Integrand size = 21, antiderivative size = 39 \[ \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx=-\frac {e (d+e x)^{-1+m} \operatorname {Hypergeometric2F1}\left (2,-1+m,m,1+\frac {e x}{d}\right )}{c^2 d^2 (1-m)} \] Output:
-e*(e*x+d)^(-1+m)*hypergeom([2, -1+m],[m],1+e*x/d)/c^2/d^2/(1-m)
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx=\frac {e (d+e x)^{-1+m} \operatorname {Hypergeometric2F1}\left (2,-1+m,m,1+\frac {e x}{d}\right )}{c^2 d^2 (-1+m)} \] Input:
Integrate[(d + e*x)^m/(c*d*x + c*e*x^2)^2,x]
Output:
(e*(d + e*x)^(-1 + m)*Hypergeometric2F1[2, -1 + m, m, 1 + (e*x)/d])/(c^2*d ^2*(-1 + m))
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1121, 27, 75}
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.
\(\displaystyle \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \frac {(d+e x)^{m-2}}{c^2 x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(d+e x)^{m-2}}{x^2}dx}{c^2}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {e (d+e x)^{m-1} \operatorname {Hypergeometric2F1}\left (2,m-1,m,\frac {e x}{d}+1\right )}{c^2 d^2 (1-m)}\) |
Input:
Int[(d + e*x)^m/(c*d*x + c*e*x^2)^2,x]
Output:
-((e*(d + e*x)^(-1 + m)*Hypergeometric2F1[2, -1 + m, m, 1 + (e*x)/d])/(c^2 *d^2*(1 - m)))
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 + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
\[\int \frac {\left (e x +d \right )^{m}}{\left (c e \,x^{2}+c d x \right )^{2}}d x\]
Input:
int((e*x+d)^m/(c*e*x^2+c*d*x)^2,x)
Output:
int((e*x+d)^m/(c*e*x^2+c*d*x)^2,x)
\[ \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c e x^{2} + c d x\right )}^{2}} \,d x } \] Input:
integrate((e*x+d)^m/(c*e*x^2+c*d*x)^2,x, algorithm="fricas")
Output:
integral((e*x + d)^m/(c^2*e^2*x^4 + 2*c^2*d*e*x^3 + c^2*d^2*x^2), x)
\[ \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx=\frac {\int \frac {\left (d + e x\right )^{m}}{d^{2} x^{2} + 2 d e x^{3} + e^{2} x^{4}}\, dx}{c^{2}} \] Input:
integrate((e*x+d)**m/(c*e*x**2+c*d*x)**2,x)
Output:
Integral((d + e*x)**m/(d**2*x**2 + 2*d*e*x**3 + e**2*x**4), x)/c**2
\[ \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c e x^{2} + c d x\right )}^{2}} \,d x } \] Input:
integrate((e*x+d)^m/(c*e*x^2+c*d*x)^2,x, algorithm="maxima")
Output:
integrate((e*x + d)^m/(c*e*x^2 + c*d*x)^2, x)
\[ \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c e x^{2} + c d x\right )}^{2}} \,d x } \] Input:
integrate((e*x+d)^m/(c*e*x^2+c*d*x)^2,x, algorithm="giac")
Output:
integrate((e*x + d)^m/(c*e*x^2 + c*d*x)^2, x)
Timed out. \[ \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,e\,x^2+c\,d\,x\right )}^2} \,d x \] Input:
int((d + e*x)^m/(c*d*x + c*e*x^2)^2,x)
Output:
int((d + e*x)^m/(c*d*x + c*e*x^2)^2, x)
\[ \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx=\frac {-\left (e x +d \right )^{m}+\left (\int \frac {\left (e x +d \right )^{m}}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) d e m x -2 \left (\int \frac {\left (e x +d \right )^{m}}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) d e x +\left (\int \frac {\left (e x +d \right )^{m}}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) e^{2} m \,x^{2}-2 \left (\int \frac {\left (e x +d \right )^{m}}{e^{2} x^{3}+2 d e \,x^{2}+d^{2} x}d x \right ) e^{2} x^{2}}{c^{2} d x \left (e x +d \right )} \] Input:
int((e*x+d)^m/(c*e*x^2+c*d*x)^2,x)
Output:
( - (d + e*x)**m + int((d + e*x)**m/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*d *e*m*x - 2*int((d + e*x)**m/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*d*e*x + i nt((d + e*x)**m/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*e**2*m*x**2 - 2*int(( d + e*x)**m/(d**2*x + 2*d*e*x**2 + e**2*x**3),x)*e**2*x**2)/(c**2*d*x*(d + e*x))