Integrand size = 20, antiderivative size = 209 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=-\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{b d^2 (1+m) (2+m+n) (3+m+n)} \]
-f*(c*f*(2+m)-d*e*(4+m+n))*(b*x)^(1+m)*(d*x+c)^(1+n)/b/d^2/(2+m+n)/(3+m+n) +f*(b*x)^(1+m)*(d*x+c)^(1+n)*(f*x+e)/b/d/(3+m+n)+(c^2*f^2*(m^2+3*m+2)-2*c* d*e*f*(1+m)*(3+m+n)+d^2*e^2*(6+m^2+5*n+n^2+m*(5+2*n)))*(b*x)^(1+m)*(d*x+c) ^n*hypergeom([-n, 1+m],[2+m],-d*x/c)/b/d^2/(1+m)/(2+m+n)/(3+m+n)/((1+d*x/c )^n)
Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.73 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\frac {x (b x)^m (c+d x)^n \left (f (c+d x) (e+f x)-\frac {f (1+m) (c f (2+m)-d e (4+m+n)) (c+d x)+(d e (2+m+n) (c f (1+m)-d e (3+m+n))-c f (1+m) (c f (2+m)-d e (4+m+n))) \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{d (1+m) (2+m+n)}\right )}{d (3+m+n)} \]
(x*(b*x)^m*(c + d*x)^n*(f*(c + d*x)*(e + f*x) - (f*(1 + m)*(c*f*(2 + m) - d*e*(4 + m + n))*(c + d*x) + ((d*e*(2 + m + n)*(c*f*(1 + m) - d*e*(3 + m + n)) - c*f*(1 + m)*(c*f*(2 + m) - d*e*(4 + m + n)))*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*x)/c)])/(1 + (d*x)/c)^n)/(d*(1 + m)*(2 + m + n))))/(d*( 3 + m + n))
Time = 0.39 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {101, 25, 27, 90, 76, 74}
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.
\(\displaystyle \int (b x)^m (e+f x)^2 (c+d x)^n \, dx\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {\int -b (b x)^m (c+d x)^n (e (c f (m+1)-d e (m+n+3))+f (c f (m+2)-d e (m+n+4)) x)dx}{b d (m+n+3)}+\frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {\int b (b x)^m (c+d x)^n (e (c f (m+1)-d e (m+n+3))+f (c f (m+2)-d e (m+n+4)) x)dx}{b d (m+n+3)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {\int (b x)^m (c+d x)^n (e (c f (m+1)-d e (m+n+3))+f (c f (m+2)-d e (m+n+4)) x)dx}{d (m+n+3)}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {\left (e (c f (m+1)-d e (m+n+3))-\frac {c f (m+1) (c f (m+2)-d e (m+n+4))}{d (m+n+2)}\right ) \int (b x)^m (c+d x)^ndx+\frac {f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d (m+n+2)}}{d (m+n+3)}\) |
\(\Big \downarrow \) 76 |
\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (e (c f (m+1)-d e (m+n+3))-\frac {c f (m+1) (c f (m+2)-d e (m+n+4))}{d (m+n+2)}\right ) \int (b x)^m \left (\frac {d x}{c}+1\right )^ndx+\frac {f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d (m+n+2)}}{d (m+n+3)}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {\frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (e (c f (m+1)-d e (m+n+3))-\frac {c f (m+1) (c f (m+2)-d e (m+n+4))}{d (m+n+2)}\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {d x}{c}\right )}{b (m+1)}+\frac {f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d (m+n+2)}}{d (m+n+3)}\) |
(f*(b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x))/(b*d*(3 + m + n)) - ((f*(c*f *(2 + m) - d*e*(4 + m + n))*(b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d*(2 + m + n)) + ((e*(c*f*(1 + m) - d*e*(3 + m + n)) - (c*f*(1 + m)*(c*f*(2 + m) - d *e*(4 + m + n)))/(d*(2 + m + n)))*(b*x)^(1 + m)*(c + d*x)^n*Hypergeometric 2F1[1 + m, -n, 2 + m, -((d*x)/c)])/(b*(1 + m)*(1 + (d*x)/c)^n))/(d*(3 + m + n))
3.10.52.3.1 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[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart [n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(b*x)^m*(1 + d* (x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !Integer Q[n] && !GtQ[c, 0] && !GtQ[-d/(b*c), 0] && ((RationalQ[m] && !(EqQ[n, -2 ^(-1)] && EqQ[c^2 - d^2, 0])) || !RationalQ[n])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
\[\int \left (b x \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{2}d x\]
\[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \]
Result contains complex when optimal does not.
Time = 8.95 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.61 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\frac {b^{m} c^{n} e^{2} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac {2 b^{m} c^{n} e f x^{m + 2} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} + \frac {b^{m} c^{n} f^{2} x^{m + 3} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} \]
b**m*c**n*e**2*x**(m + 1)*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), d*x*ex p_polar(I*pi)/c)/gamma(m + 2) + 2*b**m*c**n*e*f*x**(m + 2)*gamma(m + 2)*hy per((-n, m + 2), (m + 3,), d*x*exp_polar(I*pi)/c)/gamma(m + 3) + b**m*c**n *f**2*x**(m + 3)*gamma(m + 3)*hyper((-n, m + 3), (m + 4,), d*x*exp_polar(I *pi)/c)/gamma(m + 4)
\[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \]
\[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \]
Timed out. \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]