Integrand size = 26, antiderivative size = 204 \[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x)^2 \, dx=\frac {(d e-c f) (a d f (1+m)+b (d e-c f (2+m))) (a+b x)^{1+m} (c+d x)^{-1-m}}{b d^2 (b c-a d) (1+m)}+\frac {f (a+b x)^{1+m} (c+d x)^{-1-m} (e+f x)}{b d}-\frac {f (a d f m+b (2 d e-c f (2+m))) (a+b x)^m \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{b d^3 m} \]
(-c*f+d*e)*(a*d*f*(1+m)+b*(d*e-c*f*(2+m)))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/b/ d^2/(-a*d+b*c)/(1+m)+f*(b*x+a)^(1+m)*(d*x+c)^(-1-m)*(f*x+e)/b/d-f*(a*d*f*m +b*(2*d*e-c*f*(2+m)))*(b*x+a)^m*hypergeom([-m, -m],[1-m],b*(d*x+c)/(-a*d+b *c))/b/d^3/m/((-d*(b*x+a)/(-a*d+b*c))^m)/((d*x+c)^m)
Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.88 \[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x)^2 \, dx=\frac {(a+b x)^m (c+d x)^{-m} \left (\frac {(d e-c f) (b d e+a d f (1+m)-b c f (2+m)) (a+b x)}{d (b c-a d) (1+m) (c+d x)}+\frac {f (a+b x) (e+f x)}{c+d x}+\frac {f (-2 b d e-a d f m+b c f (2+m)) \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d^2 m}\right )}{b d} \]
((a + b*x)^m*(((d*e - c*f)*(b*d*e + a*d*f*(1 + m) - b*c*f*(2 + m))*(a + b* x))/(d*(b*c - a*d)*(1 + m)*(c + d*x)) + (f*(a + b*x)*(e + f*x))/(c + d*x) + (f*(-2*b*d*e - a*d*f*m + b*c*f*(2 + m))*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^2*m*((d*(a + b*x))/(-(b*c) + a*d))^m)))/(b *d*(c + d*x)^m)
Time = 0.35 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {101, 25, 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.
\(\displaystyle \int (e+f x)^2 (a+b x)^m (c+d x)^{-m-2} \, dx\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {\int -(a+b x)^m (c+d x)^{-m-2} (a f (c f-d e (m+1))-b e (d e-c f (m+1))-f (2 b d e+a d f m-b c f (m+2)) x)dx}{b d}+\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-1}}{b d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-1}}{b d}-\frac {\int (a+b x)^m (c+d x)^{-m-2} (a f (c f-d e (m+1))-b e (d e-c f (m+1))-f (2 b d e+a d f m-b c f (m+2)) x)dx}{b d}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-1}}{b d}-\frac {-\frac {f (a d f m-b c f (m+2)+2 b d e) \int (a+b x)^m (c+d x)^{-m-1}dx}{d}-\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (a d f (m+1)-b c f (m+2)+b d e)}{d (m+1) (b c-a d)}}{b d}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-1}}{b d}-\frac {-\frac {f (a+b x)^m \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} (a d f m-b c f (m+2)+2 b d e) \int (c+d x)^{-m-1} \left (-\frac {b x d}{b c-a d}-\frac {a d}{b c-a d}\right )^mdx}{d}-\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (a d f (m+1)-b c f (m+2)+b d e)}{d (m+1) (b c-a d)}}{b d}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-1}}{b d}-\frac {\frac {f (a+b x)^m (c+d x)^{-m} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} (a d f m-b c f (m+2)+2 b d e) \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d^2 m}-\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (a d f (m+1)-b c f (m+2)+b d e)}{d (m+1) (b c-a d)}}{b d}\) |
(f*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*(e + f*x))/(b*d) - (-(((d*e - c*f) *(b*d*e + a*d*f*(1 + m) - b*c*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d*(b*c - a*d)*(1 + m))) + (f*(2*b*d*e + a*d*f*m - b*c*f*(2 + m))*(a + b*x)^m*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^2 *m*(-((d*(a + b*x))/(b*c - a*d)))^m*(c + d*x)^m))/(b*d)
3.31.81.3.1 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*(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]
\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{-2-m} \left (f x +e \right )^{2}d x\]
\[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2} \,d x } \]
Exception generated. \[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x)^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2} \,d x } \]
\[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2} \,d x } \]
Timed out. \[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x)^2 \, dx=\int \frac {{\left (e+f\,x\right )}^2\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+2}} \,d x \]