28 A b c \,m^{2} x +54 A \,c^{2} m \,x^{2}+14 B \,b^{2} m^{2} x +108 B b c m \,x^{2}+60 B \,c^{2} x^{3}+15 A \,b^{2} m^{2}+126 A b c m x +72 A \,c^{2} x^{2}+63 B \,b^{2} m x +144 B b c \,x^{2}+74 A \,b^{2} m +180 A b c x +90 b^{2} B x +120 A \,b^{2}\right ) x^{3}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right )}\) | \(247\) |
| parallelrisch | \(\frac {60 B \,x^{6} x^{m} c^{2}+72 A \,x^{5} x^{m} c^{2}+90 B \,x^{4} x^{m} b^{2}+120 A \,x^{3} x^{m} b^{2}+108 B \,x^{5} x^{m} b c m +2 B \,x^{5} x^{m} b c \,m^{3}+2 A \,x^{4} x^{m} b c \,m^{3}+26 B \,x^{5} x^{m} b c \,m^{2}+28 A \,x^{4} x^{m} b c \,m^{2}+126 A \,x^{4} x^{m} b c m +B \,x^{6} x^{m} c^{2} m^{3}+A \,x^{5} x^{m} c^{2} m^{3}+12 B \,x^{6} x^{m} c^{2} m^{2}+13 A \,x^{5} x^{m} c^{2} m^{2}+47 B \,x^{6} x^{m} c^{2} m +B \,x^{4} x^{m} b^{2} m^{3}+54 A \,x^{5} x^{m} c^{2} m +A \,x^{3} x^{m} b^{2} m^{3}+14 B \,x^{4} x^{m} b^{2} m^{2}+15 A \,x^{3} x^{m} b^{2} m^{2}+144 B \,x^{5} x^{m} b c +63 B \,x^{4} x^{m} b^{2} m +180 A \,x^{4} x^{m} b c +74 A \,x^{3} x^{m} b^{2} m}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right )}\) | \(341\) |
A*b^2/(3+m)*x^3*exp(m*ln(x))+B*c^2/(6+m)*x^6*exp(m*ln(x))+b*(2*A*c+B*b)/(4 +m)*x^4*exp(m*ln(x))+c*(A*c+2*B*b)/(5+m)*x^5*exp(m*ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (71) = 142\).
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.06 \[ \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx=\frac {{\left ({\left (B c^{2} m^{3} + 12 \, B c^{2} m^{2} + 47 \, B c^{2} m + 60 \, B c^{2}\right )} x^{6} + {\left ({\left (2 \, B b c + A c^{2}\right )} m^{3} + 144 \, B b c + 72 \, A c^{2} + 13 \, {\left (2 \, B b c + A c^{2}\right )} m^{2} + 54 \, {\left (2 \, B b c + A c^{2}\right )} m\right )} x^{5} + {\left ({\left (B b^{2} + 2 \, A b c\right )} m^{3} + 90 \, B b^{2} + 180 \, A b c + 14 \, {\left (B b^{2} + 2 \, A b c\right )} m^{2} + 63 \, {\left (B b^{2} + 2 \, A b c\right )} m\right )} x^{4} + {\left (A b^{2} m^{3} + 15 \, A b^{2} m^{2} + 74 \, A b^{2} m + 120 \, A b^{2}\right )} x^{3}\right )} x^{m}}{m^{4} + 18 \, m^{3} + 119 \, m^{2} + 342 \, m + 360} \]
((B*c^2*m^3 + 12*B*c^2*m^2 + 47*B*c^2*m + 60*B*c^2)*x^6 + ((2*B*b*c + A*c^ 2)*m^3 + 144*B*b*c + 72*A*c^2 + 13*(2*B*b*c + A*c^2)*m^2 + 54*(2*B*b*c + A *c^2)*m)*x^5 + ((B*b^2 + 2*A*b*c)*m^3 + 90*B*b^2 + 180*A*b*c + 14*(B*b^2 + 2*A*b*c)*m^2 + 63*(B*b^2 + 2*A*b*c)*m)*x^4 + (A*b^2*m^3 + 15*A*b^2*m^2 + 74*A*b^2*m + 120*A*b^2)*x^3)*x^m/(m^4 + 18*m^3 + 119*m^2 + 342*m + 360)
Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (63) = 126\).
