Integrand size = 26, antiderivative size = 205 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx=\frac {(d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^2 (b c-a d) (2+m)}-\frac {(d e-c f) (2 a d f (2+m)-b (d e+c f (3+2 m))) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^2 (1+m) (2+m)}-\frac {f^2 (a+b x)^m \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d^3 m} \]
(-c*f+d*e)^2*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/d^2/(-a*d+b*c)/(2+m)-(-c*f+d*e)* (2*a*d*f*(2+m)-b*(d*e+c*f*(3+2*m)))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/d^2/(-a*d +b*c)^2/(1+m)/(2+m)-f^2*(b*x+a)^m*hypergeom([-m, -m],[1-m],b*(d*x+c)/(-a*d +b*c))/d^3/m/((-d*(b*x+a)/(-a*d+b*c))^m)/((d*x+c)^m)
Time = 0.34 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.90 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx=\frac {(a+b x)^m (c+d x)^{-m} \left (\frac {(d e-c f)^2 (a+b x)}{(c+d x)^2}+\frac {(d e-c f) (b d e-2 a d f (2+m)+b c f (3+2 m)) (a+b x)}{(b c-a d) (1+m) (c+d x)}+\frac {f^2 (2+m) (a+b x) \left (\frac {d (a+b x)}{-b c+a d}\right )^{-1-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{m}\right )}{d^2 (b c-a d) (2+m)} \]
((a + b*x)^m*(((d*e - c*f)^2*(a + b*x))/(c + d*x)^2 + ((d*e - c*f)*(b*d*e - 2*a*d*f*(2 + m) + b*c*f*(3 + 2*m))*(a + b*x))/((b*c - a*d)*(1 + m)*(c + d*x)) + (f^2*(2 + m)*(a + b*x)*((d*(a + b*x))/(-(b*c) + a*d))^(-1 - m)*Hyp ergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/m))/(d^2*(b*c - a*d)*(2 + m)*(c + d*x)^m)
Time = 0.35 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {100, 88, 80, 79}
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.
\(\displaystyle \int (e+f x)^2 (a+b x)^m (c+d x)^{-m-3} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^2 (m+2) (b c-a d)}-\frac {\int (a+b x)^m (c+d x)^{-m-2} \left (-d (b c-a d) (m+2) x f^2+a d (2 d e-c f) (m+2) f-b \left (d^2 e^2+2 c d f (m+1) e-c^2 f^2 (m+1)\right )\right )dx}{d^2 (m+2) (b c-a d)}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^2 (m+2) (b c-a d)}-\frac {-f^2 (m+2) (b c-a d) \int (a+b x)^m (c+d x)^{-m-1}dx-\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (-2 a d f (m+2)+b c f (2 m+3)+b d e)}{(m+1) (b c-a d)}}{d^2 (m+2) (b c-a d)}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^2 (m+2) (b c-a d)}-\frac {-f^2 (m+2) (b c-a d) (a+b x)^m \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} \int (c+d x)^{-m-1} \left (-\frac {b x d}{b c-a d}-\frac {a d}{b c-a d}\right )^mdx-\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (-2 a d f (m+2)+b c f (2 m+3)+b d e)}{(m+1) (b c-a d)}}{d^2 (m+2) (b c-a d)}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^2 (m+2) (b c-a d)}-\frac {\frac {f^2 (m+2) (b c-a d) (a+b x)^m (c+d x)^{-m} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d m}-\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (-2 a d f (m+2)+b c f (2 m+3)+b d e)}{(m+1) (b c-a d)}}{d^2 (m+2) (b c-a d)}\) |
((d*e - c*f)^2*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^2*(b*c - a*d)*(2 + m)) - (-(((d*e - c*f)*(b*d*e - 2*a*d*f*(2 + m) + b*c*f*(3 + 2*m))*(a + b* x)^(1 + m)*(c + d*x)^(-1 - m))/((b*c - a*d)*(1 + m))) + ((b*c - a*d)*f^2*( 2 + m)*(a + b*x)^m*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a *d)])/(d*m*(-((d*(a + b*x))/(b*c - a*d)))^m*(c + d*x)^m))/(d^2*(b*c - a*d) *(2 + m))
3.31.90.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{-3-m} \left (f x +e \right )^{2}d x\]
\[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3} \,d x } \]
Exception generated. \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3} \,d x } \]
\[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3} \,d x } \]
Timed out. \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx=\int \frac {{\left (e+f\,x\right )}^2\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+3}} \,d x \]