Integrand size = 29, antiderivative size = 124 \[ \int \frac {(e+f x)^m}{a c+(b c+a d) x+b d x^2} \, dx=-\frac {b (e+f x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b (e+f x)}{b e-a f}\right )}{(b c-a d) (b e-a f) (1+m)}+\frac {d (e+f x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (e+f x)}{d e-c f}\right )}{(b c-a d) (d e-c f) (1+m)} \] Output:
-b*(f*x+e)^(1+m)*hypergeom([1, 1+m],[2+m],b*(f*x+e)/(-a*f+b*e))/(-a*d+b*c) /(-a*f+b*e)/(1+m)+d*(f*x+e)^(1+m)*hypergeom([1, 1+m],[2+m],d*(f*x+e)/(-c*f +d*e))/(-a*d+b*c)/(-c*f+d*e)/(1+m)
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {(e+f x)^m}{a c+(b c+a d) x+b d x^2} \, dx=\frac {(e+f x)^{1+m} \left (b (d e-c f) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b (e+f x)}{b e-a f}\right )+d (-b e+a f) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (e+f x)}{d e-c f}\right )\right )}{(b c-a d) (b e-a f) (-d e+c f) (1+m)} \] Input:
Integrate[(e + f*x)^m/(a*c + (b*c + a*d)*x + b*d*x^2),x]
Output:
((e + f*x)^(1 + m)*(b*(d*e - c*f)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*(e + f*x))/(b*e - a*f)] + d*(-(b*e) + a*f)*Hypergeometric2F1[1, 1 + m, 2 + m , (d*(e + f*x))/(d*e - c*f)]))/((b*c - a*d)*(b*e - a*f)*(-(d*e) + c*f)*(1 + m))
Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1150, 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.
\(\displaystyle \int \frac {(e+f x)^m}{x (a d+b c)+a c+b d x^2} \, dx\) |
\(\Big \downarrow \) 1150 |
\(\displaystyle \int \left (\frac {2 b d (e+f x)^m}{(b c-a d) (2 a d+2 b d x)}-\frac {2 b d (e+f x)^m}{(b c-a d) (2 b c+2 b d x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d (e+f x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {d (e+f x)}{d e-c f}\right )}{(m+1) (b c-a d) (d e-c f)}-\frac {b (e+f x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b (e+f x)}{b e-a f}\right )}{(m+1) (b c-a d) (b e-a f)}\) |
Input:
Int[(e + f*x)^m/(a*c + (b*c + a*d)*x + b*d*x^2),x]
Output:
-((b*(e + f*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*(e + f*x))/(b *e - a*f)])/((b*c - a*d)*(b*e - a*f)*(1 + m))) + (d*(e + f*x)^(1 + m)*Hype rgeometric2F1[1, 1 + m, 2 + m, (d*(e + f*x))/(d*e - c*f)])/((b*c - a*d)*(d *e - c*f)*(1 + m))
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol ] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + b*x + c*x^2), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && !IntegerQ[2*m]
\[\int \frac {\left (f x +e \right )^{m}}{a c +\left (a d +b c \right ) x +b d \,x^{2}}d x\]
Input:
int((f*x+e)^m/(a*c+(a*d+b*c)*x+b*d*x^2),x)
Output:
int((f*x+e)^m/(a*c+(a*d+b*c)*x+b*d*x^2),x)
\[ \int \frac {(e+f x)^m}{a c+(b c+a d) x+b d x^2} \, dx=\int { \frac {{\left (f x + e\right )}^{m}}{b d x^{2} + a c + {\left (b c + a d\right )} x} \,d x } \] Input:
integrate((f*x+e)^m/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")
Output:
integral((f*x + e)^m/(b*d*x^2 + a*c + (b*c + a*d)*x), x)
\[ \int \frac {(e+f x)^m}{a c+(b c+a d) x+b d x^2} \, dx=\int \frac {\left (e + f x\right )^{m}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \] Input:
integrate((f*x+e)**m/(a*c+(a*d+b*c)*x+b*d*x**2),x)
Output:
Integral((e + f*x)**m/((a + b*x)*(c + d*x)), x)
\[ \int \frac {(e+f x)^m}{a c+(b c+a d) x+b d x^2} \, dx=\int { \frac {{\left (f x + e\right )}^{m}}{b d x^{2} + a c + {\left (b c + a d\right )} x} \,d x } \] Input:
integrate((f*x+e)^m/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="maxima")
Output:
integrate((f*x + e)^m/(b*d*x^2 + a*c + (b*c + a*d)*x), x)
\[ \int \frac {(e+f x)^m}{a c+(b c+a d) x+b d x^2} \, dx=\int { \frac {{\left (f x + e\right )}^{m}}{b d x^{2} + a c + {\left (b c + a d\right )} x} \,d x } \] Input:
integrate((f*x+e)^m/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="giac")
Output:
integrate((f*x + e)^m/(b*d*x^2 + a*c + (b*c + a*d)*x), x)
Timed out. \[ \int \frac {(e+f x)^m}{a c+(b c+a d) x+b d x^2} \, dx=\int \frac {{\left (e+f\,x\right )}^m}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c} \,d x \] Input:
int((e + f*x)^m/(a*c + x*(a*d + b*c) + b*d*x^2),x)
Output:
int((e + f*x)^m/(a*c + x*(a*d + b*c) + b*d*x^2), x)
\[ \int \frac {(e+f x)^m}{a c+(b c+a d) x+b d x^2} \, dx=\int \frac {\left (f x +e \right )^{m}}{b d \,x^{2}+a d x +b c x +a c}d x \] Input:
int((f*x+e)^m/(a*c+(a*d+b*c)*x+b*d*x^2),x)
Output:
int((e + f*x)**m/(a*c + a*d*x + b*c*x + b*d*x**2),x)