Integrand size = 24, antiderivative size = 128 \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=-\frac {(a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f m}+\frac {(a+b x)^m (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,m,1+m,-\frac {d (a+b x)}{b c-a d}\right )}{f m} \]
-(b*x+a)^m*hypergeom([1, m],[1+m],(-c*f+d*e)*(b*x+a)/(-a*f+b*e)/(d*x+c))/f /m/((d*x+c)^m)+(b*x+a)^m*(b*(d*x+c)/(-a*d+b*c))^m*hypergeom([m, m],[1+m],- d*(b*x+a)/(-a*d+b*c))/f/m/((d*x+c)^m)
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=\frac {(a+b x)^m (c+d x)^{-m} \left (-\operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )+\left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,m,1+m,\frac {d (a+b x)}{-b c+a d}\right )\right )}{f m} \]
((a + b*x)^m*(-Hypergeometric2F1[1, m, 1 + m, ((d*e - c*f)*(a + b*x))/((b* e - a*f)*(c + d*x))] + ((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[m, m, 1 + m, (d*(a + b*x))/(-(b*c) + a*d)]))/(f*m*(c + d*x)^m)
Time = 0.25 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {140, 27, 80, 79, 141}
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.
\(\displaystyle \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx\) |
\(\Big \downarrow \) 140 |
\(\displaystyle \int \frac {\left (a-\frac {b e}{f}\right ) (a+b x)^{m-1} (c+d x)^{-m}}{e+f x}dx+\frac {b \int (a+b x)^{m-1} (c+d x)^{-m}dx}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (a-\frac {b e}{f}\right ) \int \frac {(a+b x)^{m-1} (c+d x)^{-m}}{e+f x}dx+\frac {b \int (a+b x)^{m-1} (c+d x)^{-m}dx}{f}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \left (a-\frac {b e}{f}\right ) \int \frac {(a+b x)^{m-1} (c+d x)^{-m}}{e+f x}dx+\frac {b (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \int (a+b x)^{m-1} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-m}dx}{f}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \left (a-\frac {b e}{f}\right ) \int \frac {(a+b x)^{m-1} (c+d x)^{-m}}{e+f x}dx+\frac {(a+b x)^m (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,m,m+1,-\frac {d (a+b x)}{b c-a d}\right )}{f m}\) |
\(\Big \downarrow \) 141 |
\(\displaystyle \frac {\left (a-\frac {b e}{f}\right ) (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,m,m+1,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{m (b e-a f)}+\frac {(a+b x)^m (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,m,m+1,-\frac {d (a+b x)}{b c-a d}\right )}{f m}\) |
((a - (b*e)/f)*(a + b*x)^m*Hypergeometric2F1[1, m, 1 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)*m*(c + d*x)^m) + ((a + b*x) ^m*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[m, m, 1 + m, -((d*(a + b*x))/(b*c - a*d))])/(f*m*(c + d*x)^m)
3.31.65.3.1 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[((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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f ))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] || !Su mSimplerQ[p, 1]) && !ILtQ[m, 0]
\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m}}{f x +e}d x\]
\[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}} \,d x } \]
Exception generated. \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}} \,d x } \]
\[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}} \,d x } \]
Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^m} \,d x \]