Time = 0.39 (sec) , antiderivative size = 1027, normalized size of antiderivative = 14.46 \[ \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx=\begin {cases} - \frac {A b^{2}}{3 x^{3}} - \frac {A b c}{x^{2}} - \frac {A c^{2}}{x} - \frac {B b^{2}}{2 x^{2}} - \frac {2 B b c}{x} + B c^{2} \log {\left (x \right )} & \text {for}\: m = -6 \\- \frac {A b^{2}}{2 x^{2}} - \frac {2 A b c}{x} + A c^{2} \log {\left (x \right )} - \frac {B b^{2}}{x} + 2 B b c \log {\left (x \right )} + B c^{2} x & \text {for}\: m = -5 \\- \frac {A b^{2}}{x} + 2 A b c \log {\left (x \right )} + A c^{2} x + B b^{2} \log {\left (x \right )} + 2 B b c x + \frac {B c^{2} x^{2}}{2} & \text {for}\: m = -4 \\A b^{2} \log {\left (x \right )} + 2 A b c x + \frac {A c^{2} x^{2}}{2} + B b^{2} x + B b c x^{2} + \frac {B c^{2} x^{3}}{3} & \text {for}\: m = -3 \\\frac {A b^{2} m^{3} x^{3} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {15 A b^{2} m^{2} x^{3} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {74 A b^{2} m x^{3} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {120 A b^{2} x^{3} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {2 A b c m^{3} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {28 A b c m^{2} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {126 A b c m x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {180 A b c x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {A c^{2} m^{3} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {13 A c^{2} m^{2} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {54 A c^{2} m x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {72 A c^{2} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {B b^{2} m^{3} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {14 B b^{2} m^{2} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {63 B b^{2} m x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {90 B b^{2} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {2 B b c m^{3} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {26 B b c m^{2} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {108 B b c m x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {144 B b c x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {B c^{2} m^{3} x^{6} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {12 B c^{2} m^{2} x^{6} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {47 B c^{2} m x^{6} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {60 B c^{2} x^{6} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} & \text {otherwise} \end {cases} \]
Piecewise((-A*b**2/(3*x**3) - A*b*c/x**2 - A*c**2/x - B*b**2/(2*x**2) - 2* B*b*c/x + B*c**2*log(x), Eq(m, -6)), (-A*b**2/(2*x**2) - 2*A*b*c/x + A*c** 2*log(x) - B*b**2/x + 2*B*b*c*log(x) + B*c**2*x, Eq(m, -5)), (-A*b**2/x + 2*A*b*c*log(x) + A*c**2*x + B*b**2*log(x) + 2*B*b*c*x + B*c**2*x**2/2, Eq( m, -4)), (A*b**2*log(x) + 2*A*b*c*x + A*c**2*x**2/2 + B*b**2*x + B*b*c*x** 2 + B*c**2*x**3/3, Eq(m, -3)), (A*b**2*m**3*x**3*x**m/(m**4 + 18*m**3 + 11 9*m**2 + 342*m + 360) + 15*A*b**2*m**2*x**3*x**m/(m**4 + 18*m**3 + 119*m** 2 + 342*m + 360) + 74*A*b**2*m*x**3*x**m/(m**4 + 18*m**3 + 119*m**2 + 342* m + 360) + 120*A*b**2*x**3*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 2*A*b*c*m**3*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 28*A* b*c*m**2*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 126*A*b*c*m *x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 180*A*b*c*x**4*x**m /(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + A*c**2*m**3*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 13*A*c**2*m**2*x**5*x**m/(m**4 + 18*m **3 + 119*m**2 + 342*m + 360) + 54*A*c**2*m*x**5*x**m/(m**4 + 18*m**3 + 11 9*m**2 + 342*m + 360) + 72*A*c**2*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 3 42*m + 360) + B*b**2*m**3*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 3 60) + 14*B*b**2*m**2*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 63*B*b**2*m*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 90*B*b* *2*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 2*B*b*c*m**3*x...
Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.28 \[ \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx=\frac {B c^{2} x^{m + 6}}{m + 6} + \frac {2 \, B b c x^{m + 5}}{m + 5} + \frac {A c^{2} x^{m + 5}}{m + 5} + \frac {B b^{2} x^{m + 4}}{m + 4} + \frac {2 \, A b c x^{m + 4}}{m + 4} + \frac {A b^{2} x^{m + 3}}{m + 3} \]
B*c^2*x^(m + 6)/(m + 6) + 2*B*b*c*x^(m + 5)/(m + 5) + A*c^2*x^(m + 5)/(m + 5) + B*b^2*x^(m + 4)/(m + 4) + 2*A*b*c*x^(m + 4)/(m + 4) + A*b^2*x^(m + 3 )/(m + 3)
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (71) = 142\).
