Integrand size = 22, antiderivative size = 50 \[ \int \frac {(e x)^m (c+d x)}{a x+b x^2} \, dx=\frac {d (e x)^m}{b m}+\frac {(b c-a d) (e x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,-\frac {b x}{a}\right )}{a b m} \] Output:
d*(e*x)^m/b/m+(-a*d+b*c)*(e*x)^m*hypergeom([1, m],[1+m],-b*x/a)/a/b/m
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \frac {(e x)^m (c+d x)}{a x+b x^2} \, dx=\frac {(e x)^m \left (a c (1+m)+(-b c+a d) m x \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )\right )}{a^2 m (1+m)} \] Input:
Integrate[((e*x)^m*(c + d*x))/(a*x + b*x^2),x]
Output:
((e*x)^m*(a*c*(1 + m) + (-(b*c) + a*d)*m*x*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)]))/(a^2*m*(1 + m))
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {9, 88, 74}
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.
\(\displaystyle \int \frac {(c+d x) (e x)^m}{a x+b x^2} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle e \int \frac {(e x)^{m-1} (c+d x)}{a+b x}dx\) |
\(\Big \downarrow \) 88 |
\(\displaystyle e \left (\frac {c (e x)^m}{a e m}-\frac {(b c-a d) \int \frac {(e x)^m}{a+b x}dx}{a e}\right )\) |
\(\Big \downarrow \) 74 |
\(\displaystyle e \left (\frac {c (e x)^m}{a e m}-\frac {(e x)^{m+1} (b c-a d) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{a^2 e^2 (m+1)}\right )\) |
Input:
Int[((e*x)^m*(c + d*x))/(a*x + b*x^2),x]
Output:
e*((c*(e*x)^m)/(a*e*m) - ((b*c - a*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(a^2*e^2*(1 + m)))
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
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]
\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )}{b \,x^{2}+a x}d x\]
Input:
int((e*x)^m*(d*x+c)/(b*x^2+a*x),x)
Output:
int((e*x)^m*(d*x+c)/(b*x^2+a*x),x)
\[ \int \frac {(e x)^m (c+d x)}{a x+b x^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{b x^{2} + a x} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)/(b*x^2+a*x),x, algorithm="fricas")
Output:
integral((d*x + c)*(e*x)^m/(b*x^2 + a*x), x)
\[ \int \frac {(e x)^m (c+d x)}{a x+b x^2} \, dx=\int \frac {\left (e x\right )^{m} \left (c + d x\right )}{x \left (a + b x\right )}\, dx \] Input:
integrate((e*x)**m*(d*x+c)/(b*x**2+a*x),x)
Output:
Integral((e*x)**m*(c + d*x)/(x*(a + b*x)), x)
\[ \int \frac {(e x)^m (c+d x)}{a x+b x^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{b x^{2} + a x} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)/(b*x^2+a*x),x, algorithm="maxima")
Output:
integrate((d*x + c)*(e*x)^m/(b*x^2 + a*x), x)
\[ \int \frac {(e x)^m (c+d x)}{a x+b x^2} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{m}}{b x^{2} + a x} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)/(b*x^2+a*x),x, algorithm="giac")
Output:
integrate((d*x + c)*(e*x)^m/(b*x^2 + a*x), x)
Timed out. \[ \int \frac {(e x)^m (c+d x)}{a x+b x^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (c+d\,x\right )}{b\,x^2+a\,x} \,d x \] Input:
int(((e*x)^m*(c + d*x))/(a*x + b*x^2),x)
Output:
int(((e*x)^m*(c + d*x))/(a*x + b*x^2), x)
\[ \int \frac {(e x)^m (c+d x)}{a x+b x^2} \, dx=\frac {e^{m} \left (x^{m} d -\left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) a d m +\left (\int \frac {x^{m}}{b \,x^{2}+a x}d x \right ) b c m \right )}{b m} \] Input:
int((e*x)^m*(d*x+c)/(b*x^2+a*x),x)
Output:
(e**m*(x**m*d - int(x**m/(a*x + b*x**2),x)*a*d*m + int(x**m/(a*x + b*x**2) ,x)*b*c*m))/(b*m)