Integrand size = 19, antiderivative size = 68 \[ \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx=\frac {x^{1-2 r} (c x)^m \operatorname {Hypergeometric2F1}\left (2,\frac {1+m-2 r}{q-r},\frac {1+m+q-3 r}{q-r},-\frac {a x^{q-r}}{b}\right )}{b^2 (1+m-2 r)} \] Output:
x^(1-2*r)*(c*x)^m*hypergeom([2, (1+m-2*r)/(q-r)],[(1+m+q-3*r)/(q-r)],-a*x^ (q-r)/b)/b^2/(1+m-2*r)
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx=\frac {x^{1-2 r} (c x)^m \operatorname {Hypergeometric2F1}\left (2,\frac {1+m-2 r}{q-r},1+\frac {1+m-2 r}{q-r},-\frac {a x^{q-r}}{b}\right )}{b^2 (1+m-2 r)} \] Input:
Integrate[(c*x)^m/(a*x^q + b*x^r)^2,x]
Output:
(x^(1 - 2*r)*(c*x)^m*Hypergeometric2F1[2, (1 + m - 2*r)/(q - r), 1 + (1 + m - 2*r)/(q - r), -((a*x^(q - r))/b)])/(b^2*(1 + m - 2*r))
Time = 0.36 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1939, 10, 888}
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.
\(\displaystyle \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx\) |
\(\Big \downarrow \) 1939 |
\(\displaystyle x^{-m} (c x)^m \int \frac {x^m}{\left (a x^q+b x^r\right )^2}dx\) |
\(\Big \downarrow \) 10 |
\(\displaystyle x^{-m} (c x)^m \int \frac {x^{m-2 r}}{\left (a x^{q-r}+b\right )^2}dx\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^{1-2 r} (c x)^m \operatorname {Hypergeometric2F1}\left (2,\frac {m-2 r+1}{q-r},\frac {m+q-3 r+1}{q-r},-\frac {a x^{q-r}}{b}\right )}{b^2 (m-2 r+1)}\) |
Input:
Int[(c*x)^m/(a*x^q + b*x^r)^2,x]
Output:
(x^(1 - 2*r)*(c*x)^m*Hypergeometric2F1[2, (1 + m - 2*r)/(q - r), (1 + m + q - 3*r)/(q - r), -((a*x^(q - r))/b)])/(b^2*(1 + m - 2*r))
Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x _Symbol] :> Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ [e, 0]) && PosQ[s - r]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[(u_)^(m_.)*((a_.)*(v_)^(j_.) + (b_.)*(v_)^(n_.))^(p_.), x_Symbol] :> Si mp[u^m/(Coefficient[v, x, 1]*v^m) Subst[Int[x^m*(a*x^j + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, j, m, n, p}, x] && LinearPairQ[u, v, x]
\[\int \frac {\left (c x \right )^{m}}{\left (a \,x^{q}+b \,x^{r}\right )^{2}}d x\]
Input:
int((c*x)^m/(a*x^q+b*x^r)^2,x)
Output:
int((c*x)^m/(a*x^q+b*x^r)^2,x)
\[ \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx=\int { \frac {\left (c x\right )^{m}}{{\left (a x^{q} + b x^{r}\right )}^{2}} \,d x } \] Input:
integrate((c*x)^m/(a*x^q+b*x^r)^2,x, algorithm="fricas")
Output:
integral((c*x)^m/(2*a*b*x^q*x^r + a^2*x^(2*q) + b^2*x^(2*r)), x)
\[ \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx=\int \frac {\left (c x\right )^{m}}{\left (a x^{q} + b x^{r}\right )^{2}}\, dx \] Input:
integrate((c*x)**m/(a*x**q+b*x**r)**2,x)
Output:
Integral((c*x)**m/(a*x**q + b*x**r)**2, x)
\[ \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx=\int { \frac {\left (c x\right )^{m}}{{\left (a x^{q} + b x^{r}\right )}^{2}} \,d x } \] Input:
integrate((c*x)^m/(a*x^q+b*x^r)^2,x, algorithm="maxima")
Output:
c^m*(m - q - r + 1)*integrate(x^m/(a^2*(q - r)*x^(2*q) + a*b*(q - r)*e^(q* log(x) + r*log(x))), x) - c^m*x*x^m/(a^2*(q - r)*x^(2*q) + a*b*(q - r)*e^( q*log(x) + r*log(x)))
\[ \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx=\int { \frac {\left (c x\right )^{m}}{{\left (a x^{q} + b x^{r}\right )}^{2}} \,d x } \] Input:
integrate((c*x)^m/(a*x^q+b*x^r)^2,x, algorithm="giac")
Output:
integrate((c*x)^m/(a*x^q + b*x^r)^2, x)
Timed out. \[ \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx=\int \frac {{\left (c\,x\right )}^m}{{\left (a\,x^q+b\,x^r\right )}^2} \,d x \] Input:
int((c*x)^m/(a*x^q + b*x^r)^2,x)
Output:
int((c*x)^m/(a*x^q + b*x^r)^2, x)
\[ \int \frac {(c x)^m}{\left (a x^q+b x^r\right )^2} \, dx=c^{m} \left (\int \frac {x^{m}}{x^{2 q} a^{2}+2 x^{q +r} a b +x^{2 r} b^{2}}d x \right ) \] Input:
int((c*x)^m/(a*x^q+b*x^r)^2,x)
Output:
c**m*int(x**m/(x**(2*q)*a**2 + 2*x**(q + r)*a*b + x**(2*r)*b**2),x)