Time = 0.26 (sec) , antiderivative size = 340, normalized size of antiderivative = 4.79 \[ \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx=\frac {B c^{2} m^{3} x^{6} x^{m} + 2 \, B b c m^{3} x^{5} x^{m} + A c^{2} m^{3} x^{5} x^{m} + 12 \, B c^{2} m^{2} x^{6} x^{m} + B b^{2} m^{3} x^{4} x^{m} + 2 \, A b c m^{3} x^{4} x^{m} + 26 \, B b c m^{2} x^{5} x^{m} + 13 \, A c^{2} m^{2} x^{5} x^{m} + 47 \, B c^{2} m x^{6} x^{m} + A b^{2} m^{3} x^{3} x^{m} + 14 \, B b^{2} m^{2} x^{4} x^{m} + 28 \, A b c m^{2} x^{4} x^{m} + 108 \, B b c m x^{5} x^{m} + 54 \, A c^{2} m x^{5} x^{m} + 60 \, B c^{2} x^{6} x^{m} + 15 \, A b^{2} m^{2} x^{3} x^{m} + 63 \, B b^{2} m x^{4} x^{m} + 126 \, A b c m x^{4} x^{m} + 144 \, B b c x^{5} x^{m} + 72 \, A c^{2} x^{5} x^{m} + 74 \, A b^{2} m x^{3} x^{m} + 90 \, B b^{2} x^{4} x^{m} + 180 \, A b c x^{4} x^{m} + 120 \, A b^{2} x^{3} x^{m}}{m^{4} + 18 \, m^{3} + 119 \, m^{2} + 342 \, m + 360} \]
(B*c^2*m^3*x^6*x^m + 2*B*b*c*m^3*x^5*x^m + A*c^2*m^3*x^5*x^m + 12*B*c^2*m^ 2*x^6*x^m + B*b^2*m^3*x^4*x^m + 2*A*b*c*m^3*x^4*x^m + 26*B*b*c*m^2*x^5*x^m + 13*A*c^2*m^2*x^5*x^m + 47*B*c^2*m*x^6*x^m + A*b^2*m^3*x^3*x^m + 14*B*b^ 2*m^2*x^4*x^m + 28*A*b*c*m^2*x^4*x^m + 108*B*b*c*m*x^5*x^m + 54*A*c^2*m*x^ 5*x^m + 60*B*c^2*x^6*x^m + 15*A*b^2*m^2*x^3*x^m + 63*B*b^2*m*x^4*x^m + 126 *A*b*c*m*x^4*x^m + 144*B*b*c*x^5*x^m + 72*A*c^2*x^5*x^m + 74*A*b^2*m*x^3*x ^m + 90*B*b^2*x^4*x^m + 180*A*b*c*x^4*x^m + 120*A*b^2*x^3*x^m)/(m^4 + 18*m ^3 + 119*m^2 + 342*m + 360)
Time = 10.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.52 \[ \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx=x^m\,\left (\frac {A\,b^2\,x^3\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {B\,c^2\,x^6\,\left (m^3+12\,m^2+47\,m+60\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {b\,x^4\,\left (2\,A\,c+B\,b\right )\,\left (m^3+14\,m^2+63\,m+90\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {c\,x^5\,\left (A\,c+2\,B\,b\right )\,\left (m^3+13\,m^2+54\,m+72\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}\right ) \]
x^m*((A*b^2*x^3*(74*m + 15*m^2 + m^3 + 120))/(342*m + 119*m^2 + 18*m^3 + m ^4 + 360) + (B*c^2*x^6*(47*m + 12*m^2 + m^3 + 60))/(342*m + 119*m^2 + 18*m ^3 + m^4 + 360) + (b*x^4*(2*A*c + B*b)*(63*m + 14*m^2 + m^3 + 90))/(342*m + 119*m^2 + 18*m^3 + m^4 + 360) + (c*x^5*(A*c + 2*B*b)*(54*m + 13*m^2 + m^ 3 + 72))/(342*m + 119*m^2 + 18*m^3 + m^4 + 